Book of Greeks

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The

Book

of GreeksEdition L1 0

Rahul Bhattacharya

CFE School, Risk Latte Company Limited, Hong Kong

2011



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The Book of Greeks

Certificate in Financial Engineering (CFE) www.risklatte.com

Dedicated to my father, Chandranath Bhattacharya



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The Book of Greeks


What Industry Practitioners have to say about this book?

Rahul Bhattacharya has composed an indispensable collection of notes on Quantitative

Finance with an astonishingly broad coverage. The book familiarizes the reader with awide array of models used in the industry with tips on implementation in Excel™. Excel™is an excellent pedagogical tool and the material in the book invites the reader to

implement the models. The book also contains an extensive coverage of structures traded

across asset classes and even the most seasoned professional will find something new in it.

This is more than a study aid; this is an excellent reference that should be kept within reachon any derivatives desk.

Boris MangalFX Options TraderRBC Capital Markets, Tokyo

The Book of Greeks is an amazingly useful compendium. Rahul Bhattacharya has produced

an easy-reference compilation of the main results in quantitative finance, with just enough

write-ups to be able to apply them in practice, and spanning the entire field from basic stochastic processes to equity, credit, and rates, as well as exotic payoffs and numerical

methods. It distils the key results from an entire library of books, periodicals, and academic

papers into a single short volume that will save practitioners a lot of time and effort .

James de Castro

Former Head of Trading, Asia-Pacific (ex-Japan)

Merrill Lynch, Hong Kong

The Book of Greeks is a perfect companion for any risk manager, quant or derivatives

trader. It manages to have a unique blend of practical application through it's reference to

excel whilst giving the reader all the technical and academic detail he could ever require.

It takes time to give background to the world of modern derivatives and describes the

origins of advanced techniques and models we all take for granted. Everything from

pricing today's exotic derivatives to simple black-scholes greeks are covered. No one

should ever be at their desk without it.

Richard Kendrick

Market Risk ManagementGoldman SachsLondon



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The Book of Greeks


Foreward

The Book of Greeks is an excellent compendium of formulae, models and quantitative

techniques that should be in the repertoire of any derivatives trader, hedge fund manager,

derivatives structurer, academic or student of financial engineering.

Often when we work on trading desks or while structuring derivatives, we wish that we had

a book that had the answers and solutions to the myriad financial products that we have to

price or trade. Well, "The Book of Greeks" is the answer to our prayers. A wonderful, well

compiled book, it offers both the practitioner as well as the academic a one stop source of

mathematical techniques used in the world of financial engineering.

I have enjoyed going through this book and am delighted to be able to recommend it to

anyone who has an interest in quantitative finance and financial engineering.

Rajat BhatiaCEO, Neural CapitalFormer SVP, Citibank Global Asset Management, London& former VP, Lehman Brothers International, London.



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The Book of Greeks


First Few Words

This book is a collection of lecture notes that I’ve used for my CFE classes over the past 2years. These notes are intended to help the reader navigate through various formulas,

equations and algorithms that are used in the four modules of our Certificate in FinancialEngineering (CFE) course. Also, these notes will help all those students taking our CFEExam to better understand the landscape of the entire subject. However, by no means dothese notes present a comprehensive or an exhaustive list of formulas and equations that areused in the study of financial engineering and quantitative finance. At best, it is a miniscule portion of the entire discipline. If anything, these notes aim to provide to the reader aglimpse of the giant canvas on which the subject of Quantitative Finance is written.

All formulas and equations that are used in these notes are meant to be implemented on anExcel™ spr eadsheet. In our CFE course, we implement all these formulas and equations inExcel/VBA and then build quantitative models of financial derivatives and asset portfolio

pricing and risk analysis.

Financial Engineering or Quantitative Finance (and we shall use both these termsinterchangeably) is neither just physics nor applied mathematics, and much more than thesum total of the two. Quantitative Finance is concerned with the application of all those physics-like theories and advanced mathematical concepts and methodologies for thedevelopment of mathematical models and algorithms with which financial derivatives andfinancial asset portfolios can be traded, valued and their risks can be analyzed. However,such is not possible unless all those complex and abstract formulas and equations aretransformed into simpler workable format and implemented on a platform. And that platform has to be something that is very simple to understand and work on.

For our CFE course that platform is the Microsoft Excel™ spreadsheet.

These notes are exclusively meant for supplementing the knowledge and skills imparted inthe CFE course and helping students take the CFE examination. Other than that, this bookhas no other objective or pretense. This book should not be seen as an alternative orsupplement to any textbook on the subject of quantitative finance, a partial list of which isgiven at the end in the reference section. There are no substitutes for the great bookswritten by Jamil Baz, George Chacko, John Hull, Jim Gatheral, Rebonato Riccardo, PaulWilmott, to name only a few.

Rahul Bhattacharya



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The Book of Greeks


Contents

Page

Chapter

The Journey: From U of Chicago to CBOE 10

0.1 Backgroud0.2 Two Physics problems0.3 A Very Brief History of Quant Finance0.4 Timelines0.5 Black-Scholes Era0.6 Heston Era

0.7 Post-Heston Era0.8 Volatility as a Financial Asset

Chapter M

Applied Mathematics and Numerical Methods in Finance 17

Part A: Matrices & Applications

M.1 Fun with Matrices: From Quantum Mechanics to Quant FinanceM.2 Types of Matrices

M.3 Eigenvalues and EigenvectorsM.4 Cholesky matrixM.5 Finding Square Root of MatrixM.6. Applications in Quantitative Finance

Part B: Probability & Probability Distributions

M.7 Probability and Measure TheoryM.8 Is Call Option price a Probability Measure?M.9. Expectations and Jensen’s Inequality

M.10 Moments of a DistributionM.11 Gaussian distributionM.12 Skew and Fat Tails

M.13 Change of MeasureM.14 Correlation and Covariance



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Part C: Binomial & Trinomial Trees

M.15 Cox-Ross-Rubinstein (CRR) TreeM.16 Trinomial Tree

M.17 Trinomial Tree with CRR parametersM.18 Other kinds of Binomial Trees

Part D: Black-Scholes Diffusion Equaiton & Green’s Function

M.19 Valution of Vanilla OptionsM.20 Valuation of Exotic Options

Part E: Numerical Integrals & Monte Carlo Integration

M.14 Trapezoidal and other rules for Numerical IntegrationM.15 Gaussian Quadrature MethodsM.15 Monte Carlo Integration to value Non-path Dependent OptionsM.16 Routine for Implementation in Excel™

Part F: Differential Equations & Finite Difference Methods

M.17 Ordinary Differential Equations in FinanceM.18 Stochastic Differential Equations in FinanceM.19 Partial Differential Equations in FinanceM.20 Finite Difference MethodsM.21 Laplace Transform

Chapter S

Stochastic Processes for Asset Price Modeling 84 (Monte Carlo Simulation on Excel™ spreadsheet)

S.1 Stochastic Process and a Markov ProcessS.2 Random WalkS.3 Geometric Brownian motion

S.4 S.5 Reimann Zeta Function and the Brownian motionS.6 Girsanov’s Theorem S.7 Brownian motion for the Inverse of the Asset PriceS.8 Brownian motion with defaultS.9 Stochastic Process for the Relative Process of Two AssetsS.10 Arithmetic Brownian motionS.11 Mean Reverting Brownian motion



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S.12 Brownian Bridge ProcessS.13 Cox-Ross Square Root ProcessS.14 Ornstein-Uhlenbeck ProcessS.15 Vasicek Process

S.16 Cox-Ingersoll-Ross ProcessS.17 Black Derman Toy (BDT) ProcessS.18 Black Karisinski ProcessS.19 Poisson Jump Diffusion ProcessS.20 Kou’s Double Exponential Process S.21 Heston Stochastic Volatility ModelS.22 Double Mean Reverting Process for VarianceS.23 Constant Elasticity of Variance (CEV) ProcessS.24 Stochastic Alpha Beta Rho (SABR) ModelS.25 Longstaff’s Double Square Root Model

S.26 Stochastic Local Volatility (SLV) ProcessS.27 SLV Bloomberg ModelS.28 GARCH Diffusion ProcessS.29 Gibson & Schwarz Stochastic Convenience Yield ProcessS.30 Stochastic Correlation ProcessS.31 Mixture of Normals ProcessS.32 Variance Gamma (VG) ProcessS.33 Monte Carlo Simulation for VG ProcessS.34 Displaced Diffusion ModelS.35 Libor Market Model (LMM)

S.36 Homogenous Poisson ProcessS.37 Monte Carlo Simulation for Valuation of Single Asset optionsS.38 Multi-asset Stochastic ProcessS.39 Cholesky DecompostionS.40 Eigenvalue decompositionS.41 Monte Carlo Simulation of Valuation of Multi-asset optionsS.42 Cleaning Correlation MatricesS.43 Quantum Random Walk

Chapter E

Financial Products and Product Engineering (Structuring) 132

E.1 Vanilla OptionsE.2 Straddles and zero beta straddlesE.3 Binary OptionsE.4.Outperformance Digital optionsE.5 Money back options



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E.6 Fixed and Floating Strike Lookback OptionsE.7 Arithmetic Average OptionsE.8 Chooser OptionsE.9 Symmetric and Asymmetric Power Options

E.10 Forward Starting and Cliquet OptionsE.11 Reverse Cliquet OptionsE.12 Napoleon OptionsE.13 Exchange OptionsE.14 Amortizing OptionsE.15 Pyramid and Madonna OptionsE.16 Basket OptionsE.17 Best of and Worst of OptionsE.18 Himalaya, Altipano and Everest OptionsE.19 Capped Bull Note

E.20 Principal Protected Bull NoteE.21 Principal Protected Bear NoteE.22 Principal Protected Mixed NoteE.22 Equity Linked Basket NoteE.23 Note with a Short Put option embeddedE.24 Perpetual Capped Call Note (American style) with no maturityE.25 Decomposition of Structured Product through Payoff DiagramE.26 Convertible Bonds and Reverse Convertible BondsE.27 Caplet and Snowball optionsE.28 Sycurve Options

Chapter V

Volatility and Correlation 166

Part A: Implied Volatility & Volatility Surface

V.1 Numerical Estimation of Implied VolatilityV.2 Leland’s Formula V.3 Brenner-Subrahmanyam Approximations

V.4 Corrado Miller ApproximationV.5 Steven Li’s Approximation V.6 SABR VolatilityV.7 CEV VolatilityV.8 Volatility SkewV.9 Implied Volatility Surface and Interpolating Implied VolatilityV.10 Vanna Volga Methodology



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V.11 Local VolatilityV.12 Local Volatility in presence of Default

Part A: Historical Volatility

V.13 Historical Volatility using close to close priceV.14 Parkinson’s Number V.15 Garman-Klass EstimatorV.16 EWMA VolatilityV.17 GARCH Process

Part C: Model Free Volatility and Variance Swaps

V.18 Log Contract

V.19 Britten-Jones & Neuberger ModelV.20 Variance SwapV.21 VIX IndexV.22 Volatility SwapV.23 Correlation and Implied CorrelationV.24 Correlation SkewV.25 Dispersion

Chapter O

Options and Financial Derivatives Valuation 195

O.1 Vanilla Options using Black-Scholes ModelO.2 Put-Call Parity and Put-Call SymmetryO.3 Straddle OptionsO.4 Option pricing using Displaced Diffusion modelO.5 Power OptionO.6 Exchange OptionO.7 Binary OptionO.8 Barrier OptionO.9 One Touch Option

O.9 Double Barrier (Binary) OptionO.10 Fixed and Floating Strike Lookback optionsO.11 Arithmetic Average optionO.12 Forward Starting optionO.13 Caps and FloorsO.14 Swaption Valuation using Black’s formulaO.15 SYCURVE Options



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O.16 Bond Option pricing using Black’s formulaO.17 Options on Zero Coupon Bond using Vasicek’s Model O.18 Options on Variance

Chapter G

Greeks for Vanilla & Exotic Options 221

G.1 Call and Put DeltaG.2 Call and Put GammaG.3 Vega of OptionsG.4 Hedging Error due to Volatility SmileG.5 Theta and Rho of Vanilla optionsG.6 Binary Call and Put DeltaG.7 Dirac Delta Function and the Binary

G.8 Binary Gamma and VegaG.9 Variance Swap Greeks

Chapter P

Portfolio Analysis & Asset Allocation 230

P.1 Sharpe Ratio, Treynor Ration and Jensen’s Alpha P.2 Portfolio VolatilityP.3 Expected Return for Stocks and BondsP.4 Volatility of Spreads

P.5 Probability of Stocks Outperforming BondsP.6 Mean-Variance Optimization for a Total Return ObjectiveP.7 Mean-Variance Optimization by maximizing Sharpe RatioP.8 Sharpe’s Algorithm for Efficient Frontier P.9 Portfolio InsuranceP.10 Constant Proportion Portfolio Insurance (CPPI)P.11 Capital Asset Pricing Model (CAPM)P.12 Minimization of Risk and MCR AlgorithmP.13 Statistical ArbitrageP.14 Triangular Arbitrage



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Chapter 0

The Journey From the University of Chicago to Chicago Board Options Exchange……

Background

Before we talk about Quantitative Finance, talk about its brief history and then plungeourselves into the thick of this discipline with formulas, equations and theory, it isworthwhile to understand the background in which this discipline originated some 50 to 60years ago. That background is physics. The foundations of quant finance were laid based ontheories and models of physics. The twin concepts of “equilibrium” and “linearity” whichunderlie almost all of early theories and models of quant finance are both borrowed from physics, as is most of the math and the rationale.

Two Physics problems and the Birth of Quant Finance

I. Albert Einstein and the Random Walk (Diffusion) Problem

Diffusion is one of the fundamental physical processes by which material in nature moves. Itis found in biology, chemistry, geology, engineering, and above all in physics. The processfirst originated in physics in the form of Brownian motion and was studied by none other thefamous physicist Albert Einstein in 1905. Diffusion arises due to the constant thermal motionof atoms, molecules, and particles and causes material to move from areas of highconcentration to low concentration. Therefore, the final outcome of diffusion would be astate of constant concentration across space. Take a bottle of deodorant or perfume and sprayit heavily inside a small closed room. Eventually the smell spreads across the room and the

entire room starts to smell nice. This is an example of diffusion. Take an iron rod and heatone end of it. Eventually, the other end becomes hot as well, because heat was transferredfrom the hot end to the cold end.

Actually, Albert Einstein had solved the Black-Scholes problem long ago. While studyingBrownian motion of particles to complete his Ph.D. thesis, he realized that the randommotion of the molecules at the microscopic level is ultimately responsible for the process ofdiffusion that occurs at the macroscopic level. Physicists had long studied the macroscopic phenomenon of diffusion and established the governing partial differential equation, PDE,( see below for more details on what is a PDE ). However, it took Einstein’s genius to realizethat the constant coefficient of diffusion in the governing PDE of the diffusion process is

actually the same as the volatility parameter, , of the microscopic random process of themolecules.

Note: Fischer Black and Myron Scholes, while formulating the option pricing problem inearly 1970s, followed more or less the same rationale and thought process as Einstein. Themath and the physics was exactly the same. Only the context was different and certain broad



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(and sweeping) assumptions were made about the economy and the investors in relating the problem in physics to that in financial economics.

II. Richard Feynman’s “Path Integrals” and Feynman-Kac Approach

In 1948, Richard Feynman, another eminent physicist of the twentieth century made a simpleand stunning discovery. He found out that the Schrodinger’s partial differential equation(another example of a PDE) in quantum mechanics could be solved by some kind ofaveraging over the paths. This led to the reformulation of the entire quantum mechanicaltheory in physics in terms of what is known as “path integrals”. Mark Kac, a mathematicianand a colleague of Feynman at Cornell, immediately realized that the concept of “averagingof paths” could be applied to the solution of heat equation with boundary conditions andother kinds of diffusion equations in physics. In short, a diffusive partial differential equationcan be solved as an expectation, under a certain probability measure, of a function thatcontains a Brownian motion. This expectation approach has come to be known as the

Feynman-Kac solution.

Note: The expectations approach, developed by latter day quants and still used extensively by practicing quants in the banks and the academic theorists, to solve the pricing problem forvanilla and exotic options is exactly like Feynman and Kac’s approach. In f act, even quantscall it the Feynman-Kac approach. Everywhere, in finance textbooks, research papers and

other technical documents, you’ll see the expectation operator, E written as .Q E and inside

the brackets you’d see an option payoff. This is nothing but Feynman-Kac approach.

A Very Brief History of Quant Finance

The history of Quantitative Finance is essentially a history of the conquest of "Volatility".The story is about how a few exceptionally talented men across both sides of the Atlanticgrappled with the concept of volatility and ultimately tamed it. The story starts off in 1952with volatility being a statistical measure of financial risk and ends in the present day with it becoming a financial asset. That’s pretty much all there is to quantitative finance. Rest is justdetails and an awful lot of very advanced math and computer programming.

It all started in Chicago in 1952 with Harry Markowitz and ended in Chicago sometimearound 2003 when the new VIX was introduced. In that sense we may be living in the post-historical period.

Timeline

This timeline outlines how Volatility got transformed from being a measure of financial riskto financial asset. If we were to take a Jurassic Park style tour of Quant Finance then we’d betaken through the following periods:

Black-Scholes Era (1952 – 1987)

Heston Era (1988 – 2007)



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Post Heston (2007 – Present)

The History tour of Quant Finance

Black-Scholes Era (1952 – 1987)

Black Scholes era began when in 1952, Harry Markowtiz, during the course of completinghis Ph.D. dissertation realized that volatility is a statistical concept and in fact, is the same asthe concept of standard deviation. And mathematically speaking, this is the proxy forfinancial “risk”.

Constant Volatility

It was in 1973, when Fischer Black and Myron Scholes presented their seminal – perhaps,the most famous paper in all of quantitative finance – on option pricing. This could be

considered a formal beginning of the discipline of quant finance.

Here are the defining characteristics and landmarks of this era

In 1973 Black-Scholes option pricing model was introduced which modeled asset prices as following a geometric Brownian motion – a Gaussian diffusion process – with a fixed volatility parameter and option prices are determined as functions of theunderlying asset price. This fixed volatility parameter was important as this is thesame parameter that appears in the partial diffusion equation (PDE) for the option price as the “coefficient of diffusion”.

The physics of the problem was this: asset price is like the molecule following a

random walk suspended in a medium, where the “molecule” was the stock (asset) andthe “medium” was the volatility.

The fact that volatility is constant is extremely important in a Black-Scholes world, because it simplifies life enormously. Constant volatility implies and causes thefollowing:

(i) A market consisting of a stock and an interest-bearing bond is “complete”, inthe sense that all derivative payoffs can be replicated by a dynamic portfolioconsisting of the stock and the bond only.

(ii) Corollary to the above point is that the principle of no arbitrage leads to aunique equivalent martingale measure that can be used to price derivatives.

(iii) The Black-Scholes PDE (with boundary conditions) can be solved exactly as aheat equation and a neat formula for option price can be extracted (this is theBlack-Scholes option pricing formula).

In 1977 Oldrich Vasicek presented the mean reverting random walk model for asset price. This has now come to be known as the Vasicek model and over the years thismodel has found wide applications in interest rate derivatives valuation.



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In 1976 John Cox and Steven Ross presented their risk-neutral argument andintuitively one of the most curious facts of the Black-Scholes model, that is, as to whythe drift term in the asset price equation of random walk drops out of the option pricing equation and the Black-Scholes formula.

Cox and Ross also presented the square root diffusion model of asset price which onceagain opened up new ways at looking at the option pricing problem. Closely related tothe Cox-Ross model was the Cox-Ingersoll-Ross (CIR) model of asset prices, anothermean reverting, arithmetic Brownian motion model of asset price. This model alsofound wide applications in interest rate modeling. More importantly, CIR model wouldform the basis for the stochastic volatility model developed by Heston in the early1990s.

Black-Scholes era came to an end with the stock market crash of 1987 when investorsrealized that volatility is not constant. The notion of volatility smile (skew) – a varyingimplied volatility across strike prices – was born and Wall Street started becoming a very

comfortable place for the quants. Strangely, as the physics was getting out of the way ofquantitative finance and making way for something far more complex (at least in terms ofcomprehension and conceptualization), more and more physicists started getting hired onWall Street.

Heston Era (1988 – 2007)

In my opimion the Heston era began around the time when John Hull and Alan White published their one factor stochastic volatility model in 1987. But this was still not a gamechanger. The real paradigm shift happened when Steven Heston came up with his two factorstochastic volatility model. This was the first break with physics like models.

Stochastic Volatility Here are the defining characteristics and landmarks of this era

Heston postulated that not only the asset price but volatility associated with that asset price is stochastic. In other words, both the asset price and the volatility of that asset price evolve randomly over time. Even though the asset price in Heston’s modelfollows the same geometric Brownian motion, i.e. a Gaussian diffusion process, as theBlack-Scholes model, the volatility is assumed to follow a square root diffusion process, more popularly known as the Cox-Ingersoll-Ross (CIR) process. Thus, the

probability distribution of the asset price was Gaussian (Log-normal) but that of thevolatility was not.

Volatility, which varies stochastically, is also mean reverting;

And above all, the stochastic processes for the asset and the volatility are correlated.This is where there was a paradigm shift in the world view. For, there was no parallelof such a process in physics. Imagine, Einstein thinking about a diffusion process of amolecule which follows a random walk in a “medium” which is itself vibrating in arandom manner.



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Since volatility was not a tradable asset – by which we mean, an explicitly tangibletradable asset – a random volatility would give rise to a situation where a riskless portfolio cannot be constructed to replicate the option price; hence we would end up ina state where markets would no longer be complete.

And the PDE for the option got far more complex. Fortunately, Heston’s model generated a closed form solution – a formula, if you will

– with which option prices could be calculated.

And last but certainly not the least, Heston, during the course of his work on stochasticvolatility, introduced the use of Fourier transform into the quant finance domain (thistechnique would be later utilized to solve various option pricing problem extensively by many quants, most notably Peter Carr).

However, ironically, even though Heston introduced a revolutionary way of visualizingvolatility and the option pricing problem and, advertently or inadvertently, made a decisive break with physics-like models of quant finance, much of the Heston era was dominated by

the other type of volatility models, i.e. the volatility surface and local volatility. It was onlyfrom 2003 onwards that banks started using Heston’s model extensively and quants andtraders started recognizing the utility of this model.

In fact, to be precise, both concepts of “volatility surface” and “local volatility” do not meritthe addition of the word “model” after them. Vol surface and Local Vol are not reallymodels. They are just another way of looking at stochastic volatility.

