IAE University of Toulouse 2010-20111 Computational Methods in Finance Nikos Skantzos.

IAE University of Toulouse 2010-20111 Computational Methods in Finance Nikos Skantzos

IAE University of Toulouse 2010-2011 2 Course Organisation Introduction Organisation inside the dealing room Why do we need numerical methods inside a dealing room? Some reminders Derivative products Mathematics used in finance Introduction to stochastic processes and probability Introduction to VBA programming

IAE University of Toulouse 2010-2011 3 Course Organisation Evaluation of financial assets: Historical background Brownian motion: motivation and examples Black & Scholes model Greeks Other Models Numerical methods Payouts Numerical methods Analytical solutions Monte Carlo Binomial Tree Partial differential equations (PDE) Introduction to interest rate derivative products

IAE University of Toulouse 2010-2011 4 Volatility smile and market models Risk Management Calculation of VAR Introduction to credit risk Real world markets Stylised facts Pairs trading: an example strategy Kellys criterion Course Organisation

IAE University of Toulouse 2010-2011 5 Introduction Pictures from a dealing room

IAE University of Toulouse 2010-2011 6 Introduction A more realistic picture of the dealing room Cartoon by Adam Zyglis

IAE University of Toulouse 2010-2011 7 Introduction The presence and interaction of different units in a dealing room Trader Quant IT Client Sales Structurer Risk ManagementQuant, IT

IAE University of Toulouse 2010-2011 8 Inside the dealing room: Sales Sales In touch with customers They sell options and other products of the bank. Structurers design new products that are attractive to customers. Customers choose them if they offer low risk, high profit and small premium

IAE University of Toulouse 2010-2011 9 Inside the dealing room: Traders Traders Hedge the position that the structurers open. They buy sell/options to minimise the sensitivity of the banks portfolio to movements of the underlying. Prop-traders Take position based on their expectation about the markets next move.

IAE University of Toulouse 2010-2011 10 Introduction: Quants Who: Develop and implement mathematical models to price the products of structurers and calculate the risk for the bank. Where: Investment banks, hedge funds and more generally in any financial institution dealing with derivatives and market risk. Background: Mathematics, Physics, Engineering, Economy.

IAE University of Toulouse 2010-2011 11 How a bank makes money Buying low & sell high Bid-offer spread (buy price: bid, sell price: offer) Banks compete to offer best spread to customer Spread cannot go too high The customer will go to someone else Spread cannot go too low The bank will not have enough money to buy the hedge

IAE University of Toulouse 2010-2011 12 Derivative products: a reminder Main idea behind Options: pay now a small premium to have a choice in the future Example: exchange 1ml EUR for 1,3ml USD in one year What is this option worth today ? Can be used as insurance, for example: If we dont want to risk receiving less than 1,3m USD (We need the money to fund my US company) Can be used for speculation, for example If we believe that the USD will weaken

IAE University of Toulouse 2010-2011 13 Derivative products: a reminder Underlying asset: Any asset sold/bought on a stock market or trading room Example: Stocks Bonds Metals Grains Electricity Interest-rates Indices Currencies Gas Oil "Spot" Transaction: We buy or sell an underlying Example: Microsoft shares, USD Market price is known by supply and demand.

IAE University of Toulouse 2010-2011 14 Derivative products: a reminder Derivative product Its price fluctuates as a function of the value of the underlying. Requires either no or small initial investment Its settlement is made at a future date Derivative market growing rapidly since 1980s Requires numerical and heavy mathematical methods Requires strong computational power & IT infrastructure Need to process market data & produce option premium and risk Now present in the bulk of financial activity Derivative pricing Requires maths and IT

IAE University of Toulouse 2010-2011 15 Derivative products: a reminder What is the fair value of an option? Some intuition: More risk for the issuer, more expensive Longer maturity, more expensive More volatile market, more expensive

IAE University of Toulouse 2010-2011 16 Derivatives: finding the fair price In the horse races there are two horses Horse A, wins 75% of races Horse B, wins 25% of races The booker pays 100 if horse A wins 200 if horse B wins You want to buy the right to choose your horse after the end of the race How much is this option worth ?

IAE University of Toulouse 2010-2011 17 Derivatives: finding the fair price Fair price = average profit Average profit = 100 + 200 = 75 + 50 = 125 Options fair price = 125 A (75%) B (25%) 100 200 Horse race

IAE University of Toulouse 2010-2011 18 Derivatives: finding the fair price in stock options Central idea is similar: Fair price ~ Average payoff Simulate stock many times Record final value Calculate payoff for that path Average over all paths Discounting This average price is valid at maturity To calculate the equivalent price today: N in a bank account today= N e rT after T years Inversely, P at maturity = P e -rT today Option price = Discounted Average Payoff Average taken over probabilities that eliminate all risk: Risk-neutral measure

IAE University of Toulouse 2010-2011 19 Derivative products: a reminder History 6 th century BC: Greek philosopher Thales of Miletus used options to secure a low price of olives in advance of harvest. Middle Ages: futures contracts to fix in advance the price of imports of goods from Asia Holland 1637: The "Tulip Mania" one of the first speculative bubbles.

IAE University of Toulouse 2010-2011 20 Derivative products: a reminder Two most simple and popular: Call = right to buy at an agreed future date a certain amount of the underlying asset at a price fixed today. Put = right to sell at an agreed future date a certain amount of the underlying asset at a price fixed today. Terminology Agreed future date = Maturity of the option Amount of underlying = Notional Price fixed today = Strike

IAE University of Toulouse 2010-2011 21 Derivative products: a reminder The payout of an option what the option would bring to its owner at maturity (T), depends on price of the underlying at that time (S T ). Long Call payout = max(0, S T - K) Go Long a Call if you think the underlying will increase K STST Call Long ( the case of a buyer of a call) Short (the case of a seller of a call) payout = S T -K

IAE University of Toulouse 2010-2011 22 Derivative products: a reminder Long Put payout = max(0, K- S T ) Go long a Put if you think the underlying will go lower Calls and Puts are called vanillas Vanilla flavour = simple. K STST Put Long ( the case for an owener of a Put) Short (the case for a seller of a Put) payout = K- S T

IAE University of Toulouse 2010-2011 23 Derivative products: a reminder Barrier options Advantage: Cheaper than vanilla options Disadvantage: More risky K STST At maturity (T) Regular barrier Reverse barrier Knock-In = the option is activated if the spot hits the barrier Knock-Out = the option is disactivated if the spot hits the barrier

IAE University of Toulouse 2010-2011 24 Derivative products: a reminder Price of an option KSTST Call payout = S T -K At maturity (T) Today (t

IAE University of Toulouse 2010-2011 28 Some derivative strategies Call spread(K 1, K 2 ) = Call(K 1 )- Call(K 2 ) = Cheaper than a simple call Profit is limited to K 2 -K 1 for spots>K 2 +Call(K 1 ) -Call(K 2 ) K1K1 K2K2 K1K1 K2K2

IAE University of Toulouse 2010-2011 29 Some derivative strategies Straddle(K) = Call(K) + Put(K) Expensive If S T >K: gives the right to buy cheap If S T

IAE University of Toulouse 2010-2011 33 Mathematical reminder LN(e)=1 e ln(x) = x, or ln(e x ) = x Logarithm in base e Defined only for x>0 The function LN (Neperian logarithm):

IAE University of Toulouse 2010-2011 34 Mathematical reminder The derivative of a function: slope of a function at 1 point Numerical approximation: or The 2 nd derivative : curvature of a function in 1 point Numerical approximation:

