Benedetti Kevin George 4008804

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University Cattolica del Sacro Cuore di Milano Faculty of Banking, Financial and Insurance Sciences The Decline of Saras S.p.a. Benedetti Kevin Matr. 4008804 Course: Applied Statistics for Finance Prof. Iacus Prof. Zappa Accademic year 20112012

Transcript of Benedetti Kevin George 4008804

Page 1: Benedetti Kevin George 4008804

University  Cattolica  del  Sacro  Cuore  di  Milano    

 Faculty  of  Banking,  Financial  and  Insurance  Sciences  

 

The  Decline  of  Saras  S.p.a.  

 

                                                                                           

 

              Benedetti  Kevin  

                                                                                                                                             Matr.  4008804  

 

Course:  Applied  Statistics  for  Finance  

Prof.  Iacus  

Prof.  Zappa  

 

Accademic  year  2011-­‐2012  

 

 

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INDEX

Chapter 1 – Preliminary Stock Analysis

1.1 Company Characteristics 1.2 Change Point Analysis

Chapter 2 – Option Valuation

2.1 Valuation of Financial Options: introduction

2.2 The Black & Scholes Model

2.2.1 Comments on B&S Model

2.3 The Monte Carlo Method

Chapter 3 – Lévy Process

3.1 Fast Fourier Transform

3.2 Monte Carlo Approach

Chapter 4 – Greeks Analysis

4.1 Greeks

4.2 Conclusions

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Chapter 1 - Preliminary Stock Analysis – Saras S.p.a.

1.1 – Abstract 1.2 – Company Characteristics

The Saras Group, whose operations were started by Angelo Moratti in 1962, has approximately 2,200 employees and total revenues of about 11.0 billion Euros as of 31st December 2011. The Group is active in the energy sector, and is a leading Italian and European crude oil refiner. It sells and distributes petroleum products in the domestic and international markets, directly and through the subsidiaries Saras Energia S.A. (in Spain) and Arcola Petrolifera S.p.A. (in Italy). The Group also operates in the electric power production and sale, through the subsidiaries Sarlux S.r.l. and Sardeolica S.r.l.. In addition, the Group provides industrial engineering and scientific research services to the oil, energy and environment sectors through the subsidiary Sartec S.p.A.. Finally, in July 2011, the Group created a new subsidiary called Sargas S.r.l., which operates in the fields of exploration and development, as well as transport, storage, purchase and sale of gaseous hydrocarbons.

Here are the market performance of the stock integrated with an important indicator that are the volumes:

As we can notice from this picture the stock registered a small decrease in the stock price in the very first part of the graph but it went up again jast before july 2008. From this point the stock went down rapidly and it never stopped. Even today the trand of the stock in quite negative. We have to

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Volume (millions):2,239,500

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say that in the period considered for the analysis the whole world had to face the financial crisis started in 2008 and the bad situation of this company doesn’t surprise. Deslpite the negative trend on January 2012 we can observe a n incredibly high value of volumes: during this period, in fact, there was the probability for the company to be delisted.

1.3 – Change Point Analysis

Given the decline of the stock and the the pattern of the prices during this period of crisis we observe a “roller coaster” graph. Now we are going through an analysis that could help us to explain this performance in order to catch the points where a turnaround has been registered.

In this paragraph I want to use an important tool in volatility analysis: the Change Point Analysis.

Considering a process: X = {Xt, 0 ≤ t ≤ T} à dXt = b(Xt)dt + √θσ(Xt)dBt and X0 = x0, 0<θ1, θ2<∞{Bt, t ≥ 0} à Bt is a Brownian motion and the coefficients are defined and known.

The aim of this analysis is to find a point called τ0(tau0) associated to a parameter called θ(theta). R-software compute for us this kind of operation and it gives us two values of theta: θ1, the volatility just before the change point, and θ2, the volatility immediately after the change point.