Anyway, as things stand today Heston’s stochastic volatility model has become a veryimportant and essential tool in the repertoire of most quants and traders in the banks aroundthe world. This change has come about in the last 8 to 9 years.

Volatility Surface

As noted that with the demise of the Black-Scholes world view after the crash of 1987,investors and option traders discovered the volatility smile.

It was observed in the market that implied volatility of options varied across strike prices for a given maturity. From 1987 onwards, it was observed that at least forequity index options, almost always, implied volatilities increase with decreasingstrike – that is, out-of-the-money puts trade at higher implied volatilities than out-of-the-money calls. This feature is often referred to as a “negative” skew, where

skew is just another characterization of smile. It was observed that there is a term structure of volatility, i.e. implied volatility

varied across different maturities; another key observation was that long-termimplied volatilities exceed short-term implied volatilities

From this, the notion of volatility surface was born. This was a surface – a twodimensional matrix – where implied volatility was plotted against strike pricesand time to maturity. From this surface, implied volatility of any arbitrary optioncan be interpolated.



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Local Volatility in a Stochastic Volatility world

In 1994, Emanuel Derman, Iraz Kani and Bruno Dupire noted that under the condition of riskneutrality there is a unique diffusion process that is consistent with the distribution of market

prices of the European options. Derman and Kani, then at Goldman Sachs, presented thisargument using a discrete time binomial tree whereas, Dupire, working separately, presentedthe continuous time version of the same argument. Their work was based on the earlier workof Breeden and Litzenberger done in 1978.

The import of their arguments was a simple modification in the way we looked at volatility.Rather than thinking of volatility as a constant coefficient of a diffusion process, as given bythe Black-Scholes model, they assumed that volatility was now a state dependent function

expressed as: t S L , , where, the subscript L stood for the word local. The volatility

was now a function of S , the asset price and time, t . This shift in the way how volatility wasdefined by Dupire, Derman and Kani (DDK), based on what was observed in the market as

well as that of theoretical foundations of Breeden and Litzenberger’s work, presented acomplexity. How exactly can we define “local volatility”? This is a question that many seniorquants and traders in banks will have difficulty answering.

Jim Gatheral has presented a unique interpretation of this concept in his book Volatility andCorrelation. He doubts if the proponents of DDK local volatility model visualized this as amodel of evolution of volatility. In fact, he doesn’t believe that “local volatility” should forma separate class of models. Gatheral’s interpretation is that “local volatility” can be thoughtof as an average of all possible instantaneous volatilities in a stochastic volatility world .Gatheral has actually demonstrated this notion, using sophisticated mathematics, in his book.

As David de Rosa (64) points out it is important to note that even though local volatility is afunction of a stochastic variable, i.e. the asset price, in itself it is not stochastic. Thisdistinction is important to capture the essence of local volatility. Local volatility isn’t somemeasure of stochastic volatility, rather, given the existence of vanilla option prices in themarket it is a process through which quants make some simplifying assumptions to priceexotic options that are consistent with the prices of the vanilla options traded in the market.

Post Heston (2007 – Present)

One can say that the post-Heston era started with the financial crisis of 2007-2008. Duringthe ensuing market turmoil the stochastic volatility models more or less broke down. This is a

period that witnessed big spikes in volatility in many FX options market, such as the Dollar-Yen market, it caused serious problems for traders. What was really troubling, and perhaps bewildering at the same time, was the observation that somehow volatility has acquired a lifeof its own independent of that of the underlying asset. Is such a thing possible? Is it possiblefor volatility to have the freedom to rise or fall on its own accord without any change in theunderlying asset, say, the FX rate?



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Enter a new class of models known as the Stochastic Local Volatility (SLV) models. Themost important development came with the publication of a stochastic local volatility model by Ren, Madan, and Qian in 2007. As we shall see later on in these notes, this modelincorporated an independent stochastic component in the volatility process that would change

the dynamics of the process significantly. Grigore Tataru and Travis Fisher at Bloombergalso developed an SLV model in 2010 for pricing barrier options. Tataru and Fisher make akey observation in their paper, which perhaps explains the motivation for the development ofSLV models and what we have argued in the above paragraph. The authors note that pricesof barrier and path dependent options mostly depend on the dynamics of the market. Vanillaoption prices don’t matter that much for their pricing. What is important is how the market behaves and, in particular, how volatility rises and falls with the arrival of information and passage of time. SLV models try to matching this extra market dimension to obtain goodexotic option prices.

Volatility as a Financial Asset

Right from the onset in early 1970s when option trading began on the Chicago Board,investors bought or sold options for its value which was captured, mainly, by the volatility ofthe underlying asset. Therefore, implicitly at least, investors were buying and sellingvolatility via options. But this trade was an imperfect trade in volatility. Volatility in itselfwas not a tradable asset. Still, traders and sophisticated investors used various optionstrategies to create as perfect a trade as possible to trade only the volatility of the asset.

During the Heston era two fundamental developments happened that would transformvolatility into a “nearly tangible” asset. The first was the development of variance swaps (andvolatility swaps). In 1994, Anthony Neuberger introduced the Log contract. A keymathematical ingredient in the variance swap pricing mechanism was this Log contract.Variance swaps were developed mainly by Derman and his colleagues at Goldman Sachs,though many others, like Jim Gatheral, contributed to the understanding and development ofthe pricing of this product, and other related products such as volatility swaps. Varianceswap was first instrument in the history of quant finance which allowed investors to trade“pure” volatility, as if it were an asset.

The second development was the creation of the new VIX index by the Chicago BoardOptions Exchange (CBOE) in 2003. The construction of this volatility index would not have been possible without the introduction of the variance swaps. Through this new VIX indexvolatility became an exchange traded commodity and options on this VIX index have now been developed.

From being just the standard deviation of asset returns to becoming an exchanged tradedasset is the story of volatility; which, as I said before, is pretty much the story of quantitativefinance. From the University of Chicago in 1952, when Harry Markowitz had to defend hisnovel concept of risk to Milton Friedman, to the Chicago Board Options Exchange in 2003,when the new VIX was introduced, is a very short distance. But it’s been a long andfascinating journey in time.



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Chapter M Applied Mathematics and

Numerical Methods in Finance

Part A

Matrices & Applications

Fun with Matrices

From Quantum Mechanics to Quantitative Finance - I

Matrices are extremely important in the study of Quantitative Finance and they areubiquitous. While studying quant finance we come across Volatility matrix, correlation

matrix, Variance-covariance matrix, etc. Just like quantum mechanics in physics, the study ofquantitative finance would not be possible without matrices. Matrices can be easily handledand understood and most of the time they are fun to deal with. So what are matrices?

We all know what a matrix is. A 2 x 2 matrix is an arrangement of rows and columns, or inother words, arrays. A typical 2 x 2 matrix looks like:

20

11 A

The above matrix has 2 rows and 2 columns and hence its dimension is 2 x 2. A typical 3 x 3matrix will look like:

631

435.0

001

B

The above matrix has 3 rows and 3 columns and hence its dimension is 3 x 3. Matrices canalso have imaginary numbers as its components. For example, a 2 x 2 matrix can look like:

00i

i

In the above is a 2 x 2 matrix i is an imaginary number that appears on the second cell of the

first column and the first cell of the second column. The value of i is 1i . The matrix is



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designated as , which is called the “sigma” (in capital) in Greek. The above matrix forms part of something very important in quantum mechanics in physics and is known as one ofthe Pauli matrices. The above matrices are all square matrices. Why “square”? That is because the number of rows is equal to the number of columns. A 2 x 2, 3 x 3, 8 x 8 or an N

x N matrix are all examples of square matrices. In quantitative finance we only deal withsquare matrices.

Of course, non-square matrices do exist and they do have applications in other fields ofstudy, such as Cryptography. But we shall not talk about them here.

The cool thing about matrices – and from now on we only mean “square” matrices – is thatthey don’t commute. What does that mean? Take the numbers 3 and 8. Multiply 3 by 8 andyou’ll get 24. If instead, you multiply 8 by 3 you’d still get 24. It does not matter whether wemultiply 3 by 8 or 8 by 3. The result is the same 24.

248338

In algebra, if we measure a quantity by x and another by y then:

0 yx xy yx xy

If you take y time x and then subtract that quantity from x times y then result would be

zero. Any junior school student would tell you that.

However, this does not happen in the case of matrices. If a quantity is measured by say, a 2 x2 matrix (such as a correlation matrix for a two asset portfolio or for any two random

variables), P and another quantity is measured by another 2 x 2 matrix, Q then:

0 QP PQ .

Take any arbitrary square matrices, P and Q .

If

04

13 P and

80

01Q then if you multiply P with Q , you’d get

04

83 PQ .

Now, if you multiply Q with P , you’d get:

032

13QP . Therefore, the value of the

quantity given by PQ is different from the value of the quantity given by QP . How is this

possible? School algebra tell us that x times y is always equal to y times x . Yes, but that is

what we call commutative algebra. Matrices form, what we call, non-commutative algebra.



21

The Book of Greeks


It may seem trivial to many of you but this simple, non-commutative property of matrices,

i.e. the fact that 0 QP PQ gave physicists the first glimpse into the world of Quantum

Mechanics.

Fun with MatricesFrom Quantum Mechanics to Quantitative Finance - II

There is a beautiful book – perhaps, one of the best in its field – titled Quantum Mechanics in

Simple Matrix Form by Thomas F. Jordan. This article is inspired from that book.

Let us consider three special matrices:

01

101

,

0

02

i

i ,

10

013

These matrices are known as Pauli matrices in quantum mechanics and form one of the building blocks of that discipline in physics. But why do we need to know about them? Wedon’t really have to know about them, except that if we combine the Pauli matrices with an

Identity matrix, I , where

10

01 I , then we can express any arbitrary 2 x 2 matrix using

them. In that sense, we can say that Pauli matrices are the fundamental building blocks of all2 x 2 matrices in the universe.

Let’s take an arbitrary 2 x 2 matrix, A, where A is given by:

2221

1211

aa

aa A . Now take four

complex numbers, 0 z , 1 z , 2 z and 3 z . In fact, these complex numbers can be thought to be

some sort of unique complex numbers associated with every 2 x 2 matrix.

Now, given the Pauli matrices above and the Identity matrix, I we can write the matrix, A,as:

3021

2130

2221

1211

3210

2221

1211

3322110

10

01

0

0

01

10

10

01

z z iz z

iz z z z

aa

aa

z i

i z z z

aa

aa

z z z I z A

Therefore, we can express the unique complex numbers as:

221131221212211221102

1,

2

1,

2

1,

2

1aa z aai z aa z aa z



22

The Book of Greeks


As an example, take a 2 x 2 correlation matrix, M . This could be the correlation between thereturns of two stock indices. Say, one stock index is 70% correlated with the other. Then, M can be expressed as:

17.0

7.01 M

Therefore, we can now express this 2 x 2 correlation matrix in terms of the unique complexnumbers associated with it:

10 z 7.01 z , 02 z , 03 z

Every correlation matrix can be decomposed into its unique complex numbers derived fromPauli matrices.

Types of Matrices and their Applications

1. Square Matrix

In a square matrix the number of rows is equal to the number of columns. It is generally

nn mn written as matrix, where, . A matrix is a square matrix with 3 rows and 3

columns. All matrices used in financial engineering and quantitative finance are square

33 matrices. An example of a matrix would be:

320

841

102

P

An example of a square matrix in finance is the correlation matrix of asset returns. Acovariance matrix of asset returns is also an example of a square matrix.

2. Identity Matrix

I An Identity Matrix, denoted by, , has one on its diagonal and zeros in the off-diagonals.

33 It is the equivalent of number 0 in number theory. An example of a identity matrix

is:



23

The Book of Greeks


100

010

001

I

3. Diagonal Matrix

A diagonal matrix is one whose diagonals have non-zero values and the off-diagonalshave zeros. An identity matrix is a special case of a diagonal matrix. An example of a

33 diagonal matrix is:

200

060

003

D

Say, we have a three asset portfolio (1, 2, 3) with asset (return) volatilities of 15%, 10%and 12%. Then, we can write the volatilities as a diagonal matrix:

12.000

010.00

0015.0

V

4. Triangular Matrices

A triangular matrix is one where all elements above or below the diagonal are zeros. If allelements above the diagonal are zero then the matrix is known as lower triangular matrixwhereas if all elements below the diagonal are zero then the matrix is known as upper

33 triangular. Examples of lower and upper triangular matrices are:

Lower Triangular

829

011

002

L



24

The Book of Greeks


Upper Triangular

400

530681

U

5. Symmetric Matrix: Correlation & Variance-Covariance Matrices

A symmetric matrix is one where all elements above the diagonal are mirror images of

nn Athe elements below the diagonal. In other words, for any symmetric matrix, , theT Atranspose of the matrix, , is the matrix itself. The following relationship holds:

A AT

A transpose operation flips the rows of any matrix into columns and columns into rows.

33 Consider a correlation matrix of asset returns, which is a symmetric matrix:

125.055.0

25.0165.0

55.065.01

M

M All elements of below the diagonal are mirror images of the elements below thediagonal. That is obvious in the case of asset correlation matrix. The correlation of asset 1with that of asset 2 is the same as the correlation of the return of asset 2 with that of asset

12 21 1. If is the correlation of asset 1 with asset 2 then is the correlation of asset 2

2112 with asset 1 such that .

M The transpose of the asset correlation matrix, , above is given by:

M M T

125.055.0

25.0165.0

55.065.01

If we have three assets (1, 2, 3) in a portfolio then the symmetric, 33 asset correlation

matrix is given by:



25

The Book of Greeks


1

1

1

3231

2321

1312

M

If the volatilities of the three assets (1, 2, 3) are given by 1 , 2 and 3 respectively

then the symmetric, 33 asset covariance matrix is given by:

2

332233113

2332

2

22112

13311221

2

1

6. Volatility, Correlation and Variance-Covariance (VCV) Matrix of a Portfolio(Three Asset Case)

1 2 3 Given three assets (1, 2 and 3) with volatilities of , and respectively. Then the

volatility matrix can be expressed as a diagonal matrix:

3

2

1

00

00

00

V

M Given a correlation matrix of these three assets as , where,

1

1

1

3231

2321

1312

M

M jiij Where, in the matrix , we have . Then the Variance-Covariance matrix of the

three asset portfolio would be given by:

MV V T



26

The Book of Greeks


2

323321331

2332

2

21221

13311221

2

1

3

2

1

3231

2321

1312

3

2

1

00

00

00

1

1

1

00

00

00

7. Transpose of a Matrix

As explained above a transpose operation flips the rows of any matrix into columns and

33 columns into rows. Consider the following matrix.

2.030

5.021

041

B

BThe transpose of would be given by

2.05.00

324

011T B

In Excel™, the formula for Transpose is given by =TRANSPOSE(.).

8. Determinant of a Matrix

Determinant symbolizes the area or the volume enclosed by the row vectors of anymatrix. It is a scalar quantity (a single number). Determinants exist only for square

22 matrices. Consider a matrix below:

31

12C

C C C detThe determinant of this matrix, , denoted by or is given by:



27

The Book of Greeks


7)1(13231

12

C

33 For a matrix the determinant is calculated as:

251

203

121

E

E The determinant of would be given by:

9

15182101

51

03

)1(21

23

225

20

1

E

Why are determinants important?

(i) They are essential for calculating the inverse of a matrix. In fact, a determinant tellsus if a matrix can be inverted or not.

(ii) They are related to the eigenvalues and eigenvectors of a symmetric matrix(iii) They measure the area or the volume of shape defined by the row vectors of a

matrix.

Why the Determinant of a VCV or a Correlation Matrix should be Positive?

Interpretation #1

Let’s look at a simple 2 x 2 correlation matrix. The geometrical interpretation ofdeterminant is that – in a 2 x 2 framework (2 x 2 matrix) – it measures the area that isspanned by the two column vectors of the 2 x 2 correlation matrix. If both the vectorsare aligned, which means one of the vectors is linearly dependent on the other, then thedeterminant is zero. The value of the determinant becomes maximum when the angle between the vectors is 90 degrees which is equivalent to the area under a rectangleformed by the two vectors. If the determinant is a measure of the area then how can thearea of a surface outlined by two vectors be negative? Therefore, a negative determinant,though possible for any arbitrary square matrix, is geometrically incomprehensible for asymmetric, correlation or covariance matrix.



28

The Book of Greeks


Interpretation #2

The determinant of a VCV matrix is a measure of “generalized variance”, which is inessence a measure of the spread of the observations in the data set. The determinant for

a 2 x 2 VCV measures the overall information in the matrix, i.e. the total variance ofeach variable less whatever correlation (covariance) there is between the variables.Here again, a negative generalized variance cannot be defined and therefore, if thevalue of the determinant of a VCV (or correlation) matrix is negative then we are in therealm of the nonsense. And a zero variance will only apply to constants, i.e. no spreadin the observations in the data set. Therefore, zero variance is meaningless because thiswould mean all observations in the data set are same.

Determinants and Eigenvalues of a Matrix( For more on eigenvalues and eigenvectors see Chapter S, Part B)

For a symmetric variance-covariance (VCV) or Correlation matrix, the determinant of the Amatrix is equal to the product of the eigenvalues of the matrix. If is a square,

mn n m mn Asymmetric matrix with dimension ( rows and columns, with ), is the

A ....,,, 321 n Adeterminant of and are eigenvalues of then the following

relationship holds:

n A ......321

Since calculation of the determinant is much easier (and faster) than the estimation ofeigenvalues of a matrix, as a rule of thumb, a quant should always check first if the

determinant of a VCV or a correlation matrix of asset returns is positive. If it is, then thechances are good that all eigenvalues of the matrix are positive – though not necessarilyso – and hence there is a good chance that the matrix would be “valid” and not“nonsensical” (i.e. the matrix would be positive semi-definite). Of course, even with a positive determinant, a correlation matrix can be nonsensical if 2 or 4 or other evennumber of eigenvalues is negative.

In Excel™, the formula for the determinant of a matrix is given by =MDETERM(.).



29

The Book of Greeks


9. Inverse of a Matrix

A 1 A A 1 AThe inverse of a matrix, , is defined as where, multiplied by is equal to the

I I AA 1identity matrix, . Mathematically, we can write, .The inverse of a matrix is

calculated as:

A

Aadj A 1

If the determinant of a matrix is zero then the inverse of the matrix will not exist. We saythat the matrix is non-invertible. Such matrices are known as Singular matrices.

In Excel™, the formula for the inverse of a matrix is given by =MINVERSE(.).

10. Matrix Multiplication

Two matrices can be multiplied to produce a third matrix. However, two matrices canonly be multiplied if the column of the first matrix is equal to the row of the secondmatrix. The dimension of the third matrix, which is the product of the first and the secondmatrix, would be the rows of the first matrix and the columns of the second matrix.

22 A BIf we have two matrices, and given as follows

2221

1211

2221

1211

bb

bb B

aa

aa A

B A Then the product of the two matrices, is given by

2222122121222121

2212121121121111

2221

1211

2221

1211

babababa

babababa

bb

bb

aa

aa B AC

Say, we have a three asset portfolio (1, 2, 3) with asset (return) volatilities of 10%, 15%

33 and 12% respectively and expressed as a diagonal matrix as:

12.000

015.00

0010.0

V



31

The Book of Greeks


Then, the Trading Covariance matrix in Dollars per share is given by the followingmatrix multiplication:

m

T

m

mn

m

m

mm

T

p

p

p

p

p

p

P M P

00

00

00

00

00

00

1

1

1

00

00

00

00

00

00

2

1

2

1

21

221

112

2

1

2

1

12. Power of a Matrix

One of the problems of matrices is that they cannot be raised to non-integer powers. Wecan raise a square matrix to a power of 2, 3, 4….. but we cannot raise a matrix to the power of 0.5 (half) or 0.75. Thus we cannot find the square root of a matrix. Consider the

33 following matrix:

651

113

012

X

If we want to find the square of the matrix we simply multiply the matrix with itself(same as raising the matrix to the power of 2).

313623

718

137

651

113

012

651

113

0122 S S S

S 3 2S S Similarly we can raise the matrix to the power of by multiplying with ; and so

S on for higher powers of

S However, if we want to raise the matrix, , to the power of half (0.5), i.e. if we want to

S S find the square root of the matrix , we cannot do it. Square roots of the matrix, exists but they are very different from the notion of a square root that we have from number



32

The Book of Greeks


theory. In fact, a square matrix can have numerous square roots, something that we don’t

3 3see in number theory. The number, has a unique square root given by, .

13. Eigenvalues and Eigenvectors of a Matrix

Let A be an nn square matrix. There exists a number that is said to be the

eigenvalue of A if there exists a non-zero solution vector, W to the linear system ofequations:

W AW

The solution vector, W , is said to the eigenvector of A corresponding to the particular

eigenvalue, .

To solve for we set up the characteristic equation as follows:

00

0

I AW I A

W AW

Where, I A represents the determinant of I A . This characteristic equation

yields a closed form real or complex solution for the eigenvalues, . The eigenvalues, will be the solution of the polynomial (quadratic, cubic, etc.) equation generated by the

characteristic equation above. If A is a 22 matrix then there would be 2 values of , (

1 and 2 ). If A is a 33 matrix then there would be 3 values of , ( 1 , 2 and 3 )

and so on. Eigenvalues are scalar quantities and are simply real or complex numbers.

Once the eigenvalues of the matrix, A are determined then the eigenvectors, W aredetermined. Except for very rare instances, closed form solutions do not exist for

eigenvectors and the eigenvectors, W , corresponding to each eigenvalue is estimatediteratively using numerical algorithms.

If A is an nn square matrix (i.e. n rows and n columns) then there would be n

eigenvectors and W would be expressed as an nn square matrix. If A is 22 then

even W would be 22 in dimension and if A is 33 then even W would be 33 in

dimension. The matrix A and W have the same dimensions.



33

The Book of Greeks


For any arbitrary square matrix, A given by:

nnnn

n

n

aaa

aaa

aaa

A

21

22221

11211

We have the corresponding eigenvalues are n ,....,, 21 and the matrix of

eigenvectors W is given as:

nnnn

n

n

www

www

www

W

21

22221

11211

Here,

nw

w

w

W

1

12

11

1

is the eigenvector corresponding to the eigenvalue 1 ,

2

22

12

2

nw

w

w

W

is

the eigenvector corresponding to the eigenvalue 2 and so on.