IAE University of Toulouse 2010-2011 35 Some analytical derivatives

IAE University of Toulouse 2010-2011 36 Mathematical reminder Integral of a function

IAE University of Toulouse 2010-2011 37 Mathematical reminder Primitives of some commonly used functions

IAE University of Toulouse 2010-2011 38 Mathematical reminder Numerical integration of a function Method of lower rectangles Method of upper rectangles Trapezoidal method

IAE University of Toulouse 2010-2011 39 Mathematical reminder Taylor series: approximating a function around a point x 0 Converts a complex function into a simple power-series Examples exp(x) around x 0 =0: cos(x) around x 0 =0: around x 0 =0:

IAE University of Toulouse 2010-201140 Random variables and stochastic processes Basic notions

IAE University of Toulouse 2010-2011 41 Random variables and stochastic processes Random variable a number whose value is determined by the outcome of an experiment We dont know its value only how likely it is Discrete random variable: Can take on only certain separated values Example: the result of throwing a dice. The probability of every outcome is 1/6 Continuous random variable: Can take on any real value from a range Example: the price of an stock. The probability that the price is within a certain interval depends on the distribution of the random variable. Stochastic process represents the evolution in time of a random variable

IAE University of Toulouse 2010-2011 42 Properties of random variables Probability of an event: 0Prob(event) 1 Prob=0: certainty that event will not happen Prob=1: certainty that event will happen Probability of all events: Prob(ev 1 )+ +Prob(ev N ) =1 Prob(ev 1 OR ev 2 ) = Prob(ev 1 ) + Prob(ev 2 ) Example: probability that a dice is either 1 or 2 = 1/6 + 1/6 If ev 1 is independent of ev 2 then: Prob(ev 1 AND ev 2 ) = Prob(ev 1 ) Prob(ev 2 ) Example: Prob that two dice are both 1 = 1/6 1/6

IAE University of Toulouse 2010-2011 43 Random variables Characterised by: The probability density distribution function f(x) Prob that event x will happen The cumulative distribution function Prob that the outcome of the experiment will be less than x The mathematical expectation (mean) The average by repeating the experiment many times The moments (order n) : First moment is the mean Second moment is related to the variance Third moment is related to the skewness...

IAE University of Toulouse 2010-2011 44 a b Interpretation of distribution function The surface under the curve between a and b is the probability that the value of the random variable is between a and b :

IAE University of Toulouse 2010-2011 45 Central moments The central moments (of order n): remove the mean The variance (n=2), characterises the amplituded around the mean: Standard Deviation = variance,

IAE University of Toulouse 2010-2011 46 Skewness (n=3), describes the asymmetry: Kurtosis (n=4), describes the effects of fat tails: 3 : distribution leptokurtic Central moments

IAE University of Toulouse 2010-2011 47 Skewness & kurtosis Asymmetry: skewness Fat tails: kurtosis

IAE University of Toulouse 2010-2011 48 Meaning of fat tails Represents a high probability of extreme events. Catastrophic market crashes (1927, 1987) Money lost is more than of all money lost in the next 20 years Catastrophic earthquakes (Chile 1960 9.5R, Sumatra 2004 9.1R) Energy released is more than of total energy released by crust Such events are characterised by Very low probability Very high impact

IAE University of Toulouse 2010-2011 49 Examples of fat tails Fat tails means that the extreme-event probability is low, but much higher than we expect !

IAE University of Toulouse 2010-2011 50 Variance of a distribution Small variance = large certainty All distributions look the same when variance 0 Graph opposite: Lognormal vs Normal variance=0.01 Which is which ?

IAE University of Toulouse 2010-2011 51 Distribution vs cumulative Some important properties Definition or and Distribution function is normalized: Cumulative is between 0 and 1, always increasing f(x)F(x)

IAE University of Toulouse 2010-2011 52 Some important properties Integral of the distribution: probability that the random variable will be less than a certain value Probability that the random variable is between two values:

IAE University of Toulouse 2010-2011 53 Sampling from a distribution This is an important application of cumulative functions Problem: generate random variables from specific distribution Matlab, Excel, provide the uniform random number generator This selects uniformly a number between 0 and 1 We use the inverse cumulative function of the distribution Pseudo code Draw a uniform random number in [0,1] Pass it through the InvCum of the required distribution Result is a number sampled from the required distribution

IAE University of Toulouse 2010-2011 54 Use of distributions in finance Financial derivatives require us to calculate the expectation of a function of a random variable Example: a Call option where (S T ) is the distribution function of the final spot

IAE University of Toulouse 2010-2011 55 Normal Distribution Normal Distribution N(, ) Special case: = 0 and = 1 denoted N(0,1) = mean = standard deviation

IAE University of Toulouse 2010-2011 56 Normal Distribution Exercise : What are (i) the mean and (ii) the standard deviation of the index EUROSTOXX50, if we suppose that it follows a law a+bX where X follows a centered normal distribution (a and b are 2 constants) ? Calculate the mathematical expectation of e X where X follows a centered normal distribution Calculate the expectation of S=e (r-q- /2)T+x T where X follows a centered normal distribution

IAE University of Toulouse 2010-2011 57 Log-normal Distribution Very important in finance Increments in stock prices are modeled as lognormal If X follows a normal law X~N(, ), Then Y=e X is distributed log-normally. Relations between the function of X and Y, related by X = f(Y): Exercise: recover the Log- Normal distribution law

IAE University of Toulouse 2010-2011 58 Log-normal Distribution Starting from a normal distribution for X We find the log-normal law for Y=e X Exercise: Calculate the mean and variance of a log-normal function with parameters ,

IAE University of Toulouse 2010-2011 59 Central Limit Theorem This theorem is the reason why normal distributions are present so often! The sum of N independent, identically distributed random numbers is normally distributed The N numbers do not have to be normally distributed! N numbers, x 1,, x N each with mean m, variance s The random variable x 1 + x 2 + x N follows

IAE University of Toulouse 2010-2011 60 Central Limit Theorem at work For N = 5, 20, 100 Sample N random variables from some distribution (here lognormal) and sum them: x 1 ++ x N For each N, repeat many times and plot histogram Observations: For small N, only central region looks normally distributed ! For large N, the sum resembles the normal distribution very well

IAE University of Toulouse 2010-2011 61 Sum of lognormal variables Because of the Central Limit Theorem A sum of normal variables is normal A sum of lognormal variables is not lognormal In finance however we often approximate a sum of lognormal variables by a lognormal This approximation is not bad provided the number of summed variables is small.