In our case we are going to analyze a one-year period – from May 20, 2011 to May 20, 2012 – in order to observe the reaction of the market in a tough span of time: in fact from May 20, 2011 there were rumors of a probability of delisting, denied on December 1st, 2011.This announcement could have brought some “good news” fo investors in a definitely not easy period. Therefore we are going to expect a change point on around this date: from high range of volatility to a more attenuate one.

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The picture above confirms our expectations: the value τ0 – the change point – is set on January 10, 2012. In fact after the announcements, where we can observe a steep rise in stock price, we find few days more of high volatility and then, after January 10 a reduction in volatility.

Analitically speaking here are the R results:

τ0 = 2012-01-10

θ1 = 0.0444403

θ2 = 0.02976848

∆2-1=0.02976848 – 0.0444403 = -0.01467182

0.8

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1.2

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1.8S [2011-05-20/2012-05-18]

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1.8Last 0.74Bollinger Bands (20,2) [Upper/Lower]: 0.991/0.730

Mag 202011

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Dic 012011

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Apr 022012

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Chapter 2 – Option Valuation

2.1 – Valuation of Financial Options Introduction

A financial option contract gives its owner thr right (but not the obligation) to purchase or sell an asset at fixed price at some future date. Two distinct kinds of option contracts exist: call options and put options. A call option gives the owner the right to buy the asset; a put option gives the owner the right to sell the asset. The most commonly encountered option contracts are options on shares of stock: a stock option gives the holder the option to buy or sell a share of stock on or before a given date for a given price. When a holder of an option enforces the agreement and buys or sells a share of stock at the agreed-upon price, he is exercising the option. The price at which the holder buys or sells the share of stock when the option is exercised is called the strike price. There are two kinds of options. American options, the most common kind, allow their holders to exercise the option on any date up to and including a final date called the expiration date. European option allow their holders to exercise the option only on the expiration date. The price of an European Option derive basically from the difference between the reference price, the strike price(K), and the value of the underlying asset (S) plus a premium based on the remaining time until the expiration date of the option

C = max(S – K, 0) P = max(K –S, 0)

As we can easily deduce from these equations the value of a call option can’t be negative because if the value drops below zero the owner doesn’t exercise the option at the expiration date. It can be usefull having a representation of how these options work. Here is the graph of a call option and a put option with the same strike price:

   

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Payoff Functions

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Nowadays the value of an option is calculated relying on several mathematics models that help us in predicting the value of an option changes related to changing in the underlying conditions.

In evaluating options analysts should keep clearly in their mind some main conditions: first of all they surely have to consider the market price of the underlying security, the strike price of the option and the relationship that stands between them because, as we know, the option price changes a lot depending on whether the option is in the money or out of the money. Then they have to focus on the cost of holding the underlying security, the expiration date and the expected volatility of the underlying security’s price with respect to the life of the option.

2.2 The Black & Scholes Model

Most of the models that, today, are used by analysts all over the world have a common root: the model devoloped by Fisher Black and Myron Scholes (1973) called the “Black&Scholes Model” that allows, taking in consideration some assumptions, the pricing of European Call and Put option using a simple formula.

The assumptions mentioned above are:  

1)  The  stock  pays  no  dividends  during  the  option's  life  

Most companies pay dividends to their share holders, so this might seem a serious limitation to the model considering the observation that higher dividend yields elicit lower call premiums. A common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price.

2)  European  exercise  terms  are  used  

European exercise terms dictate that the option can only be exercised on the expiration date. American exercise term allow the option to be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility. This limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same.

3)  Markets  are  efficient  

This assumption suggests that people cannot consistently predict the direction of the market or an individual stock. The market operates continuously with share prices following a continuous Itô process. To understand what a continuous Itô process is, you must first know that a Markov process is "one where the observation in time period t depends only on the preceding observation." An Itô process is simply a Markov process in continuous time. If you were to draw a continuous process you would do so without picking the pen up from the piece of paper.