14. Square Root of a Matrix

K A square matrix can have many square roots. A matrix is said to be the square root of

M M M K 22 a matrix, , if the product equals to . For example, consider a matrix B

as given by

3721

2816 B

This matrix has many square roots. One of the square roots of this matrix is a matrix given

53

42 R B R R by because, .



34

The Book of Greeks


15. Methods of Finding the Square Root of a Matrix( Please see the Eigenvalues and Eigenvectors of a Matrix to understand this part fully)

th N There are many methods for finding the square root or the root of a square matrix.One of the most common and robust ways of finding the square root of a matrix is via thediagonalization method using the eigenvalues and the eigenvectors of the matrix.

X jk W If is a square matrix of dimension given by and is the matrix of the

X jk eigenvectors of with the same dimension of and is a diagonal matrix of the

X X eigenvalues of then can be decomposed as:

1 W W X

th N X Then, the root of the matrix is given by:

1

1

ˆ W W X N

th N Where, is a diagonal matrix with the diagonal elements being the root of the

k .......,,, 21 33 individual eigenvalues, . For a matrix we get the following.

33 X

333231

232221

131211

x x x

x x x

x x x

X If we have a matrix, , given by:

W And if the matrix of eigenvectors, , and the diagonal matrix of the eigenvalues, , aregiven by:

333231

232221

131211

www

www

www

W

3

2

1

00

00

00

and



35

The Book of Greeks


16. Cholesky Matrix

(See Part B, Chapter S for more on Cholesky matrix)

Cholesky is a lower triangular matrix that is obtained using the following

M nn Atransformation. If is a correlation or a variance-covariance matrix and isthe Cholesky matrix of same dimension then the Cholesky matrix is obtained by:

M AAT

The Cholesky is an important matrix in quantitative finance. A correlation matrix (or avariance-covariance matrix) of asset returns is valid, i.e. workable and usable, if andonly if the Cholesky of that matrix exists. This is related to the condition of positivesemi-definiteness. A correlation matrix is positive semi-definite if all its eigenvaluesare positive and the Cholesky of that matrix exists. If the Cholesky of a correlationmatrix does not exist then the correlation matrix is commonly known as “nonsensical”.

17. Solution of a System of Linear Equations using Matrices

Matrices can be used to solve a system of linear equations. Consider the followingsystem of three linear equations:

22

1

03

z y x

z y x

z y x

Using matrices, we can write the above as:

2

1

0

211

111

113

z

y

x

B AX The above can be written as: where,

21

0

211111

113

B z y

x

X A

The solution can be obtained using matrix multiplication and using the concept of the

B A X B AX 1inverse of a matrix:



36

The Book of Greeks


8.0

1.0

3.0

2

1

0

211

111

113 1

z

y

x

X

3.0 x 1.0 y 8.0 z Thus, , and

Applications of Matrices in Quantitative Finance

I. Hedging an Option Portfolio

An FX options trader wants to hedge a short position of 100 contracts in an OTC FXoption position that he is running that has a Gamma of 2.12, Vega of 0.1956 and a Volga

of 1.028 such that his overall portfolio (original position plus the hedge) is gamma-vega-volga neutral. He identifies three traded options in the market with the following greeks:

Option 1 Option 2 Option 3

Gamma 2.56 1.08 2.32

Vega 0.2259 0.3596 0.1862

Volga 0.3122 -0.0024 0.876

How many of each of these three traded options should he buy or sell to make his portfolio gamma-vega-volga neutral?

For gamma neutrality the equation would be:

12.232.208.156.2 321 www

For vega neutrality the equation would be:

1956.01862.03596.02259.0 321 www

For volga neutrality the equation would be:

028.1876.00024.03122.0 321 www



37

The Book of Greeks


Therefore, to solve for the weights (i.e. the number of options to buy or sell) for makingthe portfolio gamma-vega-volga neutral we need to solve the system of three linearequations given by:

12.232.208.156.2 321 www 1956.01862.03596.02259.0 321 www

028.1876.00024.03122.0 321 www

The solution is given by:

028.1

1956.0

12.2

876.00024.03122.0

1862.03596.02259.0

32.208.156.2

3

2

1

w

w

w

The matrix equation is solved as: B AW B AW 1

Therefore, the number of options to buy or sell is given by:

3257.1

1251.0

4261.0

028.1

1956.0

12.2

876.00024.03122.0

1862.03596.02259.0

32.208.156.2

3

2

1

1

3

2

1

w

w

w

W

w

w

w

W

Since the trader needs to hedge 100 contracts, he needs to buy (long) 42.61 contracts ofoption 1, sell (short) 12.51 contracts of option 2 and sell (short) 132.5 contracts ofoption 3 to make his portfolio gamma-vega-volga neutral. Of course, this will cause hisdelta to change and he needs to re-hedge his delta with FX spot contracts.

II. Estimating the Parametric Value at Risk (VaR)

A Risk Manager wants to estimate the VaR of an FX portfolio of a spot FX trader. Thespot FX positions of the trader with raw USD exposures and respective annualized

volatilities are given by:

Asset Exposure (in USD) Volatility

USD/JPY $2.50 million 8%USD/CHF $4.56 million 9%EUR/USD $1.85 million 11%GBP/USD $3.18 million 7%



38

The Book of Greeks


The asset (return) correlation matrix of the above portfolio is given by:

112.025.035.0

12.0132.045.0

25.032.0167.0

35.045.067.01

M

What is the Value at Risk (VaR) of this portfolio?

The VaR with a 66.67% confidence limit is given by:

ME E VaR T

E M In the above formula, is the vector of net exposure, is the correlation matrix andT E E is transpose of the vector, . The vector of net exposure is calculated by multiplying

the raw exposures given above with the respective volatilities. For the above portfolio E the vector of net exposure, , in USD is given by

600,222

500,203

400,410

000,200

E

The 66.66% Value at Risk of the portfolio is calculates as:

600,222

500,203

400,410

000,200

112.025.035.0

12.0132.045.0

25.032.0167.0

35.045.067.01

600,222

500,203

400,410

000,200 T

Z

272,766 Z VaR

Thus, the 66.66% VaR of the FX portfolio is $766,272. This signifies that there is a66.66% chance that the trader will not lose more than $766,272 in one year providedthat the volatilities and the correlations did not change over this period and the markets behaved normally.



39

The Book of Greeks


III. Portfolio Optimization: Strategic Asset Allocation Problems

Linear equations also appear in portfolio optimization problems (mean-variance

optimization) that appear in strategic asset allocation models used by fund managers toallocate funds between various asset classes and individual securities within assetclasses.

See Chapter P for more details on this.

IV. Applications in Algorithmic Trading (Minimization of MCR)

Matrices are also used to solve complex mathematical algorithms used by algorithmicand high frequency traders to buy and sell stocks and other assets. A particular exampleis the minimization of the Marginal Contribution of Risk (MCR) to determine the

number of stocks to buy and/or sell in a trade list.

See Chapter P for more details on this.

V. Estimating the Variance-Covariance (VCV) Matrix from Market Data

Quantitative equity analysts, fund managers and risk managers in bank need to have thevariance covariance (VCV) matrix of asset returns almost on a real time basis. How dowe estimate the VCV matrix from market data of stock (asset) prices?

The best, and the easiest way, to extract the VCV matrix from the market data is toestimate it via excess return. From a time series of stock price data we can calculate themean return of the stock and hence determine the excess return for a particular period,

i.e. the return of a particular period less the mean return. Let’s say that we have K riskyassets (stocks or stock indices, etc.) and for each of these assets we have price data

(which we can easily obtain in real time from Bloomberg) for N periods (say, 12months, 52 weeks or 5 years, etc.)

Then, here’s the algorithm for obtaining the VCV matrix of the portfolio of K assets:

1. Estimate the return on each of these assets over N periods; ideally, all percentage asset returns should be annualized

2. Calculate the mean return for each of the assets (stocks);

3. Estimate the excess return matrix; the matrix of excess returns, Rˆ is given by:



40

The Book of Greeks


K NK N N

K

K

K

r r r

r r r

r r r

r r r

R

21

33332331

22222221

11112111

ˆ

Note that the matrix of excess return, R , will have a dimension of K N and

it will not be a square matrix as N K . It is mostly likely that we may have

many more periods. N , over which return would be calculated than thenumber of assets, K .

4. Calculate the transpose of the matrix of excess returns, T Rˆ .5. Finally, the VCV matrix is given by the following matrix multiplication:

N

R RVCV

T ˆˆ

It is quite easy to implement the above algorithm in Excel™ and we can generate realtime VCV matrices for risk and portfolio analysis.

VI. Estimated Bonds Returns using Transition Matrices

Simon Benninga in his Financial Modelling using Excel™ , shows how matrices can beused to estimate the return from coupon paying bonds over multi-periods using the

rating transition matrices. A rating transition matrix, or a probability transition matrix,shows the probability that a bond that is in a particular rating category, like AAA, AA,B, etc. will migrate to a new rating category in the next period. This example

Let’s consider the return from a bond which pays annual coupon, K over several years before maturing, whereby it pays off the principal as well. For simplicity, let’s say thatin the universe of all the coupon paying bonds, there are only four rating categories, A,B, C, D and F where, A, B and C show the bond ratings of solvent bonds in decreasingorder of creditworthiness and D represents the default state. The default state is wherethe bond is in default for the first time and if this happens then it pays off R , which isthe recovery rate. F represents a state of permanent default, which means that the bond

was in a state of default in the previous period as well and therefore pays off zero inthis period.



41

The Book of Greeks


A sample transition matrix is given by:

10000

10000

0

0

0

CDCC CBCA

BD BC BB BA

AD AC AB AA

p p p p

p p p p

p p p p

Q

If the bond in consideration pays a coupon, K and if R denotes the recovery rate shouldthis bond default then the payoff vector (a column vector, in array notation) of the bond

at any time, t before maturity, T (where, T t ) and at maturity, T are given by

0

R

K

K

K

T t L and

0

1

1

1

R

K

K

K

T t L

The state vector (a row vector) of the bond will be defined as the current rating state thatit is in. If A, B, C, D and F are arranged in the decreasing order in a row and if the bondin consideration is in rating category C then its state vector will be given by:

00100S

Whereas, if the bond is currently in the rating category A, then its state vector will begiven by:

00001S

Therefore, the expected payoff of the bond is given by the following row vector:

t L J S t L E t

In the above, S is the state vector (row) of dimension 1 5, t J is the transition matrix

raised to the power of t of dimension 5 5 and t L is the bond payoff vector (column)

of dimension 5 1. The dimension of t L E is 1 5.

The expected return of the bond can be found out by calculating the Internal Rate ofReturn (IRR) using the initial investment of $1 and the expected bond payoff vector

t L E .



42

The Book of Greeks


VII. Fibonacci Numbers and the Golden Ratio – An Eigenvalue problem

Fibonacci Numbers and the Golden Ratio are used quite extensively by technical

analysts to predict stock and currency market moves. However, even though both theFibonacci numbers and the Golden Ratio are staple diet for technical analysts inanalyzing stock and currency charts, many would have trouble grasping the relationship between these two mathematical constructs.

What is the relationship between Fibonacci numbers and the Golden Ratio? As it turnsout Golden Ratio is one of the eigenvalues of a Fibonacci matrix.

A Fibonacci sequence of numbers is given by:

0, 1, 1, 2, 3, 5, 8, 13, 21, ………………

Where, each number, except for zero and one, is the sum of the previous two numbers.

In general they can be written as, with seed values, 00 f and 11 f :

21 k k k f f f

Golden Ratio is the irrational number 1.6180339……. It is generally denoted by the

Greek alphabet (psi) and is an extremely important ratio that is found in mathematics,

nature and the world of arts. Some mathematicians, represent it as:2

51 . And

interestingly, many Renaissance artists believed that the Golden Ratio was a “divine proportion”.

We can write a Fibonacci sequence (of two successive Fibonacci numbers) as a 2 x 1column vector such as:

k

k

k

k k

k

k

f

f

f

f f

f

f 1

1

1

1

2

01

11

As can be seen the above equation represents an eigenvalue problem where A is theFibonacci matrix.

01

11 A



43

The Book of Greeks


It can be easily shown that one of the eigenvalues, 1 , of A is equal to the Golden

Ratio, 1.6180339…..

Setting up the characteristic equation as 0 I A where, is the eigenvalue of A

we get the quadratic equation:

012

Solving the above, we get:2

51 where:

......6180339.12

511

and .....6180339.0

2

512

Hence, one of the eigenvalues of A, 1

is equal to the irrational number 1.6180339…which is the Golden Ratio.

A whole New Branch of Study

Random Matrices

From Nuclear Physics to Quantitative Finance - I

Matrices are arrays – an arrangement of rows and columns – of numbers. These numberscan be real or complex. For example, a 3 x 3 symmetric correlation matrix we have the pair

correlation between two variables. Say, if we have three financial assets, A, B and Carranged as A, B and C on both rows and columns, and their correlation matrix, M, is given by:

135.075.0

35.0165.0

75.065.01

M

In the above matrix, all elements are real numbers and each element represents thecorrelation between two variables. For example, 0.65 is the correlation between A and B

and 0.35 is the correlation between B and C. The matrix is symmetric because all elementsabove the diagonal are equal to all elements below the diagonal (which is how a correlationmatrix should be).

Now say a smart but a lazy quant in a bank wants to estimate the correlation matrix of thesethree financial assets. Rather than do historical analysis on the time series of the prices ofA, B and C or try to find out the correlation from the options market, he simply generates aset of uniform random numbers between 0 and 1 (using Excel spreadsheet or some other



45

The Book of Greeks


Ever since then random matrices and Random Matrix Theory have been applied to variousdisciplines including electrical engineering, quantum mechanics, sociology, econometricsand even quantitative finance.

Random MatricesFrom Nuclear Physics to Quantitative Finance - II

Consider a large N N square matrix, A, whose elements are random numbers drawn from

a certain probability distribution, say the Gaussian distribution. Then, for such a matrix, A,what can be said about the probabilities of a few of the eigenvalues or the eigenvectors ofthis matrix? This is the central problem in Random Matrix Theory (RMT) and the answerto this question has far reaching ramifications, not only in nuclear physics but also in suchdiverse areas as quantitative finance, mathematics and mechanical engineering, etc.

Let’s see how we can construct a random matrix. Take, a symmetric, N N matrix A, whose

elements are all drawn from a Gaussian (Normal) distribution with mean 0 and standard deviationof 1, i.e. all elements belong to N(0,1). Then a symmetric matrix, H, is formed as:

T A A H 2

1. The matrix, H , is known as the Wigner matrix.

For, example, using Excel™ random number generator we generate the following random matrix,A:

134.0440.0462.0

815.0302.0674.0

914.1748.0228.2

A

In the above matrix, 1,0~ N aij . Then the symmetric Wigner matrix, H , is given by:

134.0627.0725.0

627.0302.0036.0

725.0036.0228.2

H

The diagonal elements of H are distributed as i.i.d. N(0,1) and off-diagonal elements are

distributed as i.i.d. N(0,1/2). The eigenvalues of this random matrix, H , will display very

interesting properties. As N , the spacing between the eigenvalues of the matrix like H could

very accurately approximate the spacing between the energy levels of a heavy nuclei. This was

Eugene Wigner’s insight in the 1950s. Wigner originally explored a real, symmetric N N

random matrix whose diagonal elements are zero and off diagonal elements were 1 with

probability of 21 .



46

The Book of Greeks


How’s all this relevant to the study of quantitative finance?

Take an example from portfolio analysis and asset allocation. When an equity analyst in ahedge fund quantitatively analyzes a large portfolio of stocks (or any other asset) she first

and foremost estimates the correlation and/or the covariance matrix of the stock returns.Let’s say that the covariance matrix of stock returns is M and random (symmetric) matrixof the type described above (with suitable constraints) is H . If the eigenvalues of M aredistributed in a similar fashion as the eigenvalues of H then the analyst will conclude thatthe elements of M , the actual, observable covariance matrix, have considerable degree ofrandomness. This means that there is a lot of noise in the stock price data that she observesfor all the stocks in her portfolio. Random matrix theory can be used to filter out noise froma correlation or a covariance matrix in a portfolio of assets. There are many other importantapplications of random matrices in quantitative finance.

Part BProbability and Probability Distributions

Probability and Measure Theory

We consider a quantity that can take certain value, x . This quantity could be the price ofMicrosoft stock or the S&P500 index or the temperature of New York City. Let’s say that

this quantity given by, x , fluctuates randomly. In other words, x is a random number.

Now, for each possible value of x there exists a number, x p , between 0 and 1 which we

call the probability. In the simplest possible terms, we can think of x p as the likelihood

of occurrence of the variable x (though strictly speaking “likelihood” is not “probability”).The number x p can be best thought of as a “measure”. Just like the notion of “area” and

“volume” which are more general notion of measure, a probability number, x p , is also a

“measure” (of events occurring); a “ probability measure”, in fact, tells us the likelihood ofobserving any “conceivable event” in an experiment whose outcome is uncertain. However,

probability measure, x p , must assign a value 1 to the entire probability space. In fact,

probability satisfies the mathematical definition of a “measure” which is concerned withthe notion of (a) a space and (b) additivity.

Think of the area of a rectangle, which is equal to length multiplied by width, can beviewed as a product of two intervals to form a certain space and the area of the rectangle isthe measure of that space. Within this overall space, enclosed by the length and width, wecan think of many smaller rectangles enclosed and the sum of the areas of all thoserectangles will be equal to the area original space enclosed by the rectangle. Similarly, probability is also concerned with a space, known as the probability space, which can bethought of as a set of all events. Within that set lies the sub-sets of “conceivable events”that can occur and probability measures these conceivable events in such a way that thesum of the probabilities of all these conceivable events is equal to 1.



47

The Book of Greeks


The last line means that the sum of the probabilities of all possible values of x is equal toone. Therefore, the probability measure satisfies these two properties:

(i) 0 < x p < 1

(ii) 1 x

x p

The mean or the expected value of x is written as x and is denoted by the Greek letter,

mu, . This mean is given by:

x

x xp x

The uncertainty in the value of x , i.e. how much the value of x is going to differ from the

expected value (mean), x , is usually denoted by the Greek letter and is given by theexpression:

2 x x

This is called the standard deviation of x . The quantity 2 x x measures the

dispersion in the value of x and is known as the variance of x .

Is Call Option Price a Probability Measure?

There is a deep connection between the prices of financial derivatives and probability,though, within the context of derivatives we usually deal with “risk neutral” probabilities.The theory of probability and probability distributions forms the foundation for the study offinancial derivatives. It is well known that the price of a digital (binary) call or a put optionin a Black-Scholes framework is nothing but the discounted value of the probability of thestock (asset) finishing in the money. It has been shown by Peter Carr and Dilip Madan thata call or a put option price can symbolize a probability measure under a different measure.

If T S is the price of the asset at maturity and K is the strike price then the (undiscounted)

price of a call option is given by:

dS S K S K C T 0,max

In terms of the expectation operator we can write the above price as:

0,max K S E K C T



48

The Book of Greeks


Where, the above expectation is under a risk neutral probability measure, P .

Now, the value (price) of the asset today, 0S , can be represented under a suitable

probability measure as:

T S E S 0

Therefore, since 0S is constant, the normalized call price can be written as:

0,1maxˆ

0,max

0

T

T

T

S

K E

S E

K S E

S

K C

Here, the modified call price is under a different probability measure E ˆ . This is actuallyknown as the share measure.

The buyer of a call option can choose to get paid in shares instead of US Dollars and in thatcase the claim – the payoff from the call option – gets valued using the share price tiltedmeasure. The two measures are equivalent, as in the prices of the claims – call option – paid out in US Dollars can be seen as equivalent price measures where the claim is paid outin shares.

Secondly, the function

0S

K C

has a minimum value of zero and a maximum value of 1

(one). For various values of T S the normalized call price will assign a value of 1 (one) to

the entire space of payoffs. Hence, the normalized call price, 0S K C is indeed a

probability measure.

Expectation and Derivatives Valuation

The concept of expectation is an important one in quantitative finance. As we have seenabove, expectation is related to the mean of the probability distribution, i.e. the firstmoment. When we find the expectation of a function, we essentially estimate the

probability weighted average of that function. While valuing options we estimate theexpection of the payoff function under risk neutral probability measure.

dz z N K S

z N K S E Call

T

T

P

0,max

0,max



49

The Book of Greeks


Where, z is a random normal variable and 1,0~ N z . In the above z N is the measure of

the probability that the stock will finish in the money.

It can be better understood in discrete form, where the integral gets replaced by the

summation sign.

i

iT i z N K S Call 0,max ,

In a simple binomial model say, that the risk neutral probability of up move is 60% and thedown move is 40% and the current stock price is 100. In an up move the asset will finish at120 and in a down move the asset will finist at 80. Then what is the value of a 100 strikecall option.

Using the above formula for the call value as an expectation we get:

124.00,10080max6.00,100120max Call

A key point that has to be noted is that when we are estimating the option price using anexpectations approach, we always take the expectation of the option payoff and not theexpection of the asset price. This is because the option payoff is a convex function andJensen’s inequality holds.

Jensen’s Inequality

Jensen’s Inequality states that for a convex function, such as the payoff of a call option, theexpected value of the function is greater than or equal to the function of the expected value.

In other words, if x f is a convex function in x , then

x E f x f E

Paul Wilmott in his Frequently Asked Questions in Quantitative Finance and Spiegeleerand Schoutens in their The Handbook of Convertible Bonds have shown, how using aTaylor series expansion on the above expression we can explain the concepts of convexityand gamma of an option.

Jensen’s inequality is one of the most important theorems in quantitative finance and it isthe reason why financial derivatives have value embedded in their gammas. Concept ofconvexity, Jensen’s inequality, randomness and volatility of an asset price are intricatelylinked. In my opinion, study of financial derivatives should start with an understanding andthe explanation of convex functions and the Jensen’s inequality.



50

The Book of Greeks


More on Jensen’s Inequality: Explaining Convexity with Numbers

If k is a constant and x is a variable, then what is common between the functions 2 x x f

and 0,max k x x g ?

Most of us would immediately recognize the second function as the payoff from a calloption. But what about the first function? That doesn’t seem like the payoff of a financialderivative. Actually, both the above functions share the property of “convexity”. In other

words, both these functions are convex. And the function, 2 x x f is as much a payoff of

a financial derivative as the function 0,max k x x f . One of the chief reasons why

every financial derivative has inherent value (and which sometimes is also known as the“time value”) is because all financial derivatives have a convex payoff. Convexity is alsoknown as “gamma” in option parlance. It is the same as what mathematicians call“curvature” of function (or a graph).

Therefore, for financial derivatives to have value they must have a convex payoff. Everyoption, vanilla or exotic, every structured product has to have a convex payoff for it to haveinherent value based on which it can be traded. Without convexity in its payoff, thefinancial derivative would just become the underlying asset (equity, FX, interest rate, etc.)