IAE University of Toulouse 2010-2011 62 Commutation of integration & differentiation The order of integration and differentiation can be interchanged Example: the derivative of a call with respect to strike since the expectation is simply an integral

IAE University of Toulouse 2010-2011 63 Commutation of integration & differentiation We can use this trick to compute moments of a distribution Example, 2 nd moment of a central normal distribution:

IAE University of Toulouse 2010-2011 64 Commutation of expectation in a function Which is bigger? Denote and Taylor expand f(x) around x 0 Apply the expectation If then

IAE University of Toulouse 2010-2011 65 Relation between mean and variance Variance in terms of simple expectations Var[x] = E[x 2 ]-E 2 [x] Derivation:

IAE University of Toulouse 2010-201166 Basic notions of VBA Excel

IAE University of Toulouse 2010-2011 67 Basic notions of VBA Excel Enter the VBA environment : Alt+F11

IAE University of Toulouse 2010-2011 68 Basic notions of VBA Excel Header Option Explicit Option Base 1 Create a VBA function Function GetDelta(ByVal a As Integer, ByVal b As Integer, ByVal c As Integer) Dim delta As Long delta = b * b - 4 * a * c GetDelta = delta End Function Declare a variable Dim nom_variable As type_variable (double, long, string, Range)

IAE University of Toulouse 2010-2011 69 Basic notions of VBA Excel Create a VBA macro Sub SommeDeuxValeurs() 'declaration Dim nb1 As Integer Dim nb2 As Integer Dim somme As Long 'Lecture nb1 = InputBox("nbre 1") nb2 = InputBox("nbre 2") 'Traitement somme = nb1 + nb2 'Affichage MsgBox "La somme est " & somme End Sub

IAE University of Toulouse 2010-2011 70 Basic notions of VBA Excel Loops For... To... Next Function GetFactoriel(ByVal a As Integer) Dim fact As Long Dim i As Integer fact = 1 For i = 1 To a fact = fact * i Next i GetFactoriel = fact End Function

IAE University of Toulouse 2010-2011 71 Basic notions of VBA Excel Tests If... Then... Else Function EstPositif(ByVal a As Double) If a > 0 Then EstPositif = 1 ElseIf a < 0 Then EstPositif = -1 Else EstPositif = 0 End If End Function

IAE University of Toulouse 2010-2011 72 Basic notions of VBA Excel Some useful functions Tracer lhistogramme dune distribution: Utiliser la fonction frequence dans Excel In Excel ALEA() LOI.NORMALE.STANDARD( x ) LOI.NORMALE.INVERSE(x ;0;1) In VBA Excel Rnd NormaleCumul(x) (faite maison) Application.WorksheetFunction.No rmSInv( x )

IAE University of Toulouse 2010-201173 Numerical methods in finance: some background history

IAE University of Toulouse 2010-2011 74 Brownian Motion Robert Bown (botanist) Observed motion of pollen particles suspended in water (1827).

IAE University of Toulouse 2010-2011 75 Stochastic methods in finance Louis Bachelier (1870 1946) Considered as the founding father of financial mathematics. Was the first to have applied mathematical models to the analysis of financial markets Stock prices evolve according to Brownian motion

IAE University of Toulouse 2010-2011 76 Models for Brownian Motion Thorvald N. Thiele (1880), was the first to propose a mathematical theory to explain Brownian motion Danish astronomer Founder of an insurance company Louis Bachelier (1900) used Brownian motion in his thesis La thorie de la spculation to describe stock prices Albert Einstein (1905) makes a statistical theory that explains Brownian motion and allows predictions

IAE University of Toulouse 2010-2011 77 Why Brownian motion in finance? Paths resemble stock market indices Problem: Brownian motion can turn negative !

IAE University of Toulouse 2010-2011 78 How to model Brownian motion? Brownian motion is stochastic process (=sequence of r.v.) W(0), W(1), W(2),... Main properties: W(0) = 0 The increments W(2)-W(1), W(3)-W(2),... are independent of each other The increments W(t)-W(s) are normally distributed N(0,(t-s) ) This is also called Wiener process Standard Brownian motion

IAE University of Toulouse 2010-2011 79 Brownian motion: an example Bob finishes his job at 5pm and before going home he makes a stop at the bar There he drinks a bit more than he should He leaves the bar at 8pm and usually (after some zig-zags) arrives home at midnight His home is just 500m away This means he proceeds towards home with an average speed of 0.5/4 = 0.125 km/hr His friends observed that at 10pm he is on average 100m away from the straight line connecting the bar to his house

IAE University of Toulouse 2010-2011 80 Brownian motion: an example Notation: X t position at time t T=24hr t 0 =21hr X t0 =X 0 =0 Random-walk model: Position at next step X t+1 given position at previous step X t Randomness comes through the increment W t ~N(0,t) What is the meaning of and ? Bob takes first step: in this model is average speed = 0.125 km/hr Small : random walk is confined Large : random walk can make big jumps

IAE University of Toulouse 2010-2011 81 Brownian motion: an example After several steps Bob arrives home The model describes his random walk as In the limit t0: We are facing a problem: What is the meaning of an integral over a stochastic differential ? Stochastic calculus

IAE University of Toulouse 2010-2011 82 Kiyoshi It (1940s) develops stochastic calculus It integral : with stochastic differential dW Its lemma: differentiation of stochastic functions Robert Merton (1969) introduces stochastic calculus in finance to explain the price of financial products S ~ e W(t) >0 : The value of an underlying stays always positive! Stochastic calculus in mathematical finance

IAE University of Toulouse 2010-2011 83 Robert Merton, Fisher Black & Myron Scholes published the famous work on option pricing (1973) The model allows to derive analytic expression for the fair price of call and put options A significant contribution to the growth of derivatives Merton and Scholes receive the Nobel price of economics 1997 (F. Black had died in 1995) Option pricing with stochastic calculus

IAE University of Toulouse 2010-2011 84 Stochastic integral Definition: A useful property: The mean of a stochastic integral is zero Derivation Independents increments Mean of N(0,1)=0

IAE University of Toulouse 2010-201185 The Black & Scholes model

IAE University of Toulouse 2010-2011 86 The Black-Scholes model Cartoon by S Harris

IAE University of Toulouse 2010-2011 87 The Black & Scholes model Simple brownian motion dS = dW Black & Scholes model dS = S dt + S dW S : value of underlying stock, foreign exchange rate, etc : drift the price of risk-free interest rate annualised dividend: r-q (Equity) Domestic minus foreign interest risk-free rates: r dom -r for (Forex) : volatility (annualised) t : time (expressed in years) W: Wiener process (Brownian)

IAE University of Toulouse 2010-2011 88 The Black & Scholes model dt Differential equation of Black & Scholes Random variable, distributed according to a normal distribution of 0 mean & variance t Solution of the differential equation of Black & Scholes It calculus

IAE University of Toulouse 2010-2011 89 The three forms of the B&S model Stochastic differential equation Solution of the stochastic differential equation Partial differential equation governing the evolution of the price of a derivative (pricing equation)

IAE University of Toulouse 2010-2011 90 Its Lemma Its process: x solution of dx=a(x,t) dt + b(x,t) dW Consider a function G(x,t): dx = [a(x,t) dt + b(x,t) dW] 2 = ?? Some properties in differential stochastic calculus: dt. dt = 0 dW. dt = 0 dW. dW=dt Additional term from stochastic calculus

IAE University of Toulouse 2010-2011 91 Its Lemma Exercise: Black-Scholes What is the differential of ln(S) ? What is the value of S(T) ?