 

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4)  No  commissions  are  charged  

Usually market participants do have to pay a commission to buy or sell options. Even floor traders pay some kind of fee, but it is usually very small. The fees that Individual investor's pay is more substantial and can often distort the output of the model.

5)  Interest  rates  remain  constant  and  known  

The Black and Scholes model uses the risk-free rate to represent this constant and known rate. In reality there is no such thing as the risk-free rate, but the discount rate on U.S. Government Treasury Bills with 30 days left until maturity is usually used to represent it. During periods of rapidly changing interest rates, these 30 day rates are often subject to change, thereby violating one of the assumptions of the model.

6)  Returns  are  lognormally  distributed  

This assumption suggests, returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options.

7) the stock price follows a geometric Brownian motion with constant drift and volatility.

This part will be widley covered in the “comments on B&S” paragraph

 

Suppose that all of the assumptions above are verified: in this case we can plainly calculate the price of an option using the exact formula of B&S.

Here are the Call and the Put formula, respectively:

Call  price:      

Put  price:    

                     

Where:  

 

     

   

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and:  

   

What we are going to do now is try to compute the price of a Call and a Put option on “Saras S.p.a” using the Black&Scholes formula in order to verify if the results of the model are plausible with respect to the real market value of the same options.

First of all we look on a financial web site (Yahoo.fianance, Google Finance etc.) focusing on data we need to proceed in our calculations. We have to remember that the expiration date is going to be expressed as effective trading days (252). Here are the parameters:  S0= 0.75 K=0.72 T=20/252 r=0.005 σ=?  At this point of our calculations, unfortunately, we have a missed value: the volatility. This parameter is not directly observable on the market so we have to find it out by ourselves and using R it is quite immediate to obtain that the historical volatility, in the period from 2011-01-05 to 2012-01-05, is equal to 0.54220241.

SARAS S.p.a

MKT “C” B&S”C” MKT”P” B&S”P”

0.039 0.06149544 0.029 0.03120970 Looking at the results we can easily deduce that both Call and Put options prices of Saras S.p.a. computed using Black&Scholes formula are higher than the market prices of the same options. This could be caused by the value of the volatility used in the computation: in fact there is the possibility that our value is higher than the one used by analysts in the market. These results tell us that the future expected volatility is lower than the historical volatility of the past year (the one we used) and this reveals that there is a lower chance for the option to be in the money at the maturity date reflecting the possibility of lower oprion prices. Now our goal is to understand if our expectations about the value of the historical volatility are confirmed and see how much is the difference between the historical volatility and the implied one. R software gives us a function that can do this for us and the results are:

                                                                                                                         1 I wanted to be sure about the historical volatility so I checked it out using the Call-Parity equation and it has been confirmed.

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Implied Volatility for a CALL Historical Volatility

σ = 0.2468025 σ = 0.5422024

Implied Volatility for a PUT

σ = 0.5443167

 A we can see from these results the implied volatility used by analysts to price the options are different from the historical in the CALL option case and this is reflected in the difference between the market price and the one computed using the B&S formula. On the other hand the implied volatility used in pricing the PUT is extremely similar to the historical one; in fact the market price is 0.029 against the B&S one of 0.031.    

  Looking at this graph, that represents the trend of the Saras stock prices over the last year, we can see some indicators tant could help us with the volatility pattern. In order to do so some two technical tools have been included in this chart and they will surely help our comprehension of what is the overall situation. These tools are: Bollinger Bands (BBands) and the Average True Range(ATR). These indicators, combined together, are very common in fincance in order to predict the inversion of trend of a security (a stock in our case).