How do we define a convex payoff or a convex function?

If, , is a small parameter (constant) between 0 and 1 (i.e. 10 ) then, for a given

interval 21, x x , a convex function has the following property:

2121 11 x f x f x x f

Let’s take some numbers to see if the two functions above share the above property.

Let’s take 75.0 and let’s take the interval [100, 110]. Applying the above inequality on

the first function, 2 x x f , we get:

525,10506,10

11025.010075.05.102

11025.010075.0110*25.0100*75.0

f f f

f f f

Therefore, the left hand side is less than (or equal to) the right hand side. So, the function isconvex.



51

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Now, let’s take the second function, 0,max k x x g which is in fact, the payoff

function for a call option. Let’s keep the interval the same, i.e. [100, 110] and 75.0 . We

choose the constant, k (strike) to be 102. In fact, the choice of k doesn’t matter at all and

we can choose any value for k . Now, applying the property of convexity to this function

we get:

25.0

102110max*25.00,102100max*75.00,1025.102max

11025.010075.05.102

11025.010075.0110*25.0100*75.0

g g g

g g g

Once again we see that the left hand side is less than (or equal to) the right hand side. Thisshows that the function for the call option payoff is indeed convex.

In fact, we can take any values for between 0 and 1 and choose any interval 21, x x , the

above property (inequality) for convexity will always hold. We leave it as an exercise for

the readers to choose different values of in 10 and any arbitrary interval 21, x x

to test whether the inequality for convexity holds for both the above functions.

Like the previous example, let’s take two functions, x f and x g which are given by

2 x x f and 0,max k x x g respectively representing the payoff functions of

financial derivatives on the same underlying asset, say, S&P500. Here, we should note that

x is a random variable (drawn from a certain probability distribution). For example, x can

be the price of S&P500 stock index. As again, x g is the payoff of a call option, say on

SP500, with a constant strike price k . The function x f would also give the payoff of a

financial financial derivative which is given by the square of x , or in this case, the squareof the final price of S&P500 stock index. Both functions are convex.

In the above context, we should write the Jensen’s inequality as:

T T x E f x f E

The subscript T denotes the final price of the variable (SP500); for example, T could beequal to one year if the maturity of the derivative is one year.

I will now take a very simple and highly stylized example, using numbers, to prove theJensen’s inequality. Say, S&P500 is currently trading at 100 (normalized value). Now, tokeep things extremely simple let’s say that SP500 index can only take 6 (final) values at the

end of one year. The final price of S&P500, T x can be any one of these values: 100, 110,

120, 130, 140 and 150.



52

The Book of Greeks


The first financial derivative, given by the function, T x f , is the square of the final SP500

index values. The expected value of the function is given by:

917,15

6

500,22600,19900,16400,14100,12000,10

6

2

6,

2

5,

2

4,

2

3,

2

2,

2

1,

T T T T T T

T

x x x x x x

x f E

The function of the expected value is given by:

625,15

6

2

5,4,3,2,1,

T T T T T

T

x x x x x x E f

Therefore, from the above we see that T T x E f x f E .

Now consider the second function, T x g , which is a one year call option on SP500 index.

Say, the strike price, the constant, k in the function is the same as before, i.e. 102. Theexpected value of this function is given by:

33.23

60,102150max............0,102110max0,102100max

6

0,max..............0,max0,max 6,2,1,

k xk xk x x f E

T T T

T

The function of the expected value would be given by:

23

0,102

6

150..........110100max

0,6

........max

6,2,1,

k

x x x x E f

T T T

T



53

The Book of Greeks


Therefore, once again we see that T T x E f x f E . Hence we observe that given

these values of T x (all of which are assumed to be random) both the functions, x f and

x g satisfy the Jensen’s inequality.

One can take any values and a large enough set of numerical values for T x the Jensen’s

inequality for both the above functions will hold. We would urge our readers to experiment

with any random set (no matter how large is the set) of numerical values for T x to test the

Jensen’s inequality.

Since, the expected value of the function, x g , is always greater than the function of the

expected value, a call option on SP500 (or a call option on any other asset that is modeledas a random walk) will have an inherent, fundamental value at inception (time, t = 0). And

that fundamental value is given by the (discounted) expected value of the function T x g .

In our simple and stylized example above we assumed only six possible values for the

variable (SP500) T x . In reality, there will a large – infinite – number of values for T x all

drawn from a certain probability distribution (because T x is random). Thus, in real life, the

fair value of a call option on SP500 stock index at inception (t = 0) would be given by:

T

QrT x f E eCall

Where, Q is the probability measure under which the expectation, E (or, in layman’s

terms, the averaging) is being carried out. In fact, as Wilmott shows

termsorder higher x E f x f E T T

The “higher order terms” capture the gamma (convexity) of the option and the variance of

the randomness. All financial derivatives, whether they are given by functions, T x f ,

T x g or any other convex functions (of a random variable) share the above property.

Convexity terms (higher order terms) are added to the expected value (mean) of thefunction to give the financial derivative a fundamental fair value at inception (t = 0).

Probability Distribution and the Gaussian

If there is a random variable X which can take all possible values, each of which has acertain probability of occurrence, such that the sum of all probabilities is 1, then a probability distribution of X will show how the total value of 1 is distributed over all possible values. An example of a probability distribution is the famous Normal, or theGaussian, probability distribution.



54

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If z is a normally distributed random variable with mean 0 and standard deviation of 1, the probability density function (pdf) is given by:

2

2

1

2

1 z

e x f

The probability associated with this random variable is denoted by z dP , which will represent the

probability that the random variable z will fall within a small interval of length dz centred around

z . Mathematically, this probability can be written as

dyekdz z z kdz z P z dP kdz z

kdz z

y

2

2

1

2

1

Where, in the above expression, k is a constant that is less than 1. This gives us:

dz e z dP z 2

2

1

2

1

We call z dP a probability measure. z dP is a function in the probability space, P and it

satisfies all the measure properties (like additivity, etc.).

In more recognizable terms, we can write the following, for a random variable, , that is drawn

from a Normal (Gaussian) probability distribution, with mean of zero and variance of one:

x N edx x x prob x

2

2

1

21

y N dxe y prob

y x

2

2

1

2

1

y N y N dxe y prob

y x

12

1 2

2

1

In Excel™ the function, N , is estimated by the formula =NORMSDIST(.)

Radon-Nikodym Derivative: Changing a Probability measure

Now, given the above explanation for a probability measure, z dP , how can we change this

probability measure?

Let’s define a function, z , where 0 z , which is given by

84 z e z



55

The Book of Greeks


What happens if we multiply our probability measure, z dP with this new function, z ? We

would get the following result:

dz e

dz ee z z dP

z z

z z

842

1

842

1

2

2

2

1

21

If we denote, the product of z dP and z as z dQ , then, using simple algebra on the right

hand side of the above equation, we can write the expression for z dQ as:

dz e z dQ

z 24

2

1

2

1

It is obvious from the above that z dQ is a new probability measure. In fact, given the

above expression we can see that z dQ is the probability associated with a random

variable with a mean of 4 and a standard deviation of 1.

Therefore, by multiplying the original probability measure z dP by the function z we

have transformed the original Normal distribution from N(0,1) to N(4,1). The mean of thedistribution has been shifted.

The function z can be alternatively expressed as:

z dP

z dQ z

We can, therefore, think of z as mathematical derivative of Q with respect to P . These

kinds of mathematical derivatives are called Radon-Nikodym derivatives. z is a Radon-

Nikodym derivative.

Moments of a Probability Distribution

(i) Mean: The first moment of a probability distribution is known as the Mean or the

expected value. It is usually denoted by . If i x is a random variable drawn from a

probability distribution and the sample size is N then its mean is given by:



56

The Book of Greeks


N

i

i x N 1

1

(ii) Variance: The second moment of a probability distribution is known as the Variance.If i x is a random variable drawn from a probability distribution and the sample size is

N then its Variance is given by:

N

i

i x N 1

22

1

1

The standard deviation, which also measures the historical volatility of assets and is oneof the common measures of financial risk, is given by the square root of the variance:

N

i

i x N 1

2

1

1

In Excel™ the standard deviation of a time series is calculated by the formula

=STDEV(.)

(iii) Skew: The third moment of a probability distribution is known as the skewness, or

sometimes just skew. If i x is a random variable drawn from a probability distribution

and the sample size is N and the first and the second moments are and 2 then its

Skew is given by:

N

i

i x

N N

N Skewness

1

2

21

Skew measures the symmetry in the distribution. A Normal (Gaussian) distribution issupposed to be symmetrical around the mean and hence should ideally display zeroskew. However, in practice that is never the case. In the financial markets, just as inlife, probability distributions are greatly skewed either to the left or the right of themean.

The Odd Moment: More on Skewness of Probability Distributions

The notion of skewness, or simply the skew, the measure of the third central moment ofa probability distribution, is fundamental to financial markets and understanding the behaviour of agents who operate in it. If the distribution is negatively skewed then thetrader will experience frequent small gains and infrequent large losses and thedistribution will exhibit a long tail on the left hand side (relative to the mean) of thedistribution. This happens when you are long options or follow a bleed strategy. A



57

The Book of Greeks


trader or a hedge fund manager continuously but steadily buys options, usually out ofthe money options, thereby paying a small premium or in other words, bleeding cash.

His book will show frequent small losses. However, once in a while, mostly due to rare

events or one single Black Swan event, there would big gains when the out of themoney options will be in the money. This strategy of trading, which exhibits negativeskew, is known as a “bleed strategy”. A prime (but an extreme) example is that ofQuantum Fund, the hedge fund run and managed by George Soros and his traderStanley Druckenmiller. In September, 1992, these two traders bought large quantities ofout of the money put options in Pound Sterling against the Deutsche Mark. The strike price of these put options were outside the Government bank (as specified by the thenexisting Exchange Rate Mechanism, or the ERM in Europe). During the entire monththey kept on buying these options steadily thereby paying premium. This showed asteady loss on their books. Since the strike price was outside the ERM government band most traders working in the banks in London and New York, who sold these

options to Soros and Drukenmiller, thought that their own chances of their makingmoney was 99% since even 1% chance of Pound Sterling breaking the ERM bandagainst the Deutsche Mark (thereby making the GBP put options in the money) wasquite remote. However, we all now know that the unthinkable happened. GBP brokethe ERM and Soros and Druckenmiller allegedly made a billion dollars on the trade.

A positive skew on the other hand will create a long tail on the right hand side (relativeto the mean) of the distribution. This is the case with short options strategy. Whentraders steadily sell out of the money options to customers they realize immediate profits. Their estimate is that 99% of the time the options will expire worthless andhence they stand to make profits on 99% of the short option trades. However, they aresubject to infrequent loss, and sometimes a huge loss due to Black Swan and other rareevents which can wipe out their entire profits and more. The hedge fund LTCM is acase in point. Another example of a negatively skewed bet is that of banks andfinancial institutions that lend money to customers, both corporates and retail. In fact, banks lend money to countries as well who are considered sovereign clients with verylow default risk. The lending can be in the form of loans or they can buy bonds. The banks lend money over and above the risk free rate (and their own borrowing costs)thereby making a nice spread and steadily earning profits. However, there is a defaultrisk of the customer and when the customer defaults your entire earnings are wiped out.The banks think that the client is a triple A rated client with very low default risk andhence makes the loan or gives line of credit (or guarantees). The chance of the customerdefaulting is 1% and 99% of the time the banks make good money. However, when that1% happens due to some rare event – the customer suddenly goes from triple A to bankruptcy due to severe cash flow problems or due to worsening financial markets – the banks lose all their gains (on the coupon spread) and has to even write off the principal. When a bank makes a loan to a customer with a credit risk, it is essentiallyshorting a spread option (a loan can be thought of as a portfolio of short spread optionand a credit default swap, CDS). As an example, look what’s happening in Greece now.



58

The Book of Greeks


9 out of 10 people in the financial markets live the life on the “positively skewed side”of the probability distribution. They think that this is desirable and safe, unless ofcourse, an LTCM, a Lehman Brothers collapse or Greece happens. Then they look with

envy to that one guy on the other side of the mean who has lived life on the “negativelyskewed side” of the distribution all along.

(iv) Kurtosis Kurtosis is the fourth moment of a probability distribution. If i x is a random:

variable drawn from a probability distribution with the sample size N and the first and

the second moments are and 2 then the kurtosis is given by the following

expression:

32

13

321

1 24

1

N N

N x

N N N

N N Kurtosis

N

i

i

This fourth moment measures the peakedness of the distribution and the heaviness ofthe tails of the distribution. In the financial markets and option trading parlance it isrelated to the notion of “fat tails”. Heavy tails occur when the distribution exhibits positive kurtosis. In this case, most of the observations lie towards the extreme end ofthe probability distributions and there are only a few observations near the mean. Thedistribution therefore has distinct, sharp peak but it falls off very fast around the meanand make the tails heavy. Negative kurtosis is the opposite of this where most of theobservations are clustered around the mean. Here, the mean is flatter.

Fat Tails in the Markets

One of the vexing issues in the derivatives market is the phenomenon of “volatility ofvolatility” or vvol. Every trader’s volatility estimate has volatility and he knows thatthis is what gives rise to “fat tails” and the “gamma of the gamma”, the fourth momentof a probability distribution. Many seasoned traders will tell you that “fat tails” – the probability of rare events, like stock markets falling by 10% in a day, happening farmore frequently than what is predicted by the Normal probability distribution – don’thave to be necessarily caused by blown out variances. A fat tail can occur even when anasset has a low volatility but a very high volatility of volatility.

Option traders believe that fat tails are caused due to a variable volatility. In a Black-

Scholes world, where volatility is constant, there would be no fat tails. However, thefact that volatility is variable (and stochastically so) gives rise to volatility of volatility.High volatility causes a market to move towards the tail of a probability distribution.According to many traders one of the main reasons for the higher price of in the moneyand out of the money options than the Black-Scholes value is because of the existenceof volatility of volatility which causes the tails of the distribution to become fat.



59

The Book of Greeks


Correlation and Covariance

If x and y are two random variables drawn from a probability distribution and i x and

i y are two historical time series of these two random variables then the correlation

between them is given by:

N

i

N

i

yi xi

N

i

yi xi

y x

y x

y x

1 1

22

1

,

Here, y x, is known as the “correlation coefficient”. Within the context of asset prices

this is called the historical or realized correlation.

Correlation coefficient measures the association between the movements of two

random variables, such as the return on securities. If the return on security A moves inthe same direction as the return on seucurity B then AB will be positive. The value of

the correlation coefficient lies between +1 and -1. A +1 correlation coefficient signifiesa perfect positive correlation between two random variables whereas a -1 correlationcoefficient signifies a perfect negative correlation between the two random variables.In Excel™ the correlation coefficient between two time series is calculated by the

formula =CORREL(.).

Drawbacks of the Correlation measure:

(i) It only measures the linear relationships between variables. For example, if x and

y vary quadratically, via, say an equation like 2 x y , then even though there is a

perfect co-movement between the two variables the correlation coefficient will failto capture that relationship.

(ii) It only measures the direction and the degree of association between themovements of two random variables but does not take into account the magnitudeof variation in the movements. For that reason Covariance is a better estimate ofco-movement.

Covariance between two random variables, x and y is the product of correlation

coefficient and the standard deviations of variable x and y

y DevStd x DevStd y xnCorrelatio y xCov ,,

Or, we can write the covariance of x and y as xy , using the familiar notations for

correlation and standard deviations as:

y x x xy



60

The Book of Greeks


Harry Markowitz Introduces Covariance

It is interesting to note that Harry Markowitz introduced the statistical concept of

“Covariance” in the early 1950s while working on the portfolio selection problem andthe Mean-Variance optimization technique.

Markowitz was the first to think of the statistical concept of variance of a financialasset’s returns as the measure of a financial security’s “risk”, or “financial risk”. If thereturn of the asset prices follows a Gaussian (Normal) distribution then the variance ofthese returns, or the standard deviation (volatility) which is the square of the variance,measured over a historical time period is the measure of that asset’s risk. However, ifthere are two assets, each of whose returns follow a Gaussian distribution, then how dowe measure the portfolio risk? The problem is compounded because besides the twovariances (or standard deviations), one for each asset, there is also the correlation

coefficient which measures the co-movement between them. However, the correlationcoefficient is an imperfect measure of co-movement as it only captures the direction ofco-movement, but not the quantum of variation. Markowtiz’s insight was that if we areto simply take the average of the variances of the two assets then we would not be ableto capture the co-movement between them, as the correlation would be left out. By itselfcorrelation is not sufficient to capture both the direction (co-movement) and thequantum of movement of each asset. Therefore, the best way to capture portfolio riskwould be to multiply the correlation with the product of each asset’s variance. This wasthe only way in which portfolio risk of two assets can reduce to the risk of a single assetif one of them were to disappear. The covariance of a financial asset with itself willsimply become the variance of that asset because the correlation coefficient of an assetwith itself is +1.

y x then 21, x x x xy y xCov If, then

Real (Actual) versus Risk Neutral Probability

Real probability measure is the actual probability that cash flows will be large or smallwhere estimation is done using historical data and other fundamental insights about the

company. It is generally represented as

In derivatives pricing however, we are concerned not with real or actual probabilities but

with risk neutral probabilities. In an arbitrage free, frictionless market there exists a riskneutral probability measure which makes the price of any derivative today equal to theexpected value of the payoff, discounted by a risk free rate. In a complete market thisrisk neutral probability measure is unique which renders the option price unique. Risk

neutral probability is generally represented as



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The Book of Greeks


Other Probability distributions

Binomial distribution

Binomial distribution is used in the construction and analysis of a binomial tree to value

options and financial derivatives. A binomial distribution is a discrete probability

distribution (unlike the Normal distribution, which is a continuous distribution) where

the stochastic (random) variable can take on only two values. Within the context of a

binomial tree ( see below more details on binomial tree models of option valuation) these

two values can be denoted as “up-value” and “down-value” representing the stock going

up or down respectively. The random variable, t S (stock or the asset price) is called a

Bernoulli variable and has two states of existence in the next period.

If the probability of the “up state” is given by p and that of the “down state” is given by

)1( pq then the probability mass function of this distribution is given by:

mnmmnm

m

n p pmnm

n p pC m P

1

!!

!1

Where, m P , denote the probability of the stock (asset) price going up m times out of

n trials. The events in a binomial distribution are independent of each other. The above

formula is very useful in creating a binomial tree for the asset price movement and

option valuation using VBA code in Excel™.

Poisson distribution

A Poisson distribution is a discrete probability distribution and it gives the probability

that a certain number, m , of a particular event – say, the default on a bond – are going to

happen when the average occurrence, , of such an event in a given time interval is

already known. The probability of exactly m events occurring is given by:

!m

em P

m

The events in a Poisson distribution are independent of each other. Poisson distributionis quite helpful in modeling extreme events where the frequency of occurrence of the

events is less. This distribution is used in many stochastic models of asset prices which

have jumps in their paths and is associated with Poisson process and the Cox process

( see below).



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The Book of Greeks


Entropy

The concept of entropy is related to that of probability. Entropy can be defined as the measure of

“disorder” in a system. A system with high order has low entropy whereas one with a low order

(disorder) has high entropy. Entropy is a fact of life and nature. It is found in all physical, natural

and financial systems. For example, when we get older the entropy increases; when a star like

our sun burns out and becomes a red giant or a white dwarf the entropy increases; when the

pollution in our environment increases, the entropy increases.

Ludwig Boltzman was the first to derive a mathematical expression for Entropy. However, to

understand entropy fully one needs to understand a “microstate” and a “macrostate”. A

macrostate denotes the gross, overall description of a system, say our body. And, a microstate

refers to the specific arrangement of the particles of that system, in this context it will be the cells

of our body. In physics, a gas inside a container is a macrostate and the particles – the molecules

– of the gas comprise the microstate. In the context of finance, or a financial system, such a

characterization of a microstate and a macrostate may not appear that intuitive, but it still holds

true. A stock price can be considered to be a microstate and the option price or the derivatives

based on that stock can be seen as the macrostate. Or, financial assets traded in a financial

markets may be considered to be the microstates and the given financial market (asset market)

the macrostate. A macrostate subsumes a microstate and there can be, and, in most systems, there

are many microstates in a single macrostate.

Thermodynamic entropy of a macrostate is simply defined as the natural logarithm of the number

of microstates consistent with this macrostate. If n is a “macrostate” and m as the “microstate”

then the number of microstates in a given macrostate can be found out using the following

formula for Combinations:

)!(!

!

k nm

nW

The sign “!” in the above formula denotes factorial where, 12!2 and 123456!6

and so on. The above formula simply states how many ways are there to make m choices from n

objects. For example, in how many ways can we form a 3 member team from 12 candidates? The

answer is 220. Or say, if there are 200 coins then in how many ways can we get one (1) head?

The answer is 200 (because all 200 coins can result in a head once)

Boltzman defined (in fact, he derived this expression) entropy as:

W k E ln



63

The Book of Greeks


Or, in other words, we can write the expression for thermodynamic entropy as:

)!(!

!ln

k nm

nk E

The constant, k , is known as the Boltzman’s constant and is an extremely small number.

Say, we have a system where there are 200 coins. This is the macrostate, whereas the

combination of number of heads and/or tails in all these 200 coins is a microstate. Now how

many ways can we get 1 “head” out of the 200 coins in a single toss? The answer is:

200!)1200(!1

!200

. If for the sake of simplicity, if we assume the value of the constant, k , to be

one (which is not true, as k is a very small number and is equal to 231038.1 k ), then the

entropy of this system – with the possibility of one head out of 200 coins – is given by:

2983.5!199!1

!200ln1

)!(!

!ln

k nm

nk E

Now, if we want to find the entropy of the system wher e we want to have say, the 25 “heads” out

of 200 coins in a single toss, it would be given by:

65.21!195!5

!200ln1

)!(!

!ln

k nm

nk E

The entropy increases when we increase the number of “heads” from 1 to 5 in a system of 200coins. If we continue the above exercise by finding out the entropy of the above system

consisting of 200 coins and we increase the number of heads from 5, 10, 20, 100, 150, 175, 195

and all the way up to 200, we will notice an interesting fact: the entropy increases and then

decreases when we increase the number of “heads” in a single toss in a 200 coins system.



64

The Book of Greeks


Entropy and Probability

In physics, probabilistically entropy can be defined as the measure of the random ways in whichan isolated system can be organized Entropy of a system comprising random events and probability of those random events is related. Entropy can also thought of as the measure of total

randomness in a system. If p is the probability of the occurrence of a random event then the

entropy of the event is given by:

p p E log

In information theory, entropy is defined as the measure of the state of knowledge. Entropy

measures how much information is there in the system.