IAE University of Toulouse 2010-2011 92 Derivation of the Black-Scholes PDE Composition of portfolio: 1 option of value V(S,t) An amount of the underlying We adjust the amount such that the portfolio is not sensitive to risk (such as small random movements of the underlying) Putting it together, the portfolio P consists of: P = V + S The variation of the portfolio after an very small amount of time is dP = dV + dS With dS = (r q) S dt + S dw (differential equation of B&S) Classic differential calculus Additional term in stochastic differential calculus

IAE University of Toulouse 2010-2011 93 Derivation of the Black-Scholes PDE Some useful rules of the stochastic differential calculus dt dt = 0 dW dt = 0 dW dW=dt (dS) = ? dS dS = [ S dt + S dw] [ S dt + S dw] = S dt We arrive at the variation of our portfolio P:

IAE University of Toulouse 2010-2011 94 Derivation of the Black-Scholes PDE We suppress all sources of risk (risk=randomness) of the underlying (dS): delta of an option We arrive at the variation of the portfolio P The remaining portfolio contains more sources of risk: it must evolve as money placed into a "safe" savings account with interest rate r PDE of Black-Scholes

IAE University of Toulouse 2010-2011 95 Call and Put options Solution of the Black & Scholes model

IAE University of Toulouse 2010-2011 96 Derivation of the Call price for the Black-Scholes model At maturity, the call value is g(S T ) = max(0,S T -K) (S T -K) + Call price: expectation of the payoff, discounted to the value of today S (S T ): Distribution function of the random variable S T The assumed process for the random variable S T has solution where X a normal random variable (mean 0, variance 1) S T : spot K: strike e -rT : Discount factor

IAE University of Toulouse 2010-2011 97 AB Derivation of Black-Scholes call price

IAE University of Toulouse 2010-2011 98 The easy part: The more difficult part: We would like to bring this to an integral of the form Complete the square Most common way to do this is:

IAE University of Toulouse 2010-2011 99 Finaly the value of the Call: Equivalently, in the standard notation: Exercise: calculate the price of a digital option (it pays at maturity 1 unit of underlying if S T >K)

IAE University of Toulouse 2010-2011 100 Interpretation of the Black-Scholes formula N(d 2 ): probability that spot finishes in the money N(d 1 ): measures how far in the money the spot is expected to be if it finishes in the money Call price: value of receiving the stock in the event of exercise minus cost of paying the strike price

IAE University of Toulouse 2010-2011 101 Black-Scholes and risk-neutrality The Black-Scholes formula depends on the Spot, Volatility, Interest-rates and time. None of these parameters involves the risk-preference of the investor. Therefore, the B&S formula does not depend on any assumption about the risk-preferences of the investors

IAE University of Toulouse 2010-2011 102 Assumptions of the B&S model More Important Underlying evolves according to a lognormal process Volatility ( size of fluctuations) is constant and known No arbitrage opportunities exist Less important No dividends No transaction costs Risk-free rates are constant

IAE University of Toulouse 2010-2011 103 How realistic are the assumptions of the B&S model ? In real markets the size of the fluctuations is not constant The underlying can make big jumps on some economic news Calculating the volatility is not trivial The process of the underlying is typically not lognormal Interest rates are not constant All assumptions are wrong in reality ! They are made only to simplify the calculations

IAE University of Toulouse 2010-2011 104 Call-Put parity relation Call-Put = = Se -qT -Ke -rT =(F-K)e -rT The price of a call is linked to the price of a put through the forward

IAE University of Toulouse 2010-2011 105 The Black & Scholes model Solution of the Black-Scholes model for the price of a call/put with barrier Barrier in : the option is activated only if the barrier is touched Barrier out : the option is dead if the barrier is touched

IAE University of Toulouse 2010-2011 106 The Black & Scholes model Solution of the Black-Scholes model for the price of a call/put with barrier Barrier up : the barrier must be touched while the spot rises Barrier down : the barrier must be touched while the spot declines Call / Put, in / out, up / down 8 possible combinations

IAE University of Toulouse 2010-2011 107 The Black & Scholes model Parity relations: c = c ui + c uo c = c di + c do p = p ui + p uo p = p di + p do

IAE University of Toulouse 2010-2011 108 The Black & Scholes model Price of barrier options

IAE University of Toulouse 2010-2011 109 The Black & Scholes model Price of touch options One-Touch Up with S o H

IAE University of Toulouse 2010-2011 110 Important identities in the B&S model (1) and Derivation:

IAE University of Toulouse 2010-2011 111 Important identities in the B&S model (2) and and similarly and Derivation

IAE University of Toulouse 2010-2011 112 Important identities in the B&S model (3) where and Derivation We will show that Start from right-hand side

IAE University of Toulouse 2010-2011 113 The Greek Letters Delta : Gamma : Vega : Theta : The most important quantity for the daily management of the trading books

IAE University of Toulouse 2010-2011 114 The Greek Letters They represent sensitivities of the portfolio with respect to market parameters They allow us to monitor the risk of the portfolio They can be applied to a single derivative or to a portfolio of derivatives

IAE University of Toulouse 2010-2011 115 Greeks Analytic expressions for the Greeks ( here for a Call) : N(x) = (x) probability density of a normal random variable

IAE University of Toulouse 2010-2011 116 Demonstration: Delta and Derivation: Now use the fact that and And also the identity we proved: to eliminate the two right-most terms and obtain the result

IAE University of Toulouse 2010-2011 117 Example A bank has sold European call option for $300,000 on 100,000 shares of a non-dividend paying stock Market parameters are S 0 = 49 = 20%, K = 50 T = 20 weeks r = 5% The Black-Scholes value of the option is $240,000 How does the bank hedge its risk to lock in a $60,000 profit?

IAE University of Toulouse 2010-2011 118 Naked & Covered Positions Naked position Take no action Covered position Buy 100,000 shares today Both strategies leave the bank exposed to significant risk

IAE University of Toulouse 2010-2011 119 Delta Delta ( ) is the rate of change of the option price with respect to the underlying Delta small option price does not move when spot moves Delta large option price moves when spot moves Option price A B Slope = Stock price

IAE University of Toulouse 2010-2011 120 Delta: an important interpretation Remember: What does N(d 1 ) mean? To answer this: calculate probability that spot finishes in the money: Example: A call with delta=50% has roughly probability=50% that its stock price will exceed the strike at maturity. where

IAE University of Toulouse 2010-2011 121 Delta Hedging This involves maintaining a delta neutral portfolio Delta neutral: This means that if the spot makes a small change the value of the portfolio does not change Eliminates spot risk Delta hedging is done by buying/selling the underlying (e.g. cash or stocks) Black-Scholes theory shows that a Delta-neutral portfolio is possible what is the correct amount of the underlying to short

IAE University of Toulouse 2010-2011 122 Delta: an example Call option with: Premium 400 Delta 50% Spot today is at S 0 =100 This means that If spot moves to S 0 =110 (10% move) The premium will move to 420 (10%50% move) (with all other market parameters unchanged)

IAE University of Toulouse 2010-2011 123 Theta Theta ( ) is the change in value of the derivative with respect to the passage of time The theta of a call or put is usually negative. meaning: as time passes the value of the option decreases Practically, change in time is 1 day.

IAE University of Toulouse 2010-2011 124 Theta: an example Call option which today is worth: Premium 20 Theta -0.5 This means that tomorrow the premium goes to 19.5 (with all other market parameters unchanged)

IAE University of Toulouse 2010-2011 125 Gamma Gamma ( ) is the rate of change of delta ( ) with respect to the price of the underlying asset Gamma is small Delta is stable under spot movements Gamma is large Delta is not stable under spot movements Gamma neutral hedge: portfolio and Delta are stable under spot movements. better hedge than simple Delta-neutral (but more expensive!) Gamma is the second derivative of the derivative value with respect to the underlying price

IAE University of Toulouse 2010-2011 126 Interpretation of Gamma Gamma Addresses Delta Hedging Errors Caused By Curvature S C Stock price S'S' Call price C '' C'C'

IAE University of Toulouse 2010-2011 127 Relationship Between Delta, Gamma, and Theta For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q

IAE University of Toulouse 2010-2011 128 Vega Vega ( ) represents the change in value of a derivative with if market volatility moves by 1% Vega tends to be greatest for options that are close to the at-the-money Risk that volatility can move the spot out of the money

IAE University of Toulouse 2010-2011 129 Vega: an example Call option with Premium 20 Vega 0.5 Market Vol 20% This means that If market Vol goes to 21% Premium goes to 20.5

IAE University of Toulouse 2010-2011 130 Managing Delta, Gamma, & Vega can be changed by taking a position in the underlying To adjust & it is necessary to take a position in an option or other derivative

IAE University of Toulouse 2010-2011 131 Call option, strike 1.25 PriceDelta Gamma Vega Option price becomes linear for large spots Delta ~ cumulative function Convexity risk (Gamma) highest at-the-money Vol risk (vega) is highest at-the-money Spotladders: vanilla

IAE University of Toulouse 2010-2011 132 Spotladders: barrier option Knock-out option, strike 1.25, barrier 1.35 PriceDelta Gamma Vega Option price: 0 at barrier and out-of-the-money Delta, Gamma, Vega can be negative unlike vanilla!