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The Average True Range has the aim of calculate prices volatility as the breadth of their fluctuations. This mechanism is based on the idea that, during a turnaround, the volatility assume extreme values (high or low). On the other hand we have the Bollinger Bands that are based on the volatility calculation as well but there is a superior band and an inferior one so that we can have an idea of what the volatility range is. From an operating point of view the Bands give us signals of buying or selling when these conditions occure:

-­‐ when the price graph goes out of the upper band and it goes in again, this is a selling sign; this could represent an increase in price followed by an adjustment.

-­‐ When the same thing happens with respect to the lower band, we have a buying sign; this means that the price has gone down very quickly up to the point of turnaround.

Concerning Saras S.p.a. we can see some of these situations. For example we see that on August 2011 the graph crossed the lower band meaning a strong decrease in price in fact in this period we registered the peak of the Euro debt crisis. We can see that even from the volatility graph that on that date had a steep rise. The price went down up to the point of turnaround, crossing again the lower band, giving a signal of strong buying of the stock.

Conversely, on December 2011, we can see the same process that led to a steep rise in the stock price in according to the volatility graph that registered a strong increase. In this case the stock price went up and, just after having crossed the upper line, went down meaning that we were in front of a turnaround and people were selling their shares.

Looking at the very last part of the chart we can see that, in these days, the stock is having a negative trend and this is the reason why the implied volatility of the put option is higher than the call one telling us that the market expectations for the near future is still a down-trend.

2.2.1 – Comments

The Black&Scholes formula, the one we used before to price Saras S.p.a. options, doesn’t match very well with the real world we live in.

That’s why, as we have seen in the previous paragrph, is based on some assumptions that are impossible to notice in real circumstances: in particular what we are going to prove now is the soundness of the assumption that says stock price follows a geometric Brownian motion with constant drift and constant volatility. In order to understand better what we are going to talk about let’s see, first, what a geormetric Brownian motion actually is.

One of the most important assumption of the B&S model is that stock prices follow a Normal distributed process and this processi s known as the geometric Brownian motion. It defined as follows:

dSt = µSt dt + σSt dWt

where:

σ is the volatility and it is assumed constant.

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µ is the expected return.

Wt is Wiener Process that is the stochastic component of the process. In order to compute our demonstration we are not going to use the returns of the stock. Instead, we are going to use the logarithm of the returns (more reliable) given by the ratio: returns in time t over returns in time t-1. Here is the expression:

log.returns = returnst / returnst-1

 

Gen 022007

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N = 1364 Bandwidth = 0.004264

Density

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Theoretical Quantiles

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ple

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These graphs are a very clear demonstration of what we were looking for. The density function of our company are not properly ditributed and we are going to comment some indicators that tell us why: first of all, looking at the second graph(the density function), we can easily see that the tails of the function are not linear adn well defined as farther we move from central values. This fact underline the fact that the price does not follow a geometric Brownian motion so that the assumption does not hold. Finally, the third picture, should give us the explanation of why our prices does not follow a Normally distributed function. If we look at it we can see a straight line indicating the path that our prices should follow to be Normally distributed and the path they actually follow. Concerning central values they seem to be as the ideal funtion wants them to be but if we move from central value we notice that they go astray.

2.2.2 – Volatility Smiles

As we wrote before, B&S formula assesses that prices follow a normally distributed trend with constant volatiltiy. In this part we are to verify this last statement.

Here are some options (call and put) with the same expiration date:

 

Saras S.p.a.

Expiration June 15, 2012

CALL PUT

Mkt Strike(K) Mkt

0.0435 0.74 0.03

0.0415 0.76 0.037

0.032 0.78 0.047

0.0205 0.8 0.051

0.008 0.85 0.086

0.003 0.9

0.0005 0.95

0.0005 1

0.0005 1.05

For semplicity we show and comment only the graph related to Call Option.

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As we can immediatly see the stock prices are not characterized by constant volatility and, instead, it changes as the strike price changes.

The volatility seems to follow a particular path that, as analysts call it, can be similar to a “smile”. This kind of graph is called “Volatility Smile”. The reason of this trend is related to the fact that values are high when the option is deeply in, or conversely out, of the money and decrease when the option is near to the “at the money point”.