If there are many particles – equivalent to many assets in a financial system – in an isolatedsystem then the entropy function, which expresses the total randomness of the system, can beexpressed as:

i

ii p p E log

Entropy of a Binary System and Equi-probable Trees

A binary system is where two states are possible. For example, in the toss of a coin there are only

two states: either a “heads” will appear or a “tails” will appear. A binomial tree where a stock or

an asset price can either go up or down is a binary system. A model of a bankruptcy of a

company, where it can either be bankrupt or remain a “going concern” is a binary system.

A binary system can be in equilibrium only when the probability of occurrence of the event is

50%. This may seem quite intuitive from everyday experience point of view. On any given day

-2E+35

0

2E+35

4E+35

6E+35

8E+35

1E+36

1.2E+36

1.4E+36

0 50 100 150 200 250

E n t r o p y

Number of Heads in a single toss of 200 coins

Entropy

A System of 200 Coins



65

The Book of Greeks


ask a man on the street as to what is the probability of the stock market going up the next day

and he’d answer 50%. It’s always 50:50 chances in life. Toss a coin and there is a 50% chance

that “heads” will appear and 50% chance that “tails” will appear. Talking in these terms is almost

trivial. However, this 50:50 rule is the only way that markets attain maximum entropy. All

systems move towards maximum entropy. In other words, an equi-probable binomial tree (50%

chance of stock going up and 50% chance of stock going down) is in the state of maximum

entropy. Any other probabilities of up-move or a down-move will contravene the law of

maximum entropy.

Mathematically, the above can be shown as below.

Say, there is only one stock in the system and the stock’s movement is a binary function, i.e. the

stock can either go up or down in the next period. The probability of the up move is p and that

of the down move is p1 . Therefore, the entropy of the system will be:

p p p p

p p E i

ii

1log1log

log

To find the value of p , the probability of the occurrence of the random event (the up move of

the stock price), we need to take the first mathematical derivative of the above expression andequate it to zero.

01log1log

0

p p p pdpd

dp

dE

Differentiating the above expression by parts and equating it to zero gives:

2

1

01loglog

p

p p

Therefore, if a stock (asset) price is modelled as a binomial system (a binomial tree) whereby

there are two states of movement (up and down move) in the next period then for the system to be in equilibrium the probabilities of the up move and the down move have to be equal to 50%.

Part C



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The Book of Greeks


Binomial & Trinomial Trees

In a binomial tree there are two states into which an asset price can move from thecurrent price level. One is the “up” state and the other is the “down” state. There is a probability of up move and a probability of down move, such that the sum of the

probabilities is equal to one (within a risk neutral pricing framework, the probabilitiesare “risk neutral” probabilities). A binomial tree follows a binomial probabilitydistribution and in the limiting case, just as the binomial distribution approaches a Normal (Gaussian) distribution, a binomial tree (model) of the option price approachesthe Black-Scholes option price.

The simplest binomial tree is a Cox-Ross-Rubenstein (CRR) trees which is used to valueequity and FX options. A CRR tree is a recombining tree, meaning that after everyalternative node the up state and the down state converge to the same value as thestarting value of the asset. Not all binomial trees are re-combining trees and many treesused in interest rate modeling are non re-combining. It is mathematically very

challenging to handle non re-combining trees.

In a trinomial tree there are three states, the “up” state, the “down” state and the “same”state. The “same” state is the one where in the next period the asset price remains thesame as it is today. In a trinomial tree there are three corresponding probabilities, one foreach state, and all probabilities sum up to one.

1. Cox-Ross-Rubenstein (CRR) Tree

The quantum of up and down moves is given by:

ued movedown

eumoveup

t

t

1

The drift and the probabilities are given by:

updown

up

t r

p p

d u

d a p

ea

1



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The Book of Greeks


Implementing a CRR Tree in Excel™

2. Jarrow- Rudd (JR) Tree



68

The Book of Greeks


The quantum of up and down moves is given by:

11

11

2

2

t t r

t t r

eed

eeu

The most popular choice for the quantum of up and down moves for a JR tree is:

2

2

1

r

ed

eu

t t

t t

The probabilities are given by:

2

1 d u p p

3. Trinomial Tree

Trinomial Tree with CRR Parameters

If a trinomial tree can be constructed with CRR parameters, such that there are threestates: an up move, a down move and a move where the asset (stock) price remains thesame, then the quantum of respective moves will be given by:

t eu 2

t ed 2

and the respective risk neutral probabilities will be given by:

2

22

22

1

t t

t t qr

up

ee

ee p



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The Book of Greeks


2

22

2

1

2

t t

t qr t

down

ee

ee p

downup same p p p 1

For more on the above tree see, Espen Gaarder Haug’s Derivatives Models on Models.

Another common model of trinomial tree is constructed where the quantum of up move,down move and the move for remaining at the same level is given by:

t

t t

t t

em sameStay

ed down

euup

3

3

The respective risk neutral probabilities of the moves are given by:

3

2

6

1

m

d u

p

p p

4. Tian’s Equal Probability Tree

Given the following:

t qr

t

e L

e B

L L J

2

4

3

The quantum of up and down moves and that of staying the same is given by:



70

The Book of Greeks


2

3

22

22

B Lm

m L Ld

m L Lu


3

1 md u p p p

5. Kamrad and Ritchken Tree

This tree is used by many quants to value convertible bonds. There is a stretching

parameter which makes the tree recombining. This tree also posits a horizontal jump

which is given by 1m

The quantum of up move, down move and staying the same is given by:

1

m

ed

eu

t

t


2

2

2

2

2

11

2

2

1

2

1

2

2

1

2

1

m

d

u

p

t qr

p

t qr

p

The value of has to be bigger than one and is taken by many practitioners to be equal

to 23 which makes the probability of a horizontal jump equal to 31 .



71

The Book of Greeks


Part D

Black-Scholes Diffusion Equation

& Green’s Function for Valuation of Exotics

Note: This section is due to joint work with Danny Yap, CFE student, Singapore, whocovered this topic in his class presentation which was supervised and guided by me.

Also, this content appeared on Risk Latte’s website.

If you believe the stock price process is GBM,

dz t t S dt t S t dS

Then Ito’s formula and a hedging argument, leads to the Black-Scholes partial

differential equation (PDE) for t S t X , :

rX s

X t S

s

X t rS

t

X

2

222

2

1

To get the diffusion equation, make the change of variables,

K

t S r t T

r z t T

r et S t X U t T r ln,,,

2

2

2

2

2

2

2

2)(

2

2

2

2

2

2

Then the BSE becomes the Diffusion Equation (DE) for z U , ,

2

2

z

U U

In 1905, Einstein showed us that the DE arises from the Brownian motion ofmicroscopic particles. Thus, both the BSE and the DE are based on the same underlying

process. Solutions of the DE with known initial condition z U ,0 take the form,

4

' 2

4

1',,,'',0',,,

z z

e z z Gdz z U z z G z U

',, z z G or Green’s Function is called the Fundamental Solution of the DE.



72

The Book of Greeks


Option Prices

For a call option, the boundary condition at T t is,

01

00,,

y

y yh K S K S hT S T X

where )( yh is the Heaviside Step Function. Making the transformation of variables, the

boundary condition on the DE at T t becomes,

1,02

2

2

2

2

2

r Exp K

r

z h z U

z

We can put z U ,0 back into the integral, to get Black-Scholes Formula after “some”

integration

T d d T

T qr K

S

d d N Ked N eS X rT qT

12

20

1210 ,

ln

,

2

Alternatively, we can solve for z U , numerically and then transform back to get X

Initial Conditions for Other Options

K r

z h z U Put Binary

K r

z h z U Call Binary

r Exp K

r z h z U Put Vanilla

z

2

2

2

2

2

2

2

2

2

2

,0

,0

1,0



73

The Book of Greeks


Part E

Numerical Integration Techniques

& Monte Carlo Integration Routine

1. General Numerical Integration routine

2. Rectangle Rule for Numerical Integration

3. Trapezoidal Rule for Numerical Integration

The trapezoidal rule breaks up the area under the curve traced by the function intotrapezoids and evaluates the area of all those trapezoids and sums them up.

Say, we have to integrate a function, x f , between the limits, a and b :

b x

a x

dx x f I

If we divide the curve traced by the function on an xy plane, where x axis is theabscissa and y axis represents the function, x f , into small trapezoids then the area

of each trapezoid enclosed between, i x and 1i x would be given by:

1

111

2

2

1,

ii

iiiiiii

x f x f x

x f x f x x A x xTrapezoid of Area

The integral would then be the entire area contained in the curve, which represents

the sum of the areas of all the small trapezoids.



74

The Book of Greeks


b

ai

ii

b

ai

iiii

b

ai

i

x f x f x

x f x f x x

A

Trapezoids All of Areasof Sum I Integral

1

11

2

2

1

A useful closed form approximation of the integral using the trapezoidal rule is given by:

b f

ba f a f

abdx x f

b

a 24

6

Implementing Numerical Integrals using trapezoidal rule in Excel™



76

The Book of Greeks


One of the GQ methods is the Gauss-Legendre quadrature (GLQ) method. GLQmethods are based on the Legendre polynomials of the first kind. GLQ.

GLQ are constructed for integrals over the range [-1, 1]. However, we can generalize

the intervals by a suitable transformation. With a function, x f and weighting

coefficients iw , GLQ is a summation given by:

n

i

ii x f wdx x f 1

1

1

If we want to have a more general limit of integration, say, [a, b], then we use atransformation:

ab

ba x z

2

With the above transformation the integral becomes:

dz ab z ab

f ab

dx x f I

b

a

1

1 22

Using a set of weights, iw the above expression in discrete form becomes

22 1

ab z ab f w

abdx x f I

N

i i

b

a

In the GLQ method the values of the abscissas, i x , and the weights, iw , are

determined using Legendre polynomials. The Legendre polynomials, x P N , are a set

of polynomials of degree N and the roots of x P N are distinct in the interval (-1, 1)

and symmetric with respect to the origina. Legendre polynomials for N up to 100 are

already published and freely available. In the GLQ method, increasing N increasesthe accuracy of the evaluation process.

When 2 N (not at all desirable as the accuracy will be quite low) the integral can

be approximated, with a degree of precision 3, as:

3

3

3

31

1 f f dx x f



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The Book of Greeks


5. Monte Carlo Integration Routine for Non-path Dependent Options

Monte Carlo integration technique follows the same numerical integral techniques asmentioned above, with the only difference that the probability distribution has to beincorporated into the integral. This is because at maturity the call value is an

expectation given by:

0,max K S E eCall T

QrT

Where, the above expectation is taken using the risk neutral probability. Followingthe “expectations approach” we know that the above expectation is given by thefollowing definite integral:

dz z N K S eCall T

rT 0,max

Where, z is a random variable that is drawn from a Normal (Gaussian) distributionwith zero mean and unit standard deviation and z N is the cumulative probability

distribution function. The above expression is certainly not an ordinary (Newtonian)integral as we have the stochastic (random) term, z , in it and the integration is over z . However, if we let z vary deterministically over a certain range then the aboveintegral can reduce to an ordinary integral.

The same holds for the put option. The value of the put option is given by:

dz z N S K e Put T

rT 0,max

Monte Carlo Integration Routine for Valuation of a Call

If z is a random variable that is drawn from a Normal (Gaussian) distribution with

zero mean and unit standard deviation and if z N is the cumulative probability

distribution function then the value of a Call is given by:

dz z N K S eCall T

rT 0,max

If 0S is the asset price at time, 0t , then with the familiar constant parameters for

risk free rate, r , dividend yield, q and volatility, , the asset price at time, T ,

denoted by, T S , can be expressed as a stochastic equation for geometric Brownian

motion give by:



78

The Book of Greeks


z T T qr

T eS S 2

2

1

0

Therefore, we can write the definite integral for the call option as

dz z N K eS e

dz z N K S eCall

z T T qr rT

T

rT

0,max

0,max

2

2

1

0

Approximating the integral, which is the area under the curve traced by the function,

0,max K S S f T T by summation we can write,

i

i

z T T qr rt z N K eS eCall

i

0,max

2

2

1

0

The cumulative probability distribution function, i z N , can be estimated as the area

under the curve traced by the function 0,max K S S f T T , where, T S is given by

the equation above, .

Therefore, the above call valuation summation can be further broken down into:

i

i

z T T qr rt A K eS eCall

i

0,max

22

2

1

2

1

0

Where, i A , is the area under the curve or the sum of areas of all trapezoids contained in the

curve.

Implementation on Excel™

Here’s the algorithm for implementation on an Excel spreadsheet

(i) Generate the random numbers, z , in a deterministic manner; in mostroutines simply generate a sequence of numbers from -5 to +5 with equalspacing of 0.1. The limits of -5 to +5 is a proxy for the theoretical limits of

to . Therefore, there would be 101 points on the x axis (or 101

points / cells in a column in Excel). This is the x -axis or the absissca;



79

The Book of Greeks


(ii) Estimate the cumulative probability distribution function, z N for each of

these 101 points, using the Excel function =NORMSDIST( z , 0, 1, 0).

(iii) Now, these i z N , generated in step (ii) above, become the x values on the

xy plane and where, we have sliced the entire curve traced by the function,

0,max K S S f

T T

, into 101 trapezoids. We estimate the area of each

of these trapezoids using the trapezoidal rule for area (shown above);

(iv) Generate the (deterministic) asset price paths for all the 101 points using the

equation

z T T qr

T eS S 2

2

1

0 where, we input, the z values

from step (i) (v) Then using the value of T S generated for each of the 101 points, estimate the

function, 0,max K S S f T T and multiply it with i z N generated in

step (iii) above.

Numerical Integration (Monte Carlo Integration) in Excel™



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The Book of Greeks


Part F

Differential Equations

&

Finite Difference Methods

A differential equations is an equation, describing the change and rate of change of

variables, that contains terms likedx

d ,

x

,

2

2

x

,

y x

2

, etc. which are terms used in the

study of calculus. However, there are two kinds Calculus, in mathematics:

Newtonian calculus

Stochastic calculus

Newtonian calculus describes the dynamics of variables that change deterministically,

whereas, Stochastic calculus describes the dynamics of variables that change randomly(via random numbers). Ordinary differential equations (ODE) and partial differentialequations (PDE) fall in the realm of Newtonian calculus whereas stochastic differentialequations inhabit the world of stochastic calculus.

In Finance, we come across three kinds of differential equations.

Ordinary differential equations (ODE)

Partial differential equations (PDE)

Stochastic differential equations (SDE)

Ordinary Differential Equations in Finance

An ordinary differential equation (ODE) describes the dynamics of a variable that varies

with another variable in a deterministic manner. For example, if there is a variable, y ,

the change of whose value, or the rate of change of whose value, is governed by the

change in the value of another variable, x , then we can describe such a relationship viaan ODE.

Savings Account

An example of ordinary differential equation is the everyday savings account in a bankthat we use. If we deposit M as the amount of money (in Dollars) in a savings accountwhere the interest paid on the deposit is r (which is constant), then over a period of time,t , the amount in the account will be given by:



81

The Book of Greeks


t r M t M

Here, the functional form of relationship is: t f t M , where the money deposited in

the savings account is only a function of time elapsed.

The way the money will grow in the account over a time period, t , and t t , where,t is a very small (infinitesimally small) time interval, will be given by:

t Mr t t t r M t M t t M

Writing, the expression, t M t t M , as t M , where, is the delta representing a

very small (infinitesimally small) change in the value of M , the amount of money in theaccount.

Therefore, we can write the above equation for the change in the amount of money in the

savings account as:

rM t

M

t r M

M

Taking the limit as 0t , we write the above equation in the familiar differentialcalculus form as

rM dt

dM

The above is an example of an ordinary differential equation (ODE). The above ordinarydifferential equation can be easily solved.

American Capped Call Option

Another example of an ordinary differential equation is the American style capped calloption. If we consider an American style call option which is capped at a certain level,then the underlying dynamics of the call option price is described by an ordinary

differential equation as shown below. If AC is an American style capped call option, suchthat when the cap is hit, the option expires, with no maturity, such that the functional

relationship is mathematically expressed as: S f S C A . Here, the call price is only a

function of the underlying asset price, S .

The ODE for a capped American style call option is given by:



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02

12

222 r C

dS

C d S

dS

dC S qr A

At

At

Important Note

Almost always, the underlying dynamics for the change of an option price, C with theunderlying asset price, S is a function of both, S and the time, t , i.e. t S f t S C ,, . In

such cases, the differential equation describing the change in the option price with bothasset price and time will be described by a partial differential equation (PDE), like theBlack-Scholes PDE, because, the option price is partially dependent on the asset priceand partially on the time. However, in some rare cases a partial differential equation canreduce to an ordinary differential equation, like when we use the Laplace transforms tosolve derivatives problems as well as like the case above, when the option price is afunction of only one variable, i.e. the asset price.

Stochastic Differential Equation in Finance

A stochastic differential equation (SDE) is a differential equation that describes the behavior of a dependent variable with respect to an independent variable that changesrandomly. For example, the price of a financial asset (other than cash), like a stock, astock index, an FX pair or an interest rate, is assumed to follow a random processwhereby it is a function of time but the evolution of the asset price over tine is random.

An example of a stochastic differential equation is the equation of the famous geometric

Brownian motion, where, the asset price, t S , varies in a random manner:

t

t

t dW dt S

dS

It is because of the second term in the above equation, t dW , that separates this equation

from an ordinary differential equation because t dW varies in a random manner that is

unpredictable. Here, t dW varies as: t dW t t , where, t is a random number drawn

from a standard Normal distribution.

The key characteristic that separates a stochastic differential equation from an ordinary

differential equation, is the fact that the asset price, t S , is not differential with respect to

time. With the risk of oversimplification, one can say that the asset price movement overa small period of time, say, a day, is indistinguishable from the asset price movementover a smaller period of time, say, an hour. In other words, the change of asset price with

respect to time,dt

dS t , does not converge.



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Here we follow the arguments of Jamil Baz and George Chacko. Consider an arithmetic

Brownian motion for the asset price, t S , given by:

t dt dS

dW dt dS

t

t t

Taking, the discrete version of the above equation

t t S t

Dividing the above equation by t we get

t t

S t

Taking the limits as 0t , i.e. time slice becomes infinitesimally small, we get

t t Lim

t

S t Lim t

00

Hence, the change of asset price with respect to the time goes to plus or minus infinity

(depending on the random variable, ) as the time interval is made smaller and smaller.This makes the Brownian motion unpredictable over short intervals of time. This alsoshows that the asset price is not differentiable.

Partial Differential Equations (PDEs) in Finance

A partial differential equation (PDE) is an equation which has partial (mathematical)

derivatives in it. If we have a function, y which is dependent on two variables, x , which

is say, a spatial variable and t , which denotes time, such that t x f y , then a partial

mathematical derivatives of y are given by: x

y

and

t

y

. The first partial derivative

shows the variation of y with respect to x and the second partial derivative shows the

variation of y with respect to t . We can combine these two partial derivatives and write

an equation as:

2

2

x

y

t

y

The above is known as a heat equation and is an example of a PDE. This equation hasapplications in the field of financial derivatives.



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In quantitative finance, a financial derivative, such as a call option on a stock, is afunction of two variables, the underlying asset, such as the stock, and the time. We can

write the call option on the stock as t S C t S f C ,, which shows that its value

changes with the change in S and t . Therefore, the partial derivatives of a call option can be written as:

t

C

= first partial derivative of the call option with respect to time

S

C

= first partial derivative of the call option with respect to the asset price

2

2

S

C

= second partial derivative of the call option price with respect to the asset price

All these three partial derivatives can be combined to give a partial differential equationfor the call option price:

rC S

C S

S

C rS

t

C

2

222

2

1

This is the famous Black-Scholes PDE, the solution of which yields the Black-Scholes

call option pricing formula.

It has to be noted that all PDEs that we study in finance are linear in nature.

Partial differential equations (PDE) are divided into three major categories:

Parabolic Elliptic

Hyperbolic

The parabolic equation, of which diffusion and convection-diffusion equation is a subclass, is the most used and popular one in the world of derivatives. Black-Scholes PDE,the solution of which gives the celebrated Black-Scholes option pricing formula, is aconvection-diffusion equation. However, elliptic and hyperbolic (a 2nd order hyperbolicequation is a wave equation) equations are also found in some financial derivativesapplications.

Parabolic PDE

Parabolic PDEs model heat flow (financial derivatives) and fluid flow phenomenon. TheBlack-Scholes (PDE) in quantitative finance is a parabolic PDE and is given by:



85

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t S rC

S

t S C S

S

t S C rS

t

t S C ,

,

2

1,,2

222

The Black-Scholes PDE, like the heat equation stated above, is a Diffusion equation.

In the above, the Call option price, C is a function of the underlying asset price, S andthe time, t and hence is written as t S C , . The partial derivatives of the call option price

are:

t

t S C

,, which explains the variation of the call price with the time,

S

t S C

,which

explains the variation of the call price with the underlying asset (and is related to the delta

of the option) and

2

2 ,

S

t S C

which is a second mathematical derivative of the call price

with respect to the underlying asset and can be thought of as the variation of S

t S C

,

with the underlying asset.

Hyperbolic PDE

Hyperbolic partial differential equations model wave phenomenon in physics. Examplesof a hyperbolic PDE in quantitative finance are the PDEs that model deterministicinterest rates.

A wave equation for y , here t x y y , , can be written as:

2

22

2

2

x

yc

t

y

0,, t x

Elliptic PDE

Elliptic partial differential equations require specification of boundary conditions if aunique solution is desired. A important example of an elliptic PDE is the Poisson’sequation (of which Laplace’s equation is a special case) given by:

y x f y

v

x

v,

2

2

2

2

Here, y xvv , is a function of two variables, x and y . In quantitative finance, an

example of an elliptic PDE would be that of a Bond price in some of the one factorstochastic interest rate models.



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Categorization of Partial Differential Equations

Generally speaking, all partial differential equations that are used in quantitative financecan be transformed into algebraic equations, mostly notably, quadratic equations of theform:

022 cbxax

Where a, b, and c are constants and x is a generalized variable. In the case of partial

differential equations we can treat the variable x as a partial derivate of two othervariables. For example we could have a first or second order PDE with terms involving

t

y

,

2

2

t

y

and

s

y

,

2

2

s

y

and x could be a transformed variable of these.