IAE University of Toulouse 2010-2011 133 Rho Rho is the rate of change of the value of a derivative with respect to the interest rate

IAE University of Toulouse 2010-2011 134 A word on the absence of arbitrage Absence of Arbitrage (AOA) Normally there can be no profit without taking a risk. However, if an opportunity for riskless profit arises, the market reacts immediately, and soon the opportunity disappears. It is the basis of the Black-Scholes model ...and of most other derivative models. This condition allows us to determine the expectation of the underlying using risk neutrality An example

IAE University of Toulouse 2010-2011 135 AOA: example on EURUSD EURUSD = 1.3 = S o (1 EUR equals 1.3 USD) 1 EUR = underlying, USD payment currency I start with no money I borrow 1 EUR from a European bank, with 1 year maturity, interest rate q. In one year I must pay back e qT (=1 + q T + ) I convert today my EUR to USD, I receive S o USD I enter into a Forward contract (for free), allowing me to change USD into EUR within a year, at a fixed rate F o. I deposit S o USD into an american bank with interest rate r. After 1 year I receive: S o e rT After 1 year, I will have gained (without taking any risk): - e qT (money to pay back in european bank) + S o e rT / F o (money I receive from american bank in EUR) AOA implies that the forward contract has value F o = S o e (r-q)T

IAE University of Toulouse 2010-2011136 Volatility smile A practinioners introduction

IAE University of Toulouse 2010-2011 137 Black-Scholes vs market BS-price < market-price, for very low / very high strikes Plug market-price in BS formula to calculate volatility Inverse calculation implied vol Do it for all strikes Black-Scholes assumes that volatility is constant for all strikes! Here we observe a parabolic-shape looking like a

IAE University of Toulouse 2010-2011 138 Spot probability density (market) Distribution of terminal spot (given initial spot) obtained from Fat tails: Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes Main causes: Spot dynamics is not lognormal Spot fluctuations (vol) are not constant Market observable

IAE University of Toulouse 2010-2011 139 Historical data contradict Black-Scholes assumptions: Extreme events appear more often than predicted by the lognormal distribution The volatility we observe is not constant Jumps are observed in the evolution of prices Black-Scholes is based on the idea of risk neutral In reality the market is not risk neutral. For stocks, it is risk averse , it is ready to pay a significant amount for a protection against a crash. Black-Scholes volatility smile

IAE University of Toulouse 2010-2011 140 Black-Scholes volatility smile equities change Despite this, the Black-Scholes model is the standard To reflect the actual distribution of underlyings, we must adapt the model the volatility is based on the strike of the option

IAE University of Toulouse 2010-2011 141 Reflect real-world distributions: 1) Use a "naive" model (BS, vol assumed constant) in which the volatility is adapted according to the strike of the option price 2) use more sophisticated models capable of reproducing the realistic distributions Black-Scholes volatility smile

IAE University of Toulouse 2010-2011 142 Implied volatility Traders often quote vols instead of prices This means: vol price Implied vol: the vol we must put into the BS pricer to obtain the option price It is not equivalent to historical vol: measure of historical fluctuations It does not give information about the dynamics BS pricer

IAE University of Toulouse 2010-2011 143 Historical vs implied volatility Historical Volatility: Represents the size of fluctuations in the process S Implied Volatility: Represents the price of a vanilla option today

IAE University of Toulouse 2010-2011 144 Measuring historical volatility EURUSD 6-month data, closing of day Historical vol = 5.2% Implied vol in Apr2010 = 17% Measuring historical vol is not easy Which data set do we take? min, hourly, daily intervals? How do we account for low/high? Black-Scholes assumption on vol is wrong: Apr-Jun: high volatility Oct-Nov: low volatility

IAE University of Toulouse 2010-2011 145 Some more sophisticated models One way to correct the erroneous BS assumption is to consider that vol is not constant Calculations are not as elegant and simple anymore Two mainstream models Local Volatility model Volatility depends on the time and spot This model can reproduce the smile Stochastic Volatility model Spot: Geometric Brownian motion Vol: modeled as a stochastic variable that returns to a long-term mean value

IAE University of Toulouse 2010-2011 146 Local-vol vs Stochastic-vol Dupire and Heston can reproduce the vanilla-smile perfectly But can differ dramatically when pricing exotics! Rule of thumb: skewed smiles: use Local Vol convex smiles: use Heston

IAE University of Toulouse 2010-2011147 Numerical methods

IAE University of Toulouse 2010-2011 148 Models, numerical methods and payouts Payout describes the derivative product, the rights and obligations of the owner and of the issuer (no maths!). Model Assumptions concerning the evolution of the underlying in the market Numerical method The way of calculating the price of the payout, depending on the chosen model

IAE University of Toulouse 2010-2011 149 Models, numerical methods and payouts Models : Black-Scholes Stochastic Vol Local Vol Jump Diffusion Numerical methods: analytic solution Static replication Binomial tree Monte Carlo Finite differences Payout : Call, Put, barriers, european, american Callable, touch A model associated with a numerical method allows us to give the price of a payout (derivative product)

IAE University of Toulouse 2010-2011 150 Numerical methods Analytic solution: Very fast Exact result Very easy to implement Exists only for a few payouts, with some models Monte Carlo Relatively easy to implement Can be applied practically on all payouts, with all models Can be applied on payouts with several underlyings Easy to parallelize computations Slow More difficult to implement on options with American exercise Calculation of greeks is not easy

IAE University of Toulouse 2010-2011 151 Numerical methods Binomial Tree (or trinomial): Relatively easy to implement Exists for many payouts (barriers), with only some models Partial differential equation (PDE) grid Can be applied on many payouts, with most models limited to 2-3 underlyings Very stable for the calculation of the greeks Fast Difficult to parallelise computations Relatively difficult to implement

IAE University of Toulouse 2010-2011 152 Binomial Trees Binomial trees are frequently used to approximate the movements of an underlying In each small interval of time the stock price can move up by a proportional amount u move down by a proportional amount d

IAE University of Toulouse 2010-2011 153 Binomial Trees We discretise time in small steps At each time step the underlying can only have two possibilities : Increase by a factor u (>1) Decrease by a factor d (

IAE University of Toulouse 2010-2011 175 Example American Call Cells in red: Immediate exercise more interesting than keeping the option Can occur for a call if r1 (q) >0 Can occur for a put if r2 (r) >0

IAE University of Toulouse 2010-2011 176 Demo binomial tree (american)

IAE University of Toulouse 2010-2011 177 Pricing of a KO put with binomial tree KO Barrier level = 1.5

IAE University of Toulouse 2010-2011 178 Demo binomial tree (Barrier)

IAE University of Toulouse 2010-2011179 Monte Carlo Method

IAE University of Toulouse 2010-2011 180 Monte Carlo method Cartoon by S Harris

IAE University of Toulouse 2010-2011 181 Monte Carlo In most cases analytic formula is too hard to find An practical alternative is pricing via simulations We simulate the evolution of the underlying a large number of times (~10000). For every simulation we calculate the expected gain for the owner of the option Option price = (average of gains) x (disc-fact) e -rT