 

2.3 – The Monte Carlo Method

As we have already mentioned, the Black&Scholes model is used all over the world for pricing options but our paper shows that the assumptions it is based on are quite unrealistic and through our calculations we disproved them.

Now we are going to challenge another method, the Monte Carlo Method, based on a simulation: a random generation of thousand of combination of prices . The successive step is going to be the calculation of the payoff of the option for each simulation; the discounted results will be the price of the option we are looking for.

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Volatility smile SARAS S.P.A

K

smile

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In this part of the paper we want to compare two different prices of the same option: we have already calculated one of these two prices that is generated by B&S model. The second one is going to be generated by the Monte Carlo method. Then we are going to discuss our results.

As Monte Carlo simulation is based on repeated price generation we are goint to take different scenarios characterised by different number of simulation (1000, 10000, 100000, 1000000) and what we expect is that the higher are repetitions the more precise would be our price according to the market one.

SARAS S.p.a.

CALL M PUT

0.05976969 1000 0.03035832

0.06110445 10000 0.03120731

0.06154876 100000 0.0311551

0.06154721 1000000 0.03115751

As we expected the accuracy of prices with respect to ones calculated by B&S is as higher as the number of the simulations increase.

This is the main characteristic of the Monte Carlo simulation based on the “large number law” and it is plain also using a function of R software called “speed of convergence” that show us graphically the characteristic we’ve mentioned above.

To simplify our calculations, our analysis on convergency is going to be done only on one option: the characteristics of this option are S0=0.75 and the strike price(K)= 0.72. The interest rate is, as for other operations, is the interest rate of deutsch bank.

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Chapter 3 – Lévy Process

In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process that starts at 0, admits càdlàag(continue à droite, limitée à gauche) modification and has "stationary independent increments" — this phrase will be explained below. It is a stochastic analog of independent and identically distributed random variables, and the most well known examples are the Wiener process and the Poisson process.

It is defined as follows:

A stochastic process X = {Xt : t ≥ 0} is said to be a Lèvy process if,

Speed of Convergency - Saras S.p.a.

MC replications

MC

pric

e

10 100 200 500 1000

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1) X0 = 0 almost surely 2) Indipendent increments: For any 0 ≤ t1 < t2 < … < tn < ∞, Xt2 – Xt1, Xt3 – Xt2, … , Xtn – Xtn-1

are indipendent. 3) Stationary increments: for any s < t, Xt – Xs is equal in distribution to Xt-s 4) t -> Xt is almost surely right continuous with left limits.

In order to better understand what is wrong in the B&S Model we are going to discuss and represent graphically the problem. First we have to say that distribustions are not normal:

 

 

 

Looking at the graph it is evident that the distribution of the logarithm of the returns is definitely different from a normal distributed function (as we have already computed for the B&S Model). Our goal is, now, find a new model which should not be based on geometric Brownian motion like the B&S model. Well, a solution to our problem could be represented by Lévy proccesses.

Using R-software we are able to compute some Lévy processes and to plot them giving us a sketch of what could be a solution to our problem:

-0.1 0.0 0.1 0.2

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density.default(x = Ret.Saras)

N = 253 Bandwidth = 0.008547

Density

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In the above we can see:

Picture a) the NORMAL parameter estimation.

MEAN:-0.003379408 SD:0.036455639

Picture b) the NORMAL INVERSE GAUSSIAN parameter estimation

ALPHA:28.375193345 BETA:1.564509140

DELTA:0.036703176 MU:-0.005406176

Picture c) the HYPERBOLIC parameter estimation

ALPHA:43.912065188 BETA:1.505010969

DELTA:0.016400880 MU:-0.005299175

Picture d) the GENERALIZED HYPERBOLIC parameter estimation

ALPHA:3.813755900 BETA:2.074031129

DELTA:0.057400423 MU:-0.006110047

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LAMBDA:-2.229897013

Once we have introduced what the Lévy process is we have to make another important assumption: it says that Lévy markets, even if we condider simpliest one, are not complete.