The roots of the above quadratic equation are:

a

acbb x

2

The categorization of a PDE into elliptic, parabolic and hyberbolic is due the nature ofthe roots of the above quadratic equation as shown below:

Parabolic PDE: 02 acb

Hyperbolic PDE: 02 acb

Elliptic PDE: 02 acb

This is how PDEs are typically classified. Time and resource permitting we shall try todiscuss some more on PDEs and especially the non-parabolic PDEs which are not thatcommon in financial engineering applications.

Conversion of PDEs to ODEs

Partial differential equations can be (PDE) converted into ordinary differential equations(ODE) by making use of Laplace transform. PDEs are comparatively much more difficultto solve than ODEs; therefore, the use of Laplace transforms come very handy whiletransforming a PDE into an ODE and finding the solution easily.

Jamil Baz and George Chacko (51) show how to use Laplace transforms to solve PDEsfor derivatives valuation.



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Finite Difference Method

Here we follow the approach of Tung, Lai, Wong and Ng (2010) to set up the finitedifference problem as a system of linear equations and solve it using standard matrixtheory. We find this approach quite elegant and extremely easy to follow.

In a finite difference method, a partial differential equation (PDE) is transformed into afinite difference equation by converting the partial derivatives into finite differenceexpressions. Then using standard matrix method a system of linear equations is solved toget the range of values for the dependent variable.

If the call option payoff is given by: 0,max, K S T S C , where, T is the maturity

of the option and K the strike price then the Black-Scholes PDE is given by:

t S rC

S

t S C S

S

t S C rS

t

t S C ,

,

2

1,,2

222

We first convert the partial differentials into finite difference approximations. To do that

we have not note that an ordinary mathematical derivative of y with respect to x can be

expressed as:

x

x y x x y

dx

dy

We set up a two dimensional grid, a sort of ij plane, with the vertical axis, j

representing the asset price withmax

,........,,, 210 jS S S S and the horizontal axis, i

representing time withmax

,........,,, 210 it t t t . Note that T t i max

, the maturity of the option

and S jS j and t it i .

We choose an arbitrary interior point inside the grid, representing the point i j t S , and at

this point estimate the partial derivatives of the option price as:

t

t S C t S C

t

t S C

S

t S V t S C t S V

S

t S C

S

t S C t S C

S

t S C

i ji ji j

i ji ji ji j

i ji ji j

,,,

,,2,,

2

,,,

1

2

11

2

2

11



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Note that in the above, the partial derivative of the call option with time, t , given by the

third approximationt

t S C i j

, is a forward difference measure in t .

Using the above notation the Black-Scholes PDE transforms into a system of linearequations given by:

i j ji j J i j jii t S C t S C t S C t S C ,,,, 111

1........,,1,0

1.......,,2,1

max

max

ii

j j for

Where, the coefficients are given by:

t jt r

t jt r

t jt r

j j

j

j j

22

22

22

2

1

2

1

1

2

1

2

1

Given the initial inputs for volatility and risk free rate and a suitable choice of t theabove coefficients can be easily calculated and substituted in the system of linearequations.

Two more useful transformations that can be introduced here are:

t S C t S C

t S C t S C

j J I j

ii

,,

,,

maxmax1max

0010

1max j

options style American for

options style European for e t r

10

The above system of linear equations can be put in a matrix form as:

i j

i j

i

i

i j

i j

i

i

t S C

t S C

t S C

t S C

F

t S C

t S C

t S C

t S C

,

,

,

,

,

,

,

,

max

max

max

max 1

1

0

1

11

11

10



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Where, F is a tridiagonal matrix given by:

MAX J

j j j

F

00

00

0

00

111

222

111

0

maxmaxmax

The above matrix equation is iterating forward in time. However, like the binomial tree,

to solve the Black-Scholes option value, we need to work backwards. This is done byinverting the tridiagonal matrix, F and solving the difference equations:

1

11

11

10

1

1

1

0

,

,

,

,

,

,

,

,

max

max

max

max

i j

i j

i

i

i j

i j

i

i

t S C

t S C

t S C

t S C

F

t S C

t S C

t S C

t S C

The above can be easily implemented on an Excel™ spreadsheet and solved.

Laplace Transform

Laplace transform method comes handy in many areas of quantitative finance. Thesimplest application of Laplace transform is to estimate the time value of money.

Laplace transforms are used to convert time domain relationships to a set of equations

expressed in terms of the Laplace operator s . After the transformation, the solution of the

original problem is arrived at by simple algebraic manipulations in the s (Laplace)domain rather than the time domain.

But why do we need to do such a transformation? The quick answer is to simplifymathematical calculations. It is a bit like why we use logarithms in mathematicalcalculations. Logarithms simply the math considerably and makes a problem moretractable. When we take logarithms we transform numbers to the power of 10 or some

other base, say, e , which becomes natural logarithm. What we achieve by this is to



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transform mathematical manipulations and divisions to simple additions and subtractions.Similarly, Laplace transforms can be applied to linear, differential equations in a way thatthe differential equations are transformed into simple algebraic equations which can besolved easily.

The Laplace Transform of a time variable function, f (t ), is defined as:

0

dt t f et f L s F st

Now think of the present value problem in Finance. One of the most fundamental andelementary problems in finance is to estimate the present value of a future cash flow. Ifthe discount rate (interest rate) is constant and equal to r then the present value of a

future cash flow, t C , where, t C is a function of time, t is given by:

T

t

rt

T

t t

t C er t C t p

11 1

In the above we have assumed continuous compounding and present value function, t p

is a function of t. The time is bounded between 0 and some finite quantity, T . In thelimiting case, the summation is replaced by an integral and the above present valueequation can be expressed as:

T

rt dt t C er p0

Due to the presence of the integral, the domain of the computation changes from time, t to rate, r and therefore, the present value becomes a function of the rate, r . However,the boundary of the integral is still from 0 to some finite quantity, T .

Now, if we change the upper bound to infinity, i.e. T , then the definite integral will

become:

0

dt t C er P rt

Note, that equation (4) is now an exact replica of equation (1), where the “ r ” (rate)

domain acts like the Laplace domain, “ s ”. In fact, we can write equation (4) as:

0

dt t C et p Lr P rt

Therefore, the present value of a future cash flow r P is the Laplace transform of the

cash flow t C .



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Of course, one needs to be cognizant of the fact that the bounds of the integral above arefrom 0 to infinity. In other words, the Laplace transform equation, (3), exactly translatesinto the discrete time present value equation (2) only when we are considering a very

long period of time. If the cash flow is $1 then by equation (4), r

r P 1 and if the cash

flow is K (where, K is constant) then equation (4) will yield r

K r P . Actually, this can

be very easily verified using an Excel spreadsheet. Choose any interest rate, say, 5% andchoose a cash flow equal to $100. Then, over say, a 100 year period (100 years is longenough to be the real life equivalent of infinity) the present value of all the cash flows(summed up over 100 years) would be equal to $1,985. Using the Laplace transformresult, you’d get $2,000.



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Chapter S

Stochastic Process for Asset Price Modelling(For Implementing Monte Carlo Simulation in Excel™ )

A fundamental, and perhaps the most crucial, step in valuation and risk analysis offinancial derivatives is to create a particular math model for the stochastic asset price process. The following variables and constants are used in the equations in this chapter:

Random Number (from a Normal Distribution) = 1,0~ N t

Weiner Process (Random Walk) = t dW t

Discrete Weiner Process (Random Walk) = t W t

Poisson Process = t P with intensity and Jump size J

Stochastic Asset Price = t S

Asset Price today (time, 0t ) = 0S

Constant displacement in Asset price =

Stochastic Forward price = t F

Constant Volatility =

Stochastic Volatility = t

Stochastic Variance = t v

Constant rates = r

Stochastic rates = t r

Constant dividend yield = q

Drift = qr

Long term value of rates (constant) = r

Long term value of rates (stochastic) = t r

Long term value of variance (constant) = v

Long term value of variance (stochastic) =t v

Long term mean of the mean reversion parameter,

v = t m

Speed of mean reversion = k

Volatility of volatility (constant) = Volatility of the mean reversion parameter (constant) =

Correlation =

Long term mean of Correlation (constant) = Libor at time, ( T t ) = T L

Forward Libor at time ( T t ) = F

T L Maturity value of a variable is indicated by T t



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Stochastic Process

“Stochastic” means “Random”. A Stochastic process, t W , where T t t W W t , , is

defined as a sequence of random variables, t W , where the parameter, t denotes time

and flows over an index T . The state space of the stochastic process, is given by the setof all possible values for the random variables t W and each of these possible values is

called the state of the process. If T represents a countable set then W is a discrete

stochastic process and will represent discrete time measurements. The process t W is

a continuous stochastic process if the index T is a finite continuum.

Simply put, a stochastic process is variable who value over a period of time changes in arandom (stochastic) manner. A stock price is a stochastic process, as is the FX rate or theinterest rate. If we measure the random changes in the variable over discrete timeinterval, such as one month, one day, one hour, one minute, one second, etc. then it is

known as a discrete time stochastic process. Here we denote the change in the timeinterval – over which the value of the variable changes randomly – as t . If we measurethe random changes in the variable over infinitesimally small time intervals, such that

0t , then the process is known as a continuous time stochastic process. An example

of a stochastic process is a Weiner Process or what is popularly known as Random Walk.

Two World Views for Incorporating Stochasticity (Randomness)

In Finance, stochastic processes are generally of two kinds:

A process where the “asset price” is stochastic;

A process where the “time” is stochastic;

Stochastic Processes of “Asset Price” and the Levy Process

Most stochastic processes in Finance that are used to describe asset price behavior belong to the family of Levy processes. A Levy process is a generalized stochastic process that has three independent components:

(i) Weiner process (i.e. a random walk)(ii) A deterministic drift(iii) A pure jump process

Classification of Stochastic Processes for investigating key Risk Factors

Damanio Brigo Damiano Brigo, Antonio Dalessandro, Matthias Neugebauer and FaresTriki in their excellent tutorial have identified two of the key characteristic features ofasset prices that a stochastic process should incorporate, especially if an investigation is



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made from a risk management point of view. These two features of the asset price are:

Fat Tails

Mean Reversion

Consequently, he characterizes the stochastic process of asset prices into followingcategories:

Basic process: Arithmetic Brownian Motion (ABM) and Geometric BrownianMotion (GBM)

Fat tails processes: GBM with lognormal jumps, ABM with normal jumps,GARCH and Variance Gamma (VG)

Mean Reverting processes: Vasicek, Cox-Ingersoll-Ross (CIR), ExponentialVasicek

Mean Reverting processes with Fat Tails: Vasicek with jumps, ExponentialVasicek with jumps.

However, note that the above categorization is neither exhaustive nor complete. Thereare numerous other stochastic processes that describe asset prices, volatility andcorrelation that have not been included above. The above classification is only one ofthe ways to look at stochastic processes.

Stochastic Processes of “Time” and the Poisson Process

Stochastic processes where time is a random variable mostly belong to the class ofPoisson processes, which are point processes.

Poisson Process

A Poisson process is an important stochastic process used to model insurance risk andhas wide applications in both finance and insurance. There are three kinds of Poisson processes that are of interest in the field of financial engineering:

Homogenous Poisson process

Non-homogenous Poisson process

Cox process (Doubly stochastic Poisson process)

In insurance, Poisson processes are used to model claims arrival time. In finance,Poisson processes (and especially, Cox processes) are used extensively to model defaulttimes, while valuing risky bonds and credit derivatives.



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Stochastic Processes for Modelling Asset Price

1. Markov Process

The stochastic processes that mostly studies in quantitative finance are MarkovProcesses. For example, the random walk and the Brownian motion describing equity(stock) prices and currencies (FX rates) are Markovian processes. A Markov Process is astochastic process where on the present value of the variable is relevant in determiningthe future values of the variables. In other words, the future evolution of the random pathof a variable in a Markov process is solely dependent on the present state of the variable.

2. Martingale Property of a Stochastic Process

The martingale property of a stochastic process states that the conditional expectation ofthe process at any time in the future is just the current value. Suppose you are playing the

coin tossing game with an opponent and every time heads show up in a toss, you win $1and if tails show up you lose $1. The martingale property of this stochastic process statesthat the conditional expectation of your winnings at any future time is just the amount ofdollars you already hold. In terms of probability notation we can write:

i ji X i j X X E ,

3. Random Walk (Weiner Process) and Brownian Motion

All Financial Assets, like stocks, currencies, interest rates, etc. are assumed to follow a

random walk and a Brownian motion. A random walk is a simple, continuous timeMarkovian diffusion process, which is stochastic in nature, and is more formally knownas a Weiner Process.

If t W is a Weiner process, where t is time, then its behaviour can be understood by the

following two properties:

(i) Over an infinitesimally small time interval, t , the change in t W , given by t W

is given by:

t t W t t W t W t

Where, t , is a random number drawn from a Normal (Gaussian) distribution with

zero mean and unit standard deviation. In other words, 1,0~ N t . In the limiting

case, when, 0t the Weiner process is given by:



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t t dW t

(ii) For any two values of time, and s , where, s the processes W and sW

are independently distributed which makes t W a Markov process.

If we add a drift to the random walk then we get a Brownian motion. A Brownian motionis a random walk plus a drift:

Brownian motion = Random Walk + Drift.

Financial assets and securities that follow a Brownian motion have two components totheir price change. The first is the drift and the second is the random walk. The randomwalk is tied to the volatility of the asset or the security. An example of drift is the riskneutral forward rate, i.e. risk free interest rate minus the dividend yield for equities or the

domestic interest rate minus the foreign interest rate for currencies.

Random Walks have these two important properties:

(i) They are Markovian in nature

(ii) They have the Martingale property

4. Geometric Brownian Motion (GBM) with constant volatility

Geometric Brownian motion (GBM) is used to model equity, equity indices, FX rates and

Commodity prices in Finance. A geometric Brownian motion (GBM) models the returnof the stock (price), where the return is expressed as the logarithm of the price relative. Inother words, in a GBM we model the rate of change of the asset price, i.e. the percentagechange. If we assume a lognormal process for the asset price, i.e. if the natural logarithmof the asset price follows a Normal (Gaussian) probability distribution then the Stochastic

Differential Equation (SDE) for the asset price, t S is given by:

t

t

t dW dt S

dS

Here, is a constant and represents the drift of the Brownian motion and is also a

constant and represents the volatility of the asset. The volatility is the coefficient ofthe diffusion process. In a risk neutral setting, where, r is the risk free rate in the

economy and q is the dividend yield of the asset, the drift can be expressed as: qr .

Therefore, the GBM for the asset price can be written as:

t

t

t dW dt qr S

dS



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This is the process that most commercial software in banks and financial institutions usefor simulating a Geometric Brownian motion while using a Monte Carlo simulator tovalue financial derivatives. Of course, it must be added that for realistic levels ofvolatility (less than 300%) and for large number of iterations the asset price valuesgenerated by both the stochastic differential equation and the stochastic integral equationwill converge. Therefore, one can as easily use a stochastic differential equation as a

stochastic integral equation in a Monte Carlo simulation.The above GBM model issuitable for equity (stock) and stock indices whereby the drift is given by the difference between the risk free rate and the dividend yield.

For FX rate, where the drift is the difference between the domestic interest rate and theforeign interest rate, the Stochastic Differential Equation (SDE) for the GBM is given by:

t f d

t

t dW dt r r S

dS

And, the Stochastic Integral Equation for the GBM for FX rate is given by:

t f d t t r r

t t eS S 2

2

1

1

Where, t dW t t in continuous form, and 1,0~ N t .

Spurious Paths in Monte Carlo Simulation

Every once in a while the Geometric Brownian motion (GBM) breaks down due to the

problem with Euler discretization scheme in implementing a stochastic differentialequation (SDE). Spurious paths are those Monte Carlo paths which cross zero!

A GBM should never cross zero and in fact should not even come close to zero. But forsome values of random numbers it will. Why?

Take the stochastic differential equation (SDE) of a GBM:

t

t

t dW dt S

dS

In the above, dW is the Weiner Process and t S is the stochastic asset price. The Euler

scheme for the above SDE that transforms it into a difference equation is:

t t t t t t S S S 111



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Here, is a random normal number, i.e. 1,0~ N . This scheme will converge to the

mathematically correct description of GBM only in the limiting case of t becoming

infinitesimally small, i.e. 0t . But if the time slices are finite, i.e. if t are small, but

finite such as one month, one week, one day, one hour, etc. then depending on the

parameters and one can always draw a random normal number, , such that

t

t

1

There is a small but a finite probability of the above happening. In fact, if the time sliceis finite, then it is only a matter of time when the above will happen, i.e. the random

normal number, , drawn will be less than the expression on the right hand side. And if

that happens then the value of t S will become negative!

Such a path where the asset price becomes negative is called a spurious path. One caneasily test the above using an Excel spreadsheet and running a very large number of

iterations with sufficiently big time slice, i.e. 4121 or t and reasonably large

values of drift and volatility.

Is there a connection between the Riemann Zeta function and the Brownian motion ofthe asset prices?

Number Theory meets Quantitative Finance

Riemann Zeta Function and the Brownian Motion of Asset Prices

Riemann Zeta function is perhaps the most beautiful formula in Number theory andcertainly ranks at par with Euler’s formula as one of the most beautiful formulas inmathematics. The Riemann Zeta function is simply expressed as:

0n

sn s

The function is defined 1Re s . In the above, definition, s is a complex number. In a

series form the zeta function can be expressed as:

..........3

1

2

1

1

1

s s s s

All this is fine. But what has all this got to with quantitative finance?

Let’s see what happens when we take the inverse of the Riemann zeta function



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................

10

1

9

0

8

0

7

1

6

1

5

1

4

0

3

1

2

11

1 s s s s s s s s s s

The numerator of the above series are the values of the Mobius function, x , which iszero if x is square free, 1 if x has even number of prime factors and 1 if it has an oddnumber of prime factors. If we look at the values of the Mobius function the look like

.................,1,0,0,1,1,1,0,1,1,1

This can be thought of as a random walk in which a person – who is quite drunk – or afinancial asset such as a stock price of an FX pair stays where it is, i.e. 0, or takes onestep forward, i.e. +1 or takes one step backward, i.e. -1.

As David Wells writes in his excellent book, Prime Numbers, this observation was first

made by Von Sternbach in 1896, who also listed the first 150,000 values of the Mobiusfunction and estimated that the probability that x was non-zero was around 2

6

which is roughly equal to 0.608 with +1 and -1 having approximately equal probabilitiesof occurrence.

Therefore, the random walk, i.e. a Weiner process that models the asset price, is hiddendeep within the Riemann zeta function.

Let us look at some more evidence of this.

In 1997, in a seminal paper, Broadie, Glasserman and Kou, professors at ColumbiaUniversity and the University of Michigan respectively, proposed a beautiful formula forthe adjustment that needs to be made when we move from a continuous barrier to adiscrete barrier, while valuing a certain kind of financial derivative called the barrieroptions. Barrier options are extremely popular amongst sell side traders in the banks andinstitutional investors and are embedded in numerous structured products sold to retailinvestors as well.

All closed form solution for barrier options (knock-outs or knock-ins) are pricedassuming a continuous monitoring of the barrier, owing to the use of continuous timestochastic calculus in such mathematical modeling. However, in real life all optiontraders observe any barrier – a knock-out or a knock-in level of the asset, given a certain

asset price path – on a discrete basis, such as daily monitoring, weekly monitoring, etc.Therefore, adjustment needs to be made to compensate for this fact.

Broadie, Glasserman and Kou showed that if m is denotes monitoring points (number ofdays or number of months, etc.), H is the (theoretical) continuously monitored barrierlevel and H is the corresponding (practical) discrete barrier level then the two should beapproximately related by this formula (also, see Chapter O, #18 for this formula).



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m

T

He H

In the above formula, the authors showed that the constant is related to the Riemann

zeta function by

5826.0

2

21

In the above formula, s is the value of the Riemann zeta function around 21 s .

The plus and the minus sign in front of the constant represent an up barrier and a

down barrier option respectively. This beautiful formula is used today by traders tomake adjustments to the barrier level when valuing barrier options and the formula. The

function, s around 21

s has a special significance. The Riemann hypothesis statesthat the zeta function has non-real and non-trivial zeros only on the critical line for

which the real part of s is 21 .

The Riemann zeta function and the Riemann hypothesis are both related to the Primenumber theorem and the distribution of prime numbers. Prime numbers are used in many powerful mathematical algorithms to we generate random numbers. These randomnumbers are in turn used to simulate Brownian motion while modeling asset price pathsfor the valuation of financial derivatives.

5. GBM and the Log Returns in Continuous Time Finance

In continuous time finance, closely related to the notion of geometric Brownian motion(GBM) is the concept of log returns for assets. In a continuous time framework, when weslice time into smaller and smaller intervals, the return of an asset get expressed asnatural logarithm of this period’s price divided by the last period’s price . In other words,

if t S is this period’s price (say, today) and 1t S is last period’s price (say, yesterday) then

we express the return of the asset as

t

t t

S

S R ln .

It is a moot point and even though the above expression does not make intuitive sense to

the every day investors, traders and quants use this measure all the time in theircalculation. Actually, it comes from applying integral calculus to finance.

Asset return should be measured, as indeed it is by thousands of fund managers andinvestors around the world, as today’s price less yesterday’s price divided by yesterday’s

price. This is an arithmetic measure. In other words, if t S is this period’s asset price

(where the time period is discrete like one year, one month, one week, one day or even



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one minute) and 1t S is last period’s asset price then (assuming no dividend payments) the

asset return should be equal to:

1

1

t

t t t

S

S S R

The above formula gives the measure of an arithmetic return something that we are allfamiliar with. Now, if we have many periods, say, N, in one time interval (i.e. 12 monthsin an interval of one year) then the total return of the entire interval is simply the sum ofall the individual period’s returns, i.e.

N

t t

t t N

t

t T S

S S R R

1 1

1

1

However, what happens if we start to slice time into smaller and smaller intervals. Say,

we start slicing time (one year) into minutes, seconds, nanoseconds and so on until we getto the mathematical definition of an infinitesimally small interval of time. We are now

talking about the limit when delta t (the smallest measurable unit of time) goes to zero.

Mathematically speaking, we say that 0t . In the limit, the above expression for

return will reduce to:

N

t

N

t t

t

t t

t T

S

S Lt R Lt R

1 100

In the limiting case if 0t , then dS S and the summation sign will get replaced

by an integral. Therefore, the expression for the asset return (dropping the subscript fortime) becomes:

T S S

S S

T S

dS R

0

And the limits of the integral are chosen in line with the time limits, i.e. at start when

0t the asset price is 0S and at the end of the time interval (maturity), i.e. at T t , the

asset price is T S . From integral calculus, we know that x

x

dxln and therefore, we

have:

T S

S

T T T

S

S S S

S

dS R

0 0

0 lnlnln



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Therefore, in a time interval [0, T], if we assume a continuous time process, i.e. time issliced as infinitesimally small intervals, the asset return will be expressed as the naturallogarithm of the final price over the initial price.