IAE University of Toulouse 2010-2011 182 Monte Carlo Each simulation describes a randomly chosen path of the underlying The name Monte Carlo comes from the resemblance to casino games

IAE University of Toulouse 2010-2011 183 Monte Carlo method It is a method for finding the average of a function g of a random variable X: We are interested in calculating integrals of the form: where f(x) is the probability density of x in the interval [a,b] Example where (S T ) is the spot terminal density in the interval [0,] call(S T ) = max(S T -K,0)

IAE University of Toulouse 2010-2011 184 Monte Carlo method Obtain estimator of G by producing large number of realisations of x: (x 1,x 2 ,x N ). Estimator Theoretical mean The larger the N, the more accurate the estimator

IAE University of Toulouse 2010-2011 185 Monte Carlo method: an example Calculate the mean of N lognormal variables Sample N lognormal variables Sum them up Repeat for various values of N Small N: fluctuations Large N: convergence to mean How to sample at random a lognormally-distributed variable in Excel: X = RAND() Y = LOGINV(X,mean,std) where mean=mean of Lognormal distrib. where std=standard dev of Lognormal distrib.

IAE University of Toulouse 2010-2011 186 Monte Carlo Simulation and Calculate by randomly sampling points in the square? Exercice

IAE University of Toulouse 2010-2011 187 Monte Carlo Simulation and Options When used to value European stock options, Monte Carlo simulation involves the following steps: 1.Simulate one path for the stock price in a risk neutral world 2.Calculate the payoff from the stock option 3.Repeat steps 1 and 2 many times to get many sample payoffs 4.Calculate mean payoff 5.Discount mean payoff at risk free rate to get an estimate of the value of the option

IAE University of Toulouse 2010-2011 188 Sampling Stock Price Movements In a risk neutral world the process for a stock price is We can simulate a path by choosing time steps of length t and using the discrete version of this where is a random sample from (0,1) =LOI.NORMALE.INVERSE(ALEA();0;1)

IAE University of Toulouse 2010-2011 189 An alternative approach More accurate in most cases The options with a european payout require only one time step =LOI.NORMALE.INVERSE(ALEA();0;1) Often instead of using the BS stochastic differential equation, we use its solution:

IAE University of Toulouse 2010-2011 190 Extensions to several underlyings When a derivative depends on several underlying variables we can simulate paths for each of them in a risk-neutral world to calculate the values for the derivative

IAE University of Toulouse 2010-2011 191 Sampling from Normal Distribution The simplest way to sample from (0,1) : Generate 12 random numbers between 0.0 & 1.0 use the Excel function alea() (=random()) Sum them up and subtract 6.0 Exercise: calculate the mean and the variance of V=U 1 + U 2 +U 12 - 6 In Excel =LOI.NORMALE.INVERSE(ALEA();0;1) gives a random sample from (0,1)

IAE University of Toulouse 2010-2011 192 Example: pricing a call option for i=1N Generate standard normal variable U i Set S i (T) = S(0) exp[ (r- 2 )T+ T U i ] Set Call i = e -rT max(S i (T)-K,0) Call = (Call 1 ++ Call N )/N Exercise: show that this converges to the result given by the Black-Scholes formula

IAE University of Toulouse 2010-2011 193 Confidence interval Calculate the standard deviation of the Monte Carlo result For a 95% confidence interval find z /2 =N inv (1-/2) with =5% N inv is the inverse cumulative normal function 95% confidence interval means =5% and z /2 =1.96 The confidence interval is within the values Average - z /2 SD/n Average + z /2 SD/n

IAE University of Toulouse 2010-2011 194 Obtain two correlated Normal Samples Obtain independent normal samples x 1 and x 2 and set A procedure known as Choleskys decomposition =[-11] measures correlation: =1 then 1 = 2 : perfect correlation =0 then 1 = x 1 and 2 =x 1 : no correlation =-1 then 1 =- 2 : perfect anti-correlation Used when samples are required from two (or more) normal variables Exercise: show that the correlation between 1 and 2 is

IAE University of Toulouse 2010-2011 195 Application of Monte Carlo Simulation Monte Carlo simulation can deal with path dependent options (e.g. Asians, barriers,) options dependent on several underlying state variables (e.g. Forex & interest rates) options with complex payoffs It cannot easily deal with American-style options

IAE University of Toulouse 2010-2011 196 Example: pricing an Asian call option An Asian option averages the payoff spot over several intermediate dates T 1,,T N This is a path-dependent option for i=1 nbrPaths for j=1 N Generate standard normal variable U i,j Set S i (T j ) = S(T j-1 ) exp[(r- 2 )(T j -T j-1 )+ (T j -T j-1 ) U i,j ] Set meanSpot i =(S i (T 1 )++S i (T N ))/N Set Call i = e -rT max(meanSpot i -K,0) Call = (Call 1 ++ Call N )/N

IAE University of Toulouse 2010-2011 197 Monte Carlo and barrier options If the barrier monitored continuously, it requires a simulation with many points: What happens between t i and t i+1 is unknown. Was the barrier touched ? Put more points (CPU time increases!), or Smarter : Compute the pobability of touching the barrier between t i and t i+1

IAE University of Toulouse 2010-2011 198 Monte Carlo and barrier options Estimating probability of not touching barrier: Total survival probability: Knock-out option = DF Payoff(S) P surv

IAE University of Toulouse 2010-2011 199 Monte Carlo and barrier options For knock-in options we use the decomposition KI = Vanilla KO and we price the two right-hand side instruments based again on the survival probability formula

IAE University of Toulouse 2010-2011 200 Determining Greek Letters For Make a small change to asset price Carry out the simulation again using the same random numbers Estimate as the change in the option price divided by the change in the asset price Proceed in a similar manner for other Greek letters

IAE University of Toulouse 2010-2011 201 Demonstration XL

IAE University of Toulouse 2010-2011 202 Finite Difference Methods Finite difference methods represent the differential equation as a difference equation Practically speaking, we transform into and we solve for P(t): the price at the previous time step is the risk-neutral drift

IAE University of Toulouse 2010-2011 203 Finite Difference Methods: the main idea We form a grid with equally spaced time-values and stock-price values Define i,j as the value of at time i t when the stock price is j S Knowing the payoff at maturity we solve PDE backwards till T=today time Spot today maturity strike Call payoff: f f i,j i j

IAE University of Toulouse 2010-2011 204 Finite Difference Methods Explicit method Spot derivatives are calculated at t=(i+1)t Implicit method Spot derivatives are calculated at t=it

IAE University of Toulouse 2010-2011 205 Explicit method The difference equation becomes and after some re-arrangement: more compactly: For i+1=T mat the function f i+1,j is fully known Solve above equation iteratively for f i,j in every (i,j) until i=today

IAE University of Toulouse 2010-2011 206 Explicit method schematically To calculate the option value at the boundary spots S min (with j=1) S max (with j=nbrSpots) we need extra equations, the boundary conditions We obtain these by requiring that at very low and very high spots the option has no convexity: This implies: time=ittime=(i+1)t Spot = (j+1) S Spot = j S Spot = (j-1) S

IAE University of Toulouse 2010-2011 207 Explicit method at work PDE solution with 100 time steps 100 spots t = 0.005 S = 0.025 converges to the correct Black-Scholes solution

IAE University of Toulouse 2010-2011 208 Explicit method (not) at work Unstable if number of time-steps is not big enough Oscillations are produced and propagate to all spots