Now we can proceed in pricing an option and, in this particular case, this operation can follow two ways:

1) the Fast Fourier Transform 2) the Monte Carlo Approach

In order to simplify my processes we chose to proceed just for the first method giving a sketch of theory for the second one.

3.1 – Fast Fourier Transform

A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory. A DFT decomposes a sequence of values into components of different frequencies. This operation is useful in many fields but computing it directly from the definition is often too slow to be practical. An FFT is a way to compute the same result more quickly: computing a DFT of N points in the naive way, using the definition, takes O(N2) arithmetical operations, while an FFT can compute the same result in only O(N log N) operations. For example, corcerning the gepmetric Brownian motion, the characteristic function of Zt is: φ(u)= exp( iu(µ – 1/2 σ2) - σ2u2/2) where φ is known, the proce of an option can be approximated to this equation:

CT(k) ≈ e-αk/π ∑e-iv

jkψT(vj)η, where vj = η(j-1) and k = logK.

The constant alpha is considered as the dampening factor and it is usally equal to one. This model can we easily used on R and the results we obtained are:

B&S Price FFT Price ∆

0.06149541 - 0.05932576 = 0.00216965

3.2 – The Monte Carlo Approach

The first step we have to take is identifying the distribution of the returns. Then we just have to simulate the patterns of the stochastic process and apply the payoff function to the final value.

ST = S0eZT

Chapter 4 – Greeks Analysis and Conclusions

4.1 – Greeks

In mathematical finance, the Greeks are the quantities representing the sensitivities of the price of derivatives such as options to a change in underlying parameters on which the value of an

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instrument or portfolio of financial instruments is dependent. The name is used because the most common of these parameters are often denoted by Greek letters. Collectively these have also been called the risk sensitivities, risk measures or hedge parameters.

The Greeks are vital tools in risk management. Each Greek measures the sensitivity of the value of a portfolio to a small change in a given underlying parameter, so that component risks may be treated in isolation, and the portfolio rebalanced accordingly to achieve a desired exposure.

The Greeks in the Black–Scholes model are relatively easy to calculate, a desirable property of financial models, and are very useful for derivatives traders, especially those who seek to hedge their portfolios from adverse changes in market conditions. For this reason, those Greeks which are particularly useful for hedging delta, theta, and vega are well-defined for measuring changes in Price, Time and Volatility. Although rho is a primary input into the Black–Scholes model, the overall impact on the value of an option corresponding to changes in the risk-free interest rate is generally insignificant and therefore higher-order derivatives involving the risk-free interest rate are not common.

The most common of the Greeks are the first order derivatives: Delta, Vega, Theta and Rho as well as Gamma, a second-order derivative of the value function. The remaining sensitivities in this list are common enough that they have common names, but this list is by no means exhaustive.

FIRST ORDER GREEKS

Delta

Delta, , measures the rate of change of option value with respect to changes in the underlying asset's price. Delta is the first derivative of the value of the option with respect to the underlying instrument's price .

For a vanilla option, delta will be a number between 0.0 and 1.0 for a long call (and/or short put) and 0.0 and −1.0 for a long put (and/or short call) – depending on price, a call option behaves as if one owns 1 share of the underlying stock (if deep in the money), or owns nothing (if far out of the money), or something in between, and conversely for a put option.

Vega

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Vega measures sensitivity to volatility. Vega is the derivative of the option value with respect to the volatility of the underlying asset.

Vega can be an important Greek to monitor for an option trader, especially in volatile markets, since the value of some option strategies can be particularly sensitive to changes in volatility. The value of an option straddle, for example, is extremely dependent on changes to volatility.