6. Girsanov’s Theorem and Exotic Options Pricing

One of the fundamental concepts in asset price modeling for valuation of exotic optionsis the Girsanov’s theorem. It is closely related to the notion of Radon-Nikodymderivative that we learnt above. Barrier option traders come face to face with both theseconcepts quite regularly.

How can we explain Girsanov’s theorem?

According to one veteran exotic options trader Girsanov’s theorem makes the driftdisappear from an asset (diffusion process) making it a martingale. In other words, aBrownian motion with drift becomes a drift-less random walk changing the probability

of the asset hitting a barrier. In other words, we look at the problem of a barrier being hitnot under the real (physical) probability measure but a risk neutral one. The real(physical) probability of the barrier being hit may be too low or immeasurable but undera risk neutral probability measure the problem becomes tractable.

Another trader’s refrain was: options can be very easily priced using the difference between the two reflected processes. Reflection principle helps us in estimating the riskneutral probability of all random paths that reach a given point without going through a barrier. Under a real (physical) probability measure the Brownian motion of the asset price will have drift and hence the two Weiner processes will not be equivalent. Thedrift needs to get out of the way for the Weiner processes to be equivalent. Actually, the

probability takes care of the drift.

Say, you are a barrier options trader and you are investigating an asset price diffusion process given by:

t dW t dt t t dX

Here, t is the drift of the asset and t is the coefficient of diffusion (i.e. volatility).

You are interested in estimating the probability of t X , the asset price, entering a

certain space, say, B (the region around the barrier) in the time interval T ,0 . The only

way to do this is to run Monte Carlo simulations. So you set up an Excel™ spreadsheetsimulator and start to run simulations.

However, you are quite sure that the probability of this event happening is very small,

say, 510 . In other words, you’d have to run 100,000 iterations (realizations of asset price path) to measure this probability.



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Now say, you could somehow change the drift of the asset diffusion process, shown

above, in such a way that the probability of the above event, i.e. asset price, t X

entering the barrier space, B in the time interval T ,0 , becomes significantly larger

than 510 . Therefore, you’d need a lot less number of iterations (realizations of the asset

price paths) to measure this probability. This is music to a trader’s ears! Girsanov’stheorem does precisely that.

Two probability measures, Q and P are equivalent if Q is absolutely continuous with P

and P is absolutely continuous with Q . A probability measure Q is absolutely

continuous with P if the Radon-Nikodym derivative, shown below, exists.

dP

dQ

Say we have an Ito diffusion process (such as a Geometric Brownian motion) for anasset price given by

t dW t dt t t dX

Where, t W is a Weiner process (a random walk) under a probability measure, P . In

the above equation, t is the drift and t is the coefficient of diffusion (equivalent

to volatility).

Note that t X is a martingale if the above process is driftless, i.e. the drift, 0 . That

it, we would need the stochastic differential equation to be of the form t dW t dX .

How can we do so?

Rewrite the above diffusion equation as:

t W d t t dX

t dW t

t t t dX

The resultant equation also looks like a geometric Brownian motion. However, the new

Weiner process, t W is given by:

t

dst

t t W t W

dt t

t t dW t W d

0



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The Poisson process is related to the default intensity, where the probability that the

value of dQ will be 1 is equal to dt .

9. Geometric Brownian Motion for the Inverse of the Asset Price

This process comes handy in understanding the movements of FX rates (currency pairs).They can help us to investigate how the process of USD/JPY (Dollar-Yen) differs fromthe process for JPY/USD (Yen-Dollar).

Given the Jensen’s inequality

X E f x f E

The geometric Brownian motion for the inverse of the asset price, S Y 1 , is given by

t

t

t dW dt Y

dY 2

An example would be, if t S is the USD/JPY (Dollar-Yen) price then t t S Y 1 , would be

the JPY/USD (Yen-Dollar) price.

10. Stochastic Process for the Relative Performance of Two Assets

If there are two assets (two stocks or two stock indices), 1S and 2S such that their

respective stochastic processes (geometric Brownian motions) are given by:

dt dW dW where

dW dt S

dS

dW dt S

dS

t t

t

t

t

t

t

t

21

2

22

,2

,2

1

11

,1

,1

,

Where, the respective Weiner processes are correlated via the correlation coefficient, .

Now, consider a relative process (can be thought of as “relative performance”) given by:

2

1

21 ,S

S S S R

It can be shown that this relative process, R , also follows a geometric Brownian motion

given by:



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t R R

t

t dW dt R

dR

Where, 21

2

221 R and 21

2

2

2

1 2 R

11. Arithmetic Brownian motion with constant volatility

(See Vasicek and CIR process for Rates Modelling )

Arithmetic Brownian motion (ABM) are primarily used to model short rates (interestrates). Also, in stochastic volatility models, such as Heston (see below), the variance ofan asset price is assumed to follow a mean-reverting arithmetic Brownian motion.

In an arithmetic Brownian motion (ABM), we model the change in the value of the asset price and therefore, the value of the asset price can be negative in this model.

t t dW dt qr dS

12. Geometric Brownian Motion for the “Square of the Asset”

The GBM for the square of the asset is given by:

t t t t dW S dt S qr S d 2222 22

13. Geometric Brownian Motion for the thn Power of the Asset

The GBM for the n th power of the asset is given by:

t

N

t

N

t

N

t dW S N dt S N N qr N S d

212

1

14. Mean Reverting Geometric Brownian Motion

Geman (2005) describes a mean reverting geometric Brownian motion that is quite

popular in modeling commodities such as energy and agricultural commodities. If t S is

the asset price and S is the long term average of the asset price, k is the speed of mean

reversion, is the volatility of the asset and t W is a Weiner process such that

t dW t t and 1,0~ N t then the stochastic differential equation for the asset price

is given by:

t t

t

t dW dt S S k S

dS

ln



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Making the following transformations:

t t S G ln

t

rT

t GeY

and integrating between the limits T t , , we get the stochastic integral equation for the

asset price within the time limit T t , , where the value t Y at time t is observed and is the

starting value and T Y is a function of the Weiner process, t W .

t

kt kT kt kT

t T W k

eeee

k S Y Y

22

222

15. Brownian Bridge Process

A Brownian bridge is a “tied down” Brownian motion. It is used to model Treasury bonds where the asset redeems at par. In a Brownian bridge the final asset price reverts back to the par value or the starting price and is therefore known in advance. However,

between 0t and maturity, T t , the process is stochastic.

A Brownian bridge is described by the following process:

T t

T W T

t t W t B

,0

)(*)()( T W t t W t B

Where, t W is the corresponding Weiner process. The stochastic integral equation for

the asset (bond), with two known values, the starting price, 0S , the price at maturity, T S ,

T t

T W T

t W

T

t

s

S

t eS S

0

ln

0

16. Cox-Ross Square Root Process

A Cox-Ross, square root stochastic process is given by:

t t t t dW S dt S dS



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Where, t dW is a Weiner process given by t dW t . This is an extremely useful

process because of the additive property which also makes it a desirable process forvaluation of Asian options. But the most important property of a Cox-Ross process is thatit explains the volatility skew observed in the market. Let’s write the above square root

process differently and express the left hand side as a return process (as opposed to achange in price process).

dz S

dt S

dS

t t

t

It is obvious from the above return process that the instantaneous variance of the return2

t

t

S

dS is equal to

t S

2 . This is an inverse function of the equity (asset) price. Thus,

lower the equity (asset) price the higher the variance and vice-versa. This fact isempirically found to be true. Therefore, the Cox-Ross square root process is a better process than a geometric Brownian motion (GBM) when it comes to explaining thevolatility skew.

In fact, along with the square root process, Cox and Ross also introduced the ConstantElasticity of Variance (CEV) stochastic model of asset prices which till this day is popular amongst many FX option traders. CEV process can be viewed as a moregeneralized form of the Cox-Ross process.

17. Ornstein-Uhlenbeck Process

Ornstein-Uhlenbeck (OU) are a class of mean reverting, stochastic processes which findwide application in the modeling of interest rates and commodities. A generic OU process can be written as:

t t t dW dt S k dS

Where, t dW is a Weiner process given by t dW t t , where, 1,0~ N t is a random

number drawn from a Gaussian (Normal) distribution with mean zero and standarddeviation of one.

Discretization of OU process for Monte Carlo Simulation in Excel™

t t t t t t S k S S 11



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A constraint on the above formula is that the time step, t should be quite small. Forlarge time steps the simulation often breaks down. A more exact discrete formula for thestochastic process is given by:

t

t k t k

t

t k

t

t

t k

t k t

t k t

t k

eeS eS

dW k

eeS eS

2

11

211

2

1

2

1

Ornstein-Uhlenbeck Process in Interest Rate Models

Deutsche Bank quants, Marcus Overhaus, Hans Buehler, Ana Bermudez, AndrewFerraris, Christopher Jordinson and Aziz Lamnouar in their book Equity Hybrid

Derivatives, have shown that within the context of short rate modeling if the short rate(one period annualized interest rate), t r , is a function of the variable t y and its mean, t y

such that t y y f r t t t ,, , then the short rate can be expressed as a generalized mean

reverting OU process given by:

t t t t t dW dt yk dy

This generalized OU process yields most of the well-known single factor short rate

models. For example, if t t t y yr , we recover the Vasicek or the Hull-White model; if

we let the short rate take exponential form of the type t t y y

t er then we recover the

Black-Karasinkski model. And we such an exponential form for short rate, if we make

0k then we retrieve the Black-Derman-Toy model.

Further, if we want to have a better grip on the functional relationship between the short

rate and its volatility then a parameter can be introduced by interpolating between anormal and a lognormal model so that the short rate is expressed as:

1t t y y

t

er

Here, in the limit when 0 , we recover the Hull-White model and when 1 , weretrieve the Black-Karasinski model.



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18. Mean Reverting Vasicek Process for Interest Rates

In 1977 Vasicek introduced a continuous time mean reverting random walk, which wasarithmetic in nature, to model the interest rate. In this model he proposed modelling thethe term structure of rates through the changes in the spot rate. Vasicek’s process is an

extremely important stochastic process because it is one of the first models to have laidthe foundations for modeling interest rates as a mean reverting process. The idea of meanreversion – the notion that the underlying variable, the rate, reverts to a long term averagevalue – finds strong acceptance amongst many economists and central bankers who takethe Vasicek model as the classical model for short rates.

The stochastic differential equation for a Vasicek process is given by:

t t t dW dt r r k dr

In the above stochastic process, the short rate, t r is being modeled as a mean revertingarithmetic Brownian motion (ABM) where,

r is the long term average value of the

short rate and k is the speed of mean reversion. Since the process follows an arithmeticBrownian motion, the short rate can become negative. This is the biggest drawback of aVasicek process.

The discretized process (for implementing it on an Excel™ spreadsheet) will be given by:

t t t t

t t t

t t r r k r r

t t r r k r

11

In a Vasicek model, the following formula gives the probability that the rates will benegative:

t sk

t sk

t

s

ek

er r r N r

21

2

0Pr



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Oldrich Vasicek

Oldrich Vasicek was one of the first professionals in the field of quantitative finance whothought that asset prices could follow a process other than a classical geometric Brownianmotion. His revolutionary paper on the dynamics of yield curve appeared in 1977 and forthe first time modeled an asset – the short rate – as an Ornstein Uhlenbeck process. Thishas come to be known as the Vasicek model. He would also probably be one of therichest academic quant (outside the banking world) alive today. Along with StephenKealhofer, John McQuown, he founded a company called KMV in 1989 which was soldto Moody’s in 2002 reportedly for a sum of US$210 million.

19. Mean Reverting Cox-Ingersoll-Ross (CIR) process for Interest Rates

This is a very important process in quantitative finance. Besides being used in thevaluation of interest rate derivatives, it is also a process which has inspired the stochasticvolatility process of Heston. The process is significant improvement over the Vasicek process as the possibility of obtaining negative rates is significantly redunced.

t t t t dW r dt r r dr

20. Hull-White Process

t t t t t dW dt r r k dr

21. n Dimensional Bessel Process

t

t

t dW dt S

ndS

2

1

22. Black-Derman-Toy (BDT) Process

BDT (1990) is a one factor, mean reverting lognormal model of interest rates where thespeed of mean reversion depends entirely on the volatility function. In practice, a BDT is

implementd in a discrete form using a tree. A continuous version of the BDT stochastic process can be written as:

t t t t t

t t dW r dt r r

dt

d dr

lnln

ln



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Fischer Black, Emanuel Derman and William Toy, all three working at Goldman Sachs& Co. in 1990 explain the key features of this model in their original paper as:

(i) The fundamental variable of the model is the short rate, the one period annualizedinterest rate which drives all security prices. This is the single (and the only) factor

of the model;(ii) The two key inputs to the model are an array of long term interest rates, i.e. the yield

on zero-coupon Treasury bonds for various maturities, known as the yield curve andan array of the yield volatilities of these bonds, known as the volatility curve.

(iii) According to BDT, the model varies “an array of means and an array of volatilitiesfor the future short rate to match the inputs”. The future mean reversion changeswith the future volatilities.

Usefulness of BDT Model

Out of all the one factor models of short rate for pricing options, the BDT model has been

the most popular and for the entire decade of 1990s BDT was one of the dominantmodels of short rate used by banks. It is still used by some banks. There are three reasonsfor its usefulness and popularity:

(i) The model is capable of pricing an arbitrary set of discount bonds that trade in themarket;

(ii) The assumption of log-normal distribution for the short rates which facilitates thecalibration to caplet volatilities;

(iii) Translated into a tree form the model can be very easily implemented.

23. Black-Karisinski (BK) Process

BK is an explicitly mean reverting model of short rate and the dependence on time is via

t k k t , t t and volatility of the short rate, t t . The model is described by

the following stochastic differential equation.

t t t t t t dW dt r k r d lnln

In the above process the speed of mean reversion, t k and the volatility of the short rate,

t are independent.

24. Poisson Jump Diffusion Process

A Poisson Jump diffusion process for an FX rate can be written as:

t t d f

t

t JdP dW dt r r S

dS



114

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Where, an independent jump factor is introduced to the Brownian motion. The equity

counterpart of this equation will have risk free rate and the dividend yield in the drift

process.

25. Kou’s Double Exponential Process

(Stochastic Integral Equation)

)(

1

2

1 )(2

1exp

t N

i

it t Y t W t r S S

111 2

2

1

1

q p

2

2

1r

11 = Mean size of upward jump

21 = Mean size of downward jump

26. Heston Stochastic Volatility Process

Heston (1993) stochastic volatility is a two factor model whereby, besides the asset price

following a stochastic GBM, the variance of the asset varies stochastically as well, albeitvia a mean reverting Cox-Ingersoll-Ross (CIR) process. Heston’s stochastic volatilitymodel is used to price many exotic equity options, especially those where there is aforward skew, such as cliquets and reverse cliquets.

Asset Price Process with Stochastic Variance

The asset price, t S follows a GBM and the variance of the asset price, t v follows a mean

reverting CIR process, with a long term average value of the variance as: v .

2

1

t t t

t t t

t

dW dt vvk dv

dW vdt S

dS

dt dW dW 21



115

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Where, the Weiner processes are given by: t dW t t 1 and t dW t 22

Asset Price Process with Stochastic Volatility

Zhu (2008) has shown that a stochastic process for volatility can be derived from thestochastic variance process of Heston (1993). Using the relationship: t t v and

making use of the Taylor Series expansion, Zhu derives the two factor stochasticvolatility process as:

dt dW dW

k where

dW dt k d

dW vdt S

dS

t

t t t

t t

t

t

21

2

2

1

4,

Where, the Weiner processes are given by: t dW t t 1 and t dW t 22

Finite Difference Discretization of the Variance Process using Euler scheme

t vt vvk vv t t t t 1

Given the fact that the stochastic variance process in Heston’s model follows anarithmetic random walk, there is a possibility of observing negative variances if wefollow Euler scheme of discretization as given above. In a Monte Carlo simulator, thissituation is often mitigated by using the following rule:

conditionreflecting vvthenvif

conditionabsorbing vthenvif

t t t

t t

,0

0,0

However, a better discretization process is the Milstein’s scheme which follows secondorder Ito-Taylor expansion to arrive at the finite difference equations from the

stochastic differential equations.

Finite Difference Discretization of the Variance Pr ocess using Milstein’s scheme isgiven by:



116

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t t vvk t vv

t t vt vvk vv

t t t

t t t t

42

14

22

1

22

1

Main Drawbacks of a Stochastic Volatility Model

(i) The model’s fit with European option prices is not good

(ii) The volatility of volatility (vvol) is determined using calibration to option prices

and the fit is also not good.

(iii) A stochastic volatility model does not produce accurate deltas for hedging

purposes

(iv) In times of great market stress, a stochastic volatility model mostly fails.

For the above reasons, some experts have proposed a Stochastic Local VolatilityModels ( see below).

27. Double Mean Reverting Process for Variance

Given below is the Double Mean Reverting Stochastic Volatility (DMR-SV) model. Tothe best of our knowledge, the first DMR-SV model was introduced by Jim Gatheral. In amore generalized case, a DMR-SV model can be based on Buehler’s “Variance CurveModels”. In a Double Mean Reverting Stochastic Volatility model (DMR-SV) the shortvariance is a mean reverting stochastic process whose “mean reversion” itself isstochastic. Such behaviour is quite often seen in the financial markets.

m

t t t

v

t t t t t

v

t t t t t

t t

t

t

dW mdm

dW vvm pdv

dW vdt vvk dv

dBvS

dS

In the above process the Weiner processes are given by

42

32

,

2

,

2

22

1

1

11

1

t mt m

m

t

t vvt vvvt v

v

t

t vt v

v

t

t t

W BW

W W BW

W BW

W B



117

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In the above stochastic model, the variance is a CIR process that reverts to a long term

average value, t v , which itself varies stochastically. This “long term average variance”

also follows a mean reverting CIR process and the mean of this process is itself astochastic variable.

Link between DMR-SV Model and Buehler’s Variance Curve Model

The DMR-SV model is closely related to the concept of Variance Curve modelsintroduced by Hans Buehler in 2006. The key idea is to get a spot stochastic volatilitymodel based on observable Variance Swap curves in the market.

A Variance curve – the floating leg of a variance swap – is observable in the market andtherefore, forward variance is observable from the variance swap curves. This can benumerically approximated. From forward variance we can get the implied shortvariance. By following an approach very similar to the Heath Jarrow Morton (HJM)

approach for modelling the forward interest rates Buehler derives implied spot volatilityfrom the forward variance. The beauty of Buehler’s approach is to show that areasonable specification of variance swaps in the market is the key to modellingstochastic volatility.

In the above DMR-SV model the calibration of the initial states along with both themean reversion speeds of the “variance”, as well as the “long term average variance” aredone by fitting a double linearly mean reverting variance curve functional to theobserved variance swap market data. This is where Buehler’s Variance Curve model andthe techniques to estimate Variance Curve functionals come in.

28. Constant Elasticity of Variance (CEV)

A Constant Elasticity of Variance (CEV) process for an asset, t S , with a constant

volatility parameter, , is given by:

t t t dW S dt S dS

A parameter, is important and it determines the nature of the distribution and it varies

from 0 to 1. For 1 we get the geometric Brownian motion (GBM), i.e a lognormal

process. For 0 we get an arithmetic Brownian motion, i.e. a normal process. For

21 , we get the Cox-Ross square root process.

29. SABR (Stochastic Alpha Beta Rho) Process

The stochastic alpha beta rho (SABR) model was introduced by Hagan, Kumar,

Lesniewski and Woodward in 2002. This model has mainly been used to model the short

rate and price interest rate derivatives (swaptions, caps, floors). SABR process is



118

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described by the following stochastic differential equations, one for the forward rate and

the other for the volatility of the rate.

2

1

t

t t t

dW d

dW F dF

dt dW dW t t 21

In the above model the forward rate follows a CEV (constant elasticity of variance) type

process; however, the volatility, , of the forward rate, is now stochastic and has a

volatility of volatility equal to . Here, t dW t t or in discrete form t W t t

The key feature of this model is that via the parameter, , it can capture and preserve the

downward sloping shape of the volatility smile. Interest rate markets exhibit downward

sloping volatility smile and hence SABR is a good fit for modeling the short rates.

To implement SABR in Monte Carlo, we use Euler Discretization with BackwardDifference Method:

2111

111

t F F F

t

t t t t

t t t

30. Longstaff’s Double Square Root Model

This model, which was originally proposed by Longstaff in 1989 for rates and by Zhu forthe variance process in 2000, is similar to the Heston’s model for variance except that thedrift term has a mean reversion in volatility and not variance.

2

1

t t t

t t

t

t

dW dt vk dv

dW vdt S

dS

dt dW dW 21

Where, is the long term value (mean) of volatility of the asset.



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31. Stochastic Local Volatility (SLV) Process

In a Stochastic Local Volatility model we mix stochastic volatility with local volatilitymodels. In most “stochastic local volatility” model, volatility has the freedom to rise andfall independent of the underlying spot.

One form of a stochastic local volatility process is given by:

2

1,

t t t

t

v

t t L

t

t

dW dt kvdv

dW eS dt S

dS t

In this model, the second equation follows an Ornstein-Uhlenbeck log-variance process.

Ren, Madan and Qian (2007) have introduced a stochastic local volatility process given

by:

kt

t

t t t t

t t t L

t

t

ek

v

dW dt Z vk Z d

dW Z t S dt S

dS

22

2

1

12

lnln

,

dt dW dW t t 21

In the above model, k is the speed of mean reversion and is the volatility of volatility.

The long term drift, t v of t Z is deterministic and due to the last equation for t v , the

unconditional expectation of 2

t Z is unity. The two Brownian motions, 1

t dW and 2

t dW are

corrected via the correlation coefficient, . Also, it is assumed that 10 Z .

Ren, Madan and Qian’s model is important in one key aspect. In this model, there is an

independent stochastic component, t Z , t Z is not the variance process, though it does

depend on the long term average variance, t v , which is given by the third equation above.

In this basic form, however, the two Brownian motions are not correlated.

The asset price process, given by the first stochastic differential equation, almostmakes the asset price a local volatility process;

The process of t Z gives a lot of freedom for volatility to move regardless of the

underlying asset price’s move.