IAE University of Toulouse 2010-2011 209 Implicit method More complex but avoids instabilities of explicit method The difference equation becomes and after some re-arrangement: more compactly: For i+1=T mat the function f i+1,j is fully known Solve above equation iteratively for f i,j in every (i,j) until i=today

IAE University of Toulouse 2010-2011 210 Implicit method schematically 1 equation, 3 unknowns ! We have to solve the entire system of equations for each time step Linear algebra methods LU decomposition Boundary conditions remain as before time=ittime=(i+1)t Spot = (j+1) S Spot = j S Spot = (j-1) S

IAE University of Toulouse 2010-2011 211 Explicit vs Implicit methods In practise we use a combination of the two methods Crank-Nicolson method Combines efficiency and stability

IAE University of Toulouse 2010-2011 212 Interest-rate products: introduction More difficult than derivatives of equities/Forex: The behavior of a rate is more complex than the price of a stock or exchange rate (political, macro-economics) The underlying is a curve and not a price Every point on this curve can have a different volatility

IAE University of Toulouse 2010-2011 213 Bonds (obligations) Bond with one unique payment at maturity (zero-coupon) PV=C T /(1+r) T where PV (present value) is the value of the bond today. C T is the capital payed at maturity r is the interest rate payed over a given composition (annual, monthly) T is the number of periods in the composition of the interest rate

IAE University of Toulouse 2010-2011 214 Bonds: example Bond of maturity 2 years, face value 100 (it pays C T =100 at maturity) A) interest = 3% per year, annual composition PV=100/(1+0.03) 2 = 94.26 B) interest = 3% per year, monthly composition PV=100/(1+0.03/12) 24 =94.18 C) interest = 3% per year, continuous composition PV=100 exp(-0.03 x 2) = 94.176

IAE University of Toulouse 2010-2011 215 Bonds with periodic coupons Bond with coupons + payment at maturity where PV (present value) is the value of the bond today. C i is the amount of the i th coupon (where the reimbursement occurs of the face value if i=T) r est le taux dintrt pay sur une priode de composition donne (annuelle, mensuelle ) T is the number of compounding periods = number of interest payments

IAE University of Toulouse 2010-2011 216 Example / Exercise Bond of maturity 2 years, Face value 100 Coupons 10% / year, semi-annual payment Interest rate (composition semestriel) 4% /year PV = 111.42

IAE University of Toulouse 2010-2011 217 Bonds: sensitivities The duration D expresses the sensitivity of the PV of a bond compared to a change of the rate. We often use the duration Mc Aulay : For a bond, D is 0

IAE University of Toulouse 2010-2011 218 Exercise: a portfolio of interest-rate products has a McAulay duration of 15 years, and is currently worth 10 millions , what does its value become (approximately) if the interest rate goes from 4% to 3.9% ? Answer : 10,144,231

IAE University of Toulouse 2010-2011219 Risk management and calculation of VAR

IAE University of Toulouse 2010-2011 220 VAR (Value At Risk) VAR is a measure of market risk on a group of assets. Def: Maximum loss that can be reached in x days such that there is a small probability p that the realised loss is bigger. It can be calculated at different levels: single portfolios, small group of portfolios, bank portfolios, It is not additive (diversification effect) It computes the amount of capital the bank must hold to cover its risks Bassel accord: p=1%, x=10 days.

IAE University of Toulouse 2010-2011 221 VAR: historical approach identify the parameters of the market that influence the value of the portfolio: V=f(S 1, S 2, ..) S i : Forex spots, swap rates, market vols, etc on a large sample of historical data (two or more years), calculate the daily returns of these market parameters:

IAE University of Toulouse 2010-2011 222 Apply these returns from the past to todays market data and recalculate the value of the portfolio V j =f(S 1 1 (t 0 -j), S 2 2 (t 0 -j), ..) j=1 N (number of daily observations) For each scenario replayed, calculate the profit or loss: PL j = V j - V 0 Order the PnL from the smaller (great loss) to the larger (great gain) VAR: historical approach

IAE University of Toulouse 2010-2011 223 p=5% Var is the largest value such that at least (1-p) of observations are above it

IAE University of Toulouse 2010-2011 224 Temporal extrapolation The VAR obtained in this way corresponds to a horizon of 1-day Assuming the daily increments are i.i.d. Independent Identically distributed

IAE University of Toulouse 2010-2011 225 Quantile extrapolation The VAR previously obtained are for p=5% Assuming the observations of PnL are normally distributed N -1 (p): inverse cumulative normal function

IAE University of Toulouse 2010-2011 226 Example The 5% VAR of 1-day is 42,000$, what is the value of 10-day 1% VAR?

IAE University of Toulouse 2010-2011 227 Disadvantages of historical VAR It is based on historical data. Implicitly assumes that the markets will behave in the future as they behaved in the past. It reduces the measure of risk to a single digit. This does not necessarily represent the potential damage The two distributions have the same VAR!

IAE University of Toulouse 2010-2011 228 Conditional VAR (CVAR) Measurement of the average loss exceeding VAR The two distributions do not have the same CVAR !

IAE University of Toulouse 2010-2011 229 VAR: different possible implementations Historical simulation Advantages Easy to calculate Matches data distributions Disadvantages Depends on limited experience (past data) not enough extreme events Monte-Carlo simulation: daily returns are randomly sampled based on a model Advantages Can generate lots of data & scenarios Disadvantages Introduces model risk: dependence on the assumed distibution of daily returns

IAE University of Toulouse 2010-2011 230 VAR for a continuous distribution VAR: some useful identities

IAE University of Toulouse 2010-2011 231 Full revaluation VAR VS. Linear / Quadratic VAR

IAE University of Toulouse 2010-2011232 Introduction to Credit Risk

IAE University of Toulouse 2010-2011 233 Credit Loss (loss by default), definition b i : binary indicator: 1 if default, 0 if not CE i : credit risk exposure f i : recovery rate in case of default

IAE University of Toulouse 2010-2011 234 Two possible measures of the default probability: Actuarial: we measure the credit risk on statistical basis of default of payment. Data produced by rating agencies. Implicit: deducing the default risk of certain market prices (more complex).

IAE University of Toulouse 2010-2011 235 Actuarial measure of the default risk (1)

IAE University of Toulouse 2010-2011 236 Actuarial measure of the default risk (2)

IAE University of Toulouse 2010-2011 237 Actuarial measure of the default risk (3) Marginal default rate during a period T: Probability of default during the year T, given that no default has occurred in previous years d T Cumulative rate of default between 0 and T: probability that at least one default occurs between 0 and T: C T Link between C T and d T Survival rate between 0 and T : St=(1-d 1 ) (1-d 2 ) (1-d T )

IAE University of Toulouse 2010-2011 238 Actuarial measure of the default risk (4) The measurement of default rates over a long period of time may be problematic (small sample) A more robust approach: Transition probability from one state to another: Example : a company with a rating B has a probability of 12% to be upgraded to A within a year.