Theta

Theta, θ , measures the sensitivity of the value of the derivative to the passage of time: the "time decay.”

The mathematical result of the formula for theta is expressed in value per year. By convention, it is usual to divide the result by the number of days in a year, to arrive at the amount of money per share of the underlying that the option loses in one day. Theta is almost always negative for long calls and puts and positive for short calls and puts. An exception is a deep in-the-money European put. The total theta for a portfolio of options can be determined by summing the thetas for each individual position.

The value of an option can be analysed into two parts: the intrinsic value and the time value. The intrinsic value is the amount of money you would gain if you exercised the option immediately, while the time value is the value of having the option of waiting longer before deciding to exercise.

Rho

Rho, , measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk free interest rate (for the relevant outstanding term).

Except under extreme circumstances, the value of an option is less sensitive to changes in the risk free interest rate than to changes in other parameters. For this reason, rho is the least used of the first-order Greeks.

Rho is typically expressed as the amount of money, per share of the underlying, that the value of the option will gain or lose as the risk free interest rate rises or falls by 1.0% per annum (100 basis points).

Lambda

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Lambda, , omega, , or elasticity is the percentage change in option value per percentage change in the underlying price, a measure of leverage, sometimes called gearing.

SECOND ORDER GREEKS

Gamma

Gamma, , measures the rate of change in the delta with respect to changes in the underlying price. Gamma is the second derivative of the value function with respect to the underlying price. All long options have positive gamma and all short options have negative gamma. Gamma is greatest approximately at-the-money (ATM) and diminishes the further out you go either in-the-money (ITM) or out-of-the-money (OTM).

When a trader seeks to establish an effective delta-hedge for a portfolio, the trader may also seek to neutralize the portfolio's gamma, as this will ensure that the hedge will be effective over a wider range of underlying price movements. However, in neutralizing the gamma of a portfolio, alpha (the return in excess of the risk-free rate) is reduced.

Thanks to R, calculating the coefficients of Greeks on Saras Call options is quite easy and using a simple function here are the results we obtained:

SARAS S.p.a

As the Underlying Stock Price Changes - Delta and Gamma

Delta measures the sensitivity of an option's theoretical value to a change in the price of the underlying asset. It is normally represented as a number between -1 and 1, and it indicates how much the value of an option should change when the price of the underlying stock rises by one euro. As an alternative convention, the delta can also be shown as a value between -100 and +100 to show the total euro sensitivity on the value 1 option, which comprises of 100 shares of the underlying.

Call options have positive deltas and put options have negative deltas. At-the-money options generally have deltas around 50. Deep-in-the-money options might have a delta of 80 or higher, while out-of-the-money options have deltas as small as 20 or less.

In our case if we multiply the value of Delta we obtain: 0.6354121*100=63.54121.

Delta 0.6354121 Gamma 3.279769 Theta 0.06149541 Vega 0.07938833 Lambda 7.749506 Rho 0.03294156

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The value can be considered standing around 50 so we are describing an option that stands just beyond the “at the money” zone but, at the same time, it is not in “in the money” zone. Furthermore, if the underlying price changes by 1 euro the related variation in the option proce is going to be almost 0.64.

Another thing we are interested in is how delta may change as the stock proce moves: Gamma measures the rate of change in the delta for each one-point increase in the underlying asset. It is a valuable tool in helping you forecast changes in the delta of an option or an overall position. Gamma will be larger for the at-the-money options, and gets progressively lower for both the in- and out-of-the-money options. Unlike delta, gamma is always positive for both calls and puts.

In our case this indicator helps us to better understand what kind of option we are dealing with in fact as we said above lower values of gamma indicates a “in the money” option (almost 3.28). Besides, it tells us that for one point our delta is going to change by 3.28.