120

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32. Stochastic Local Volatility (SLV) / Bloomberg Model

Grigore Tataru and Travis Fisher at Bloomberg have proposed a model of asset priceswhich combines the features of a stochastic volatility process and a local volatilitysurface. Tataru and Fisher’s SLV process is given by:

dt dW dW

dW V dt V V k dV

dW V t S dt r r S

dS

t t

t t t t t

t t t L f d

t

t

21

2

1

.

,

The above model is specifically tailored towards modeling FX rates and valuation of

barrier options. In the above process, d r and f r are domestic and foreign interest rates.

In the above model the first equation describes the process for the asset price, t S and the

second equation describes the process for stochastic volatility, t V , where, t V is the long

term average of the volatility, k , is the speed of mean reversion, is the volatility of

volatility and t S t L , represents the local volatility, which Tataru and Fisher call the

“leverage surface”. The asset price and the stochastic volatility processes are correlated

via the correlation coefficient, .

Important Aspects of the SLV Model

In the above SLV model, if we make the volatility of volatility zero, i.e. 0

then SLV model degenerates into a pure local volatility model. On the other hand

making the local volatility (leverage surface) equal to one, i.e. 1, t S t L

recovers a purely stochastic volatility model.

Since both local volatility and stochastic volatility is present in the model, there is

a mixing fraction, which is controlled by the volatility of volatility parameter, .

33. GARCH Diffusion Process for Volatility

The GARCH diffusion model is the continuous-time limit of many GARCH-type

processes and in such a process volatility is modeled as a stochastic process given by:

t t t t dW dt bd



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The Book of Greeks


Where, t dW is a Weiner process given by: t dW t and 1,0~ N . The mean

reversion in the volatility process is captured by the constant drift parameters, b and .

Since, , has dimensions of inverse time, the parameter 1 can be thought of as the

“half -life” of volatility.

Heston-Nandi(1997) has derived a degenerate case for the Heston(1993) stochasticvolatility process as a limiting case for a particular GARCH type process.

34. Mean Reverting Random Walk for Commodity Prices

Geman (2005) describes a mean reverting geometric Brownian motion that is quite popular in modeling commodities space such as energy and agricultural commodities.Here, mean reversion is introduced in a geometric Brownian motion (GBM), which isvery different from equity or FX processes. Due to the lognormal evolution the modeldoes not allow negative values for commodity prices

dW dt S S k S

dS t

t

t ln

Using the variables t t S G ln and t

rT

t GeY we get

t

kt kT kt kT

t T W k

eeee

k S Y Y

22

222

35. Gibson & Schwarz (1990) Stochastic Convenience Yield Process

The forward price of the commodity (Oil), at time, t for a contract maturing at t time, T

is related to the spot price by:

t T yr et S T t F ,

Where, r is the continuously compounded interest rate prevailing at t , for maturity, T

and y is convenience yield on the commodity. Gibson and Schwarz (1990) present the

following two factor model for pricing of contingent claims on Oil is given by:

2

2

1

1

t t t

t

t

t

dW dt yk dy

dW dt S

dS

dt dW dW t t 21.



122

The Book of Greeks


The rationale for a stochastic process for convenience yield is the realization, based ontheoretical and empirical findings by the authors, that a key factor impacting therelationship between the spot and futures commodity prices is the convenience yield.

A Note on Convenience Yield

Convenience yield can be defined as a benefit that “accrues to the owner of the physical commodity but not to the holder of a forward contract” (Kalder(1939)and Working (1948, 1949)). The analogy here is with the dividend yield which is paid to the owner of a stock but the owner of a derivative contract on the samestock does not benefit from it (Geman (2005)).

Convenience yield can be viewed as an “embedded timing option attached to thecommodity” since the inventory, like a gas storage facility, gives us the flexibilityto put the commodity in the market depending on whether the prices are high orlow (Geman (2005)).

In the stochastic models of commodity, the convenience yield is expressed as arate. It is defined as the gain on a physical commodity less the cost of storage(Geman (2005)).

36. Two Factor Model for Oil Prices

Eydeland & Geman (1998) have proposed a Heston type stochastic volatility model forcommodities, especially to model gas or electricity prices. There is mean reversion in thespot price which follows a GBM. The volatility of the asset (commodity) follows a meanreverting CIR process (Heston) and is correlated to the underlying commodity price process.

dt dW dW

dW vvvdv

dW dt S k S

dS

t t

t t t t

t t t

t

t

21

2

2

.

ln

Where, t t and 2t t vvt

37. Dynamics of Forward Price and Valuation of Commodity Derivatives

Many top banks who are important players in the commodity derivatives market valuethese derivatives using a two factor or three factor PCA model. Using PrincipalComponents Analysis (PCA) – which is done using Eigensystem decomposition of acovariance matrix of forward rates – one can identify the most important risk factors thatimpact a commodity forward rate.



123

The Book of Greeks


Keeping the geometric Brownian motion (GBM) framework assuming a varying

volatility in time t and T many practitioners have proposed a model of the forward rateas ( see 43):

motions BrownianW

factorsrisk of number N

dW T t T t F

T t dF

i

t

N

i

i

t i

:

:

,,

,

1

The risk factors are identified through Principal Components Analysis (PCA) of theforward curve / futures.

Financial Engineering Associates (FEA) Model for PCA

In particular, we look at the model proposed by Carlos Blanco, David Soronow & PaulStefiszyn of FEA in 2002.

N

j

N

j

j jij jij t ct c

et F

t dF 1 1

2

2

1

The discretized version of the above continuous time stochastic equation is

N

j

N

j

j jij jij t ct ct F t t F 1 1

2

2

1exp

PCA is done on the covariance matrix of the forward curve / futures. After that the factor

scores and factor loadings are calculated. In the above, ’s denote the factor scores and c’s denote the factor loadings.Then the forward price is simulated using the continuoustime GBM framework shown above

38. Joint Mean Reverting Model of Spot and Forward for Commodities

In the same paper referenced above ( see 63), Carlos Blanco, David Soronow & PaulStefiszyn discuss a stochastic process that models the joint mean reverting movements of

the spot and the forward price of a commodity.

s

t t s dW dW t dt F

t S a

F

t S Log d 21

11

1log



124

The Book of Greeks


Where,

1 F : prompt moth forward price

a : spot mean reverting rate

: correlation between the spot and the prompt month : volatility of the spot price at time, t

39. Stochastic Correlation Process

In a stochastic correlation process the correlation between asset returns vary randomly

within the boundary 11 . The stochastic correlation never crosses the +1 or -1 on

either side of the boundary but within this boundary it varies randomly.

t t t t t dW dt k d 11

The stochastic correlation is modeled as a Jacobi process which is bounded between two

limits. If t X is a random variable that follows a Jacobi process then the process is

described by

t t t t t dW X X cdt bX adX 1

Where, a , b , and c are constants and t dW is a Weiner process. If we transform t X as:

12 t t X

Where, t is the correlation then we can retrieve the above stochastic correlation

process. However, the process imposes the following constraint for the correlation not to breach the bounds:

1122

k k

40. Mixture of Normals / Gamma Process

Here, we follow the arguments presented in T.W. Epps. Understanding this process can be a very good precursor to understanding the Variance Gamma process, explained later.This process is the mixture of Normals which can also explain excess kurtosis. We

assume that the asset price, t S , follows a discrete time stochastic process of the kind

shown below.



126

The Book of Greeks


The Need for a Variance Gamma (VG) Process

It is observed in real life that rather than following a purely diffusive process like ageometric Brownian motion, asset prices follow a diffusive process which has jumps init. It is observed that there are sizable jumps that occur over the path of the asset price.

These jumps are casued due to the arrival of news and information, including financialnews, in the market. Small pieces of news and information arrive quite frequently in the

market causing frequent jumps. Big news and large pieces of information are rare andtherefore they arrive infrequently in the market. This leads to small frequent jumps andlarge infrequent jumps in the asset prices over a particular time period. With theexception of interest rates, this fact is well borne out by almost all other asset marketssuch as equity, FX and commodities. Therefore, we need a mathematical model whichcan incorporate this fact of life in financial markets.

Another way of looking at the above is to note that perhaps “economic” time does not

have uniform speed due to the impact of “unpredictable news” and information whicharrives in market. As we move forward the flow of time accelerates or decelerates in arandom manner. “Time change” therefore should become a key feature of a stochasticasset price model.

What is so Unique about a Variance Gamma Process?

In most of the stochastic processes volatility is intricately linked to randomness. In aGBM or ABM, volatility, even though constant, is tied to the Weiner process, in GARCHstochastic process, even though conditional volatility is not random, the unconditionalvolatility is random; in stochastic volatility model volatility is itself a random process. In

other jump diffusion models, jumps add new source of randomness over and above theWeiner process. However, in all these processes the flow of time is not changed.

In a VG process, however, “financial time” is made random via a new process called the“subordinator”. Here, the “financial time” or the “market activity time” gets randomlytransformed from calendar time.

Concept of the Variance Gamma Process

We start with a standard Brownian motion that explains a pure diffusive process, withconstant or deterministic volatility:

t W t t B ,;

Note, that in the above, we have altered the flow of flow of time in a purely deterministic

manner via a constant volatility, . However, in a Variance Gamma setting we have tomove one step further and specify a “time change” process that is stochastic. This is



127

The Book of Greeks


because of the uncertain arrival of news and information that induces jumps and alters theflow of time in an uncertain manner. As Rebonato explains, this is accomplished byintroducing the condition that a Brownian motion should be evaluated at a particular time

by a gamma process, , with unit mean rate and a variance equal to v , i.e. vt t ,1, .

This gives us a process, t X , which is a kind of Levy process given by:

t W B X t t t

This new stochastic process, t X , is the Variance Gamma (VG) process. A key

observation to make here is that the “stochastic time” is present in both the drift term aswell as the stochastic term.

Distinguishing features of a VG Process

The following are the distinguishing features of a VG process

(i) The variance gamma process (VG) is a Lévy process and is characterized by arandom time change.

(ii) There diffusion component in the VG process is absent and it is instead a pure jump process.

(iii) The increments are independent and follow a Laplace distribution.

Difference between Brownian motion and the Variance Gamma (VG) Process

A Levy process is a kind of a generalized stochastic process which has independent andstationary increments. These increments are given by independent and identicallydistributed random numbers. Both the Brownian motion and the Variance Gamma (VG) process are Levy Process. In a standard Brownian motion (also known as a Weiner process), the mean of these stochastic increments is zero and the variance is proportional

to the size of the increment. If 2, N denotes a Normal (Gaussian) distribution with

mean, , and standard deviation, , then we can write a Weiner process as:

t N t W t t W t dW ,0~

In a Gamma process, the stochastic increments are gamma distributed and if , is a

gamma distribution with mean, and variance, 2 then the process is given by:

1,~ t t t t t d



128

The Book of Greeks


Given the above definition of a Gamma process a VG process can be characterized in twoalternative ways:

(i) It can be thought of as a time-changed, subordinated Brownian motion, wherethe subordinator is a gamma process.

(ii) A VG process can be thought of as a difference of two gamma processes.

Math Model for VG and Monte Carlo Simulation

Simulating VG as a Stochastic Differential Equation

Following the arguments used by Brigo, et al, if we consider a time changed Brownian

motion with mean, ˆ and volatility, and another constant then we can write the

stochastic differential equation for an asset price, t S , which follows a VG process as:

t g dW t dg dt S d t ˆˆlog

Where, .dW is a modified Weiner process. The only way that the above SDE differs

from a standard Brownian motion is that it contains the term t g , which, in probabilistic

term, is known as a subordinator. This term characterizes the market activity time. The

market activity time, t g , is a positive increasing random process where

st s g t g E . This reconciles the market activity time with the real time between

the time, t and s .

t s g t g s g W t g W ˆ~ˆ

Where, t , is a random normal number drawn from a Normal (Gaussian) distribution

with mean 0 and standard deviation of 1, i.e. 1,0~ N t .

Finally, VG model assumes, that the process t g is related to the Gamma process as

v

v

t t g ,~ , v being the variance of the gamma process. Here, the increments are

independent and stationary random variables drawn from a gamma distribution.

Simulating VG as a Stochastic Integral Equation

Take a Brownian motion, t B (which is a Weiner process, t W plus a deterministic drift,

) given by:

t t W t B



129

The Book of Greeks


In the above equation, qr , where, r is the risk free rate and q is the dividend yield

of the asset. Thereafter, we construct VG as Levy process, t X , that characterizes a time

changed Brownian motion with three parameters, the drift of the asset , the volatility of

the asset, and a constant given by:

t W B X t t t

Where, t , is a gamma process with unit mean and a variance parameter, v . Then the

equation for the asset price, t S , with risk free interest rate, r , and dividend yield q , will

be given by

t X t qr

t eS S

0

Where, , is a constant that makes the discounted asset price a martingle measure and isrelated to the characteristic function of the Levy process. This is the equation we woulduse to simulate the asset price.

In the above equation, is given by:

21ln1 2

42. Displaced Diffusion Model

Displaced diffusion model for asset prices was introduced by Rubinstein in 1983.

If t S is the asset price at time, t then in a displaced diffusion process, the quantity t S

follows a geometric Brownian motion, where, is a constant.

t

t

t dW S

S d

and are the drift and the volatility of the process, t S .

Advantages

It’s extremely simple to model

The process can be made to mimic a Constant Elasticity of Variance (CEV) process while retaining the analytical properties of a geometric Brownian motion.



130

The Book of Greeks


A closed form solution, similar to the Black-Scholes formula, can be found outeasily.

A displaced diffusion process may be quite suitable to model inflation andtherefore can find use in valuation of inflation derivatives;

Disadvantages

Whereas, the quantity t S , which follows a GBM in guaranteed to be positive,

there is no guarantee that the asset price, t S , will be positive. In this process, t S ,

is bounded in the range , .

43. Simplified Libor Market Model

(Single factor with Lognormal interest rates)

We assume that spot LIBOR, t L is stochastic and follows a lognormal distribution, i.e. a

geometric Brownian motion of the type:

t L L

t

t dW dt L

dL

However, instead of the above SDE, we use the stochastic integral equation for the GBMto model stochastic LIBOR. Also, we assume that LIBOR diffuses over the time interval

T t , . Therefore,

t l L L t T T T

t T e L L

2

2

1

Here, L is the LIBOR volatility which could be calibrated to caplet volatilities and L is

the drift of the LIBOR. Also, we assume that the forward LIBOR rates are not correlatedwith each other.

Given the fact that LIBOR has a well defined, observable forward curve at all points, thedrift of the LIBOR rate is given by:

F

t

F

T

L

L

L

Substituting the above in the equation for LIBOR above and doing some algebraicmanipulation we get the equation for LIBOR as:



131

The Book of Greeks


t L L t T

F

t

F

T t T e

L

L L L

2

2

1

In the above equation, t is a random number drawn from a Normal distribution with

zero mean and unit standard deviation.

44. Homogenous Poisson Process

This is a process for modeling stochastic time. A homogeneous Poisson process is a

continuous-time stochastic process 0: t P t with an intensity rate), , such that 0

with the following conditions:

(i) t P is a point process,

(ii) the times between events are i.i.d. (random variables) with an exponential

distribution, i.e. exponential with mean .

1.

A Poisson process with a parameter, ( 0 ) can be defined as an integer-valued,

continuous time stochastic process 0: t P t such that the following conditions hold:

(i) 000 P P ;

(ii) For all values of time, nt t t t ......210 , the increments, t dP given by

01 t P t P , 12 t P t P , ….., are independent random variables; (iii) For 0t , 0 and non-negative integers m , the increments have a Poisson

distribution given by:

!Pr

m

et m P t P

t m



132

The Book of Greeks


References:

1. Notes on Solving the Black-Scholes Equation, EOLA Investments, LLC, Oct 2009

2. Solution to the Black-Scholes Equation, S. Karim, MIT, May 20093. The Diffusion Equation – A Multidimensional Tutorial , T.S. Ursell, Caltech, 20074. Mathematical Methods in Earth Sciences, Lectures by Prof. Francis Nimmo, Department

of Earth and Planetary Sciences, University of California, Santa Cruz5. Equity Hybrid Derivatives, Marcus Overhaus, Ana Bermudez, Hans Buehler, Andrew

Ferraris, Christopher Jordinson and Aziz Lamnouar (John Wiley & Sons, Inc. 2007)6. The Volatility Surface, Jim Gatheral (John Wiley & Sons, Inc. 2006)7. Exotic Options and Hybrids, Mohamed Bouzoubaa, Adel Osseiran (John Wiley & Sons,

Ltd. 2010)8. Dynamic Hedging , Nassim Taleb (John Wiley & Sons, Inc. 1997)9. The Complete Guide to Option Pricing Formulas, Espen Gaarder Haug (McGraw-Hill,

2007)10. Volatility and Correlation, Riccardo Rebonato (John Wiley & Sons, Ltd. 2004)11. A New Formula for Computing Implied Volatility, Steven Li, School of Economics and

Finance, Queensland University of Technology, Brisbane, Australia12. A Primer for the Mathematics of Financial Engineering, Dan Stefanica (FE Press, NY)13. Frequently Asked Questions in Quantitative Finance, 2nd Edition, Paul Wilmott (John

Wiley & Sons, Limited)14. Asset Allocation for Institutional Portfolios, Mark P. Kritzman, Richard D.Irwin, 1990.15. Modern Portfolio Theory and Investment Analysis, 4

th Edition, Edwin J.Elton and Martin

J.Gruber, John Wiley & Sons, 1994.16. The Handbook of Exotic Options, Instruments, Analysis and Applications, Edited by

Israel Nelken, Irwin Professional Publishing, 1996.17. The Handbook of Convertible Bonds, Pricing, Strategies and Risk Management, Jan DeSpiegeleer and Wim Schoutens, John Wiley & Sons, Ltd. 2011.

18. FX Options and Structured Products, Uwe Wystup, John Wiley & Sons, Ltd. 2006.19. Demystifying Exotic Products, Interest Rates, Equities and Foreign Exchange, Chia

Chiang Tan, John Wiley & Sons, Ltd. 2010.20. Modeling Derivatives in C++, Justin London, John Wiley & Sons, Inc. 2005.21. Optimal Trading Strategies, Robert Kissell and Morton Glantz, AMACOM, 2003.22. Foreign Exchange Option Pricing, A Practitioner’s Guide, Iain J.Clark, John Wiley &

Sons Ltd., 201123. Structured Equity Derivatives, Harry M.Kat, John Wiley & Sons, Ltd. 2001.24. Paul Wilmott on Quantitative Finance, Volume 2, Second Edition, John Wiley & Sons,

Ltd., 2006.25. Modelling FX Skew, Dharminder Kainth and Nagulan Saravanamuttu, Royal Bank of

Scotland26. Quantum Mechanics in Simple Matrix Form by Thomas F. Jordan27. Derivatives Sensitive to Volatility Surface, Ser-Huang Poon, Manchester Business

School, April 26, 200828. High Frequency Trading, Irene Aldridge, John Wiley & Sons, 2010.



133

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29. Probability of Exercise in the Call Price, Fabrice Douglas Rouah, www.FRouah.com 30. Prime Numbers by David Wells (John Wiley & Sons, 2005)31. A Continuity Correction for Discrete Barrier Options, Mark Broadie, Paul Glasserman

(Columbia Univerity), Steven Kou (University of Michigan), Mathematical Finance, Vol

7, No. 4 (October 1997)32. On Pricing of Discrete Barrier Options, S.G.Kou, Columbia University, April 2001 /

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40. Consistent Variance Curves, Hans Buehler, Finance and Stochastics (2006);41. A Double Mean Reverting Stochastic Volatility Model with Concurrent Jumps by Chenxu

Li, Columbia University, October 200942. The Modigliani-Miller Theorem, The New Palgrave Dictionary of Economics, Anne P.

Villamil, University of Illinois.43. Commodities and Commodity Derivatives, Helyette Geman, John Wiley & Sons, 2005.44. http://quantlabs.net/labs/quant-books/cat_view/37-quant-simplified-math;

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Journal of Business, 63:511 – 524, 1990.

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Department of Decision & Information Technologies, University of Maryland

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and Mary-Ann Gatto, The Journal of Finance (1979).

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Group, Bloomberg.

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Chacko, Cambridge University Press (2004).

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http://www.frouah.com/



http://www.globalriskguard.com/







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The Book of Greeks


53. http://www.nas.ewi.tudelft.nl/people/Piet/CUPbookChapters/PACUP_Poisson.pdf

54. An Introduction to the Simulation of Risk Processes, Krzysztof Burnecki, Wolfgang

Härdle and Rafal Weron, Hugo Steinhaus Institute for Stochastic Methods, Institute of

Mathematics, Wroclaw University of Technology.55. Pricing Digital Outperformance Options with Uncertain Correlation, Jacinto Marabel

Romo, BBVA, Ph.D. Dissertation, University of Alcalá (UAH), Spain.

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Lectures, Steven Lalley, Department of Statistics, University of Chicago

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58. http://www.iasonltd.com/FileUpload/files/pdf_res_3.pdf

59. Process Dynamics: Laplace Transforms by M Tham, Department of Chemical and

Process Engineering, University of Newcastle upon Tyne

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State University.61. Option Valuation under Stochastic Volatility, Alan L. Lewis, Finance Press, Newport

Beach, California, USA (2000).

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John Wiley & Sons (2007).

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Economics (2009).

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Edition, David deRosa, John Wiley & Sons (2011)

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Paul Stefiszyn , FEA, September 2002

http://www.nas.ewi.tudelft.nl/people/Piet/CUPbookChapters/PACUP_Poisson.pdf



http://www.scienceisart.com/A_Diffus/DiffusMain_1.html



http://www.iasonltd.com/FileUpload/files/pdf_res_3.pdf








135

The Book of Greeks

About the Author

Rahul Bhattacharya is the CEO of Risk Latte Company Limited. He has taught and conductedtraining sessions for traders, quants, structuring and sales professionals and risk managers in top

tier global banks and investments all over the world, such as Goldman Sachs, HSBC, Citigroup,Barclays Capital, Credit Suisse, RBS, to name only a few. He teaches the CFE course for theCFE School, which is the education and learning division of Risk Latte Company.

He has more than 15 years experience in options and FX trading, commodity structuring and riskadvisory and analytics and has worked all over the world. He holds an B.Sc. (Hons) in Physics,an M.Sc. in Nuclear Physics and an MBA in Finance.

He can be reached at [email protected]

CFE School

CFE School is the Education and Learning division of Risk Latte Company in Hong Kong andconducts the public course Certificate in Financial Engineering (CFE) and other specializedquantitative finance courses for banking and finance professionals all over the world.

For any queries on the CFE course or other programmes of the school, please send your requestto [email protected].

mailto:[email protected]








Book of Greeks

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