IAE University of Toulouse 2010-2011 239 What is the cumulative probability that a company currently rated as A faces default in the next 3 years? Exercise

IAE University of Toulouse 2010-2011240 Trading in the real world

IAE University of Toulouse 2010-2011 241 Classical theory of financial markets Efficient market hypothesis Assumes: All information concerning a financial asset is already incorporated into the current price Implies: risk-free profit is impossible, traders are completely rational Asset increments are Independent from one tick to the next Identically distributed Normally distributed

IAE University of Toulouse 2010-2011 242 Market empirical (stylized) facts Fat tails The market-realised distribution of log-returns is not Normal Opposite graph S&P500 density of log-returns Normal density with same mean and variance Y-axis in log-scale Example: Probability of a daily move of -6% Market: 0.02% Normal: 0.000005%

IAE University of Toulouse 2010-2011 243 Market empirical (stylized) facts Volatility clustering Periods of high volatility Periods of low volatility Not reproduced by a time series of normal N(0,1) increments

IAE University of Toulouse 2010-2011 244 Market empirical (stylized) facts Decaying autocorrelations Dependence of market-returns between different times Graph opposite as function of where

IAE University of Toulouse 2010-2011 245 A simple trading strategy: Pairs trading Find two stocks that are consistently correlated Wait till one of them breaks the pattern Then buy the cheap one, sell the expensive one Wait till the trend reverses to the normal pattern Then close the position

IAE University of Toulouse 2010-2011 246 Pairs trading at work Several implementations exist. A possible one: Measure distances between stocks, S a and S b, across timeseries When the distance is too far away from the mean: trade Backtest the algorithm and optimise through modifying Distance threshold (based on e.g. multiple of the standard deviation) Size of data Asset classes of stocks The measure of distance (alternative to above can be correlation)

IAE University of Toulouse 2010-2011 247 Pairs trading at work: an example Algorithm gives signals for distances higher than 1.5standard deviation of the mean

IAE University of Toulouse 2010-2011 248 Kellys criterion You are a gambler You know your game and you win with probability 55% How much of your capital should you bet each time ? Historical background J L Kelly (1956) Bells labs USA Develops analysis for maximizing expected capital Mathematician Ed Thorp uses the analysis at Las Vegas casinos Reportedly made fortune Author of best-seller book Beat the Dealer 1962 700,000 copies sold Founder of hedge fund

IAE University of Toulouse 2010-2011 249 Kellys criterion for coin-tossing Notation You played N times Number of times you won: W Number of times you lost: L Win probability p=W/N Lose probability q=1-p Initial capital X 0 Strategy Each time you bet a fraction of your remaining capital f Example: 1 st time: Capital to bet: f X 0 Capital that remains: (1- f) X 0 This time you lose 2 nd time: Capital to bet: f (1- f )X 0 Capital that remains: (1- f) (1- f) X 0 After n rounds Capital that remains: (1- f) L (1+ f) W X 0

IAE University of Toulouse 2010-2011 250 Kellys criterion for coin-tossing Remaining capital after n rounds X n =(1- f) L (1+ f) W X 0 Ratio (in logarithm): Take expectations:

IAE University of Toulouse 2010-2011 251 Kellys criterion for coin-tossing Choose f opt maximizes the Kelly function This is the optimal fraction that leads to the maximal expected capital Avoid Ruin fraction f ruin that leads to a negative capital: you lose all your money f opt =p-q

IAE University of Toulouse 2010-2011 252 References Options, futures and other derivatives J. Hull (2008) Prentice Hall Monte Carlo Methods in Finance P. Jckel (2003) Wiley Stochastic Calculus for Finance II: Continuous-Time Models S. Shreve (2004) Springer Finance Pricing Financial Instruments: The finite-difference method D. Tavella and C. Randal (2000) Wiley Monte Carlo methods in financial engineering P. Glasserman (2000) Springer Paul Wilmott on Quantitative Finance 3 Vol Set Paul Wilmott (2000) Wiley Financial Risk Manager Handbook P. Jorion (2009) Wiley Finance

IAE University of Toulouse 2010-2011 253 Exercises: 1. Decompose the following strategies into simple Call and Put positions (short or long). Discuss advantages and disadvantages of each of the strategies 2. Integrate numerically the function exp(-x/2) between 4 and +4, using an interval of dx=0.01. 3. Differentiate numerically and analytically the function exp(-x/2). 4. Write a program in VBA that calculates the functions min(a,b) and max(a,b) using the min / max of two numbers. 5. Write a program in VBA to generate a brownian motion W(t). The input parameters are: the number of time steps, the final time. As an output, the function should return the simulated trajectory.

IAE University of Toulouse 2010-2011 254 Exercices: 6. Use the function of exercise 4 to calculate the variance of the final value of a brownian trajectory (10 time steps spaced by 3 sec), on the basis of 1000 realisations. 7. Show that the variance of random variable is given by V(X) = E(X)- (E(X)) 8. What are (i) the mean (ii) the standard deviation of returns of the index EUROSTOXX50, if we consider that it follows the law a+bX where X is a normal gaussian variable (a and b are 2 constants) ? 9. Calculate the mean and the variance of e aX where X is a guassian normal random variable 10. Calculate the expectation of S=e (r-q- /2)T+X T where X is a guassian normal random variable 11. Write a programe in VBA to compute a Black-Scholes pricer (analytic formula) for a Call option: Call(S, K, , r, q, T).

IAE University of Toulouse 2010-2011 255 Exercises: 12. Compare the price of a simple call option to the price call with a barrier where the barrier level H increases. 13. What is the value of a 3m call on EUR/USD, r EUR = 4%, r USD = 5% vol=25%, K=1.3 for different values of the spot. For each point of the curve calculate the Delta using finite differences and the analytic formula. If S=1.27, what is the cost of an option on 1,000,000 EUR notional? And on an option on 1,000,000 USD notional? 14. Show that for small t, the relations 15. Derive the density function of a logNormal random variable. 16. Calculate the mean and the variance of a log-normal density with parameters , . are solutions of 2 t = pu 2 + (1 p )d 2 e 2(r-q) t u = 1/ d

IAE University of Toulouse 2010-2011 256 Exercises: 17. Calculate with Monte Carlo the value of an Asian put option and compare with the value of the corresponding vanilla put. How do you explain the difference in the prices? 18. Calculate the number using a Monte-Carlo method 19. Programm a VBA function allowing the pricing of a Call with Monte-Carlo: Call(S, K, s, r, q, T, Nsimu). Compare with the exact solution from Black- Scholes formula 20. Show that the variables 1, 2 obtained from Choleskys decomposition have a correlation equal to 21. Compute analytically the Delta, Gamma and Vega of a Put option

IAE University of Toulouse 2010-2011 257 Exercises: 22. Using Its lemma, and starting from the differential equation of Black-Scholes dS=Sdt+ SdW, calculate the differential of ln(S). Derive an expression for S(t). 23. Using Its lemma compute the stochastic differential of the variable Z=X/Y where X and Y are stochastic variables 24. Calculate the price of a digital option (at maturity it pays 1 unit of underlying if S T >K). Write a VBA program that calculates with Monte Carlo simulations. 25. Calculate the price of a knock-out option using Monte Carlo and the formula for the surviving probabilities 26. Price a put option using the explicit PDE method and compare the result to the Black-Scholes formula. 27. Bachelier vs Black-Scholes: Price a call option with the monte carlo method using (i) brownian motion (Bachelier model) and (ii) geometric brownian motion (Black-Scholes model).

IAE University of Toulouse 2010-2011 258 Exercises: 28. Find the stochastic derivatives of the process: X t =W t 2 -t and X t =W t 2 -W t t 29. Write a Monte Carlo program in VBA that simulates a coin-tossing game and verify that the optimal fraction of capital f opt proven by Kelly leads to the maximum expected capital 30. Demonstrate that if W t is a brownian motion then E[(W t -W s ) 2 ]=t-s

IAE University of Toulouse 2010-20111 Computational Methods in Finance Nikos Skantzos.

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Transcript of IAE University of Toulouse 2010-20111 Computational Methods in Finance Nikos Skantzos.