Changes in Volatility and the Passage of Time - Theta and Vega

Theta is a measure of the time decay of an option, the euro amount that an option will lose each day due to the passage of time. For at-the-money options, theta increases as an option approaches the expiration date. For in- and out-of-the-money options, theta decreases as an option approaches expiration. Theta is one of the most important concepts for a beginning option trader to understand, because it explains the effect of time on the premium of the options that have been purchased or sold. The further out in time you go, the smaller the time decay will be for an option. If you want to own an option, it is advantageous to purchase longer-term contracts. If you want a strategy that profits from time decay, then you will want to short the shorter-term options, so that the loss in value due to time happens quickly.

In our case theta is low (almost 0.06) indicating that our option approaches the expiration and it actually does because the expiration of the option we considered in the calculation expires within a month(June 15, 2012). Since the expiration date is not far this indicator tells us that the amount money we are going to loose up to the expiration is small.

Vega measures the sensitivity of the price of an option to changes in volatility. A change in volatility will affect both calls and puts the same way. An increase in volatility will increase the prices of all the options on an asset, and a decrease in volatility causes all the options to decrease in value. However, each individual option has its own vega and will react to volatility changes a bit differently. The impact of volatility changes is greater for at-the-money options than it is for the in- or out-of-the-money options. While vega affects calls and puts similarly, it does seem to affect calls more than puts. Perhaps because of the anticipation of market growth over time.

Since we are analyzing an “in the money” option even vega has a low value indicating a small impact of volatility.

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Changes with respect to interest rate - Rho

Since it measures the sensitivity with respect to the interest rate we are going to multiply by 100 the value we obatained on R: 0.03294156*100=3,294156. This value in the gain of our option related to a variation of 1.0% of the interest rate.

Elasticity - Lambda

Finally we are going to discuss the only second order greek that gives us a measure of leverage. In our case it is 7.749506 and this is the percentage variation (almost 7.75%) in our option per percentage change in the underlying asset value.

4.2 - Conclusions

To be honest working on this paper actually helped me to better understand the reality of the financial market but, first of all, the reality of an incredibly important company as Saras within the financial market. My technical skills were low at the beginning of the process but this lack helped me consolidating my theory and my practical skills. Concerning the results obtained I have to say that I am satisfied. According to these results the company is facing a tough crisis period and the stock prices reflect it. If we look at the options value, the price of a put option still worths more than a call indicating the trend is not going to have turnaround. The differences in prices is not very consistent due to the different methods we have implented: for example Lévy process should have given a price improving the Black & Scholes formula based on the geometric Brownian motion but the difference between the two prices is very low. Afterall, going through my analysis I found one of the most powerfull company in the field of refining facing the difficulties in a very bad way. Besides, the problems are not coming only from financial markets in facts the company is working to fix the problem of its employers’ death while on working . The overall situation is very clear and it tells that this company is going down, the stock price registered a decreasing trend and I don’t think is going to stop.

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SCRIPT

## Chapter 1 ##

library(quantmod) get.symbols(“SRS.MI”, from=”2007-05-20”, to=”2012-05-20”) chartSeries(“SRS.MI”, theme="white", TA=c(addVo(), addBBands()))

##Change Point##

library (tseries) S <- get.hist.quote("SRS.MI", start = "2011-05-01", end="2012-04-30") chartSeries(S, TA = c(addVo(), addBBands()), theme="white") S <- S$Close require(sde) cpoint(S) addVLine = function(dtlist) plot(addTA(xts(rep(TRUE, + + NROW (dtlist)), dtlist), on = 1, col = "red")) addVLine(cpoint(S)$tau0)

##Chapter 2##

##Option Valuation##

f.call <- function(X) sapply(X, function(X) max(X-K,0))) f.put <- function(X) saplly(X, function(X) max(c(K-X,0))) K<-1 curve(f.call, 0,2,main=”Payoff Functions”,col=”Blue”, Ity=4, lwd=4,ylab=expression(f(x))) curve(f.put,0,2,col=”red”,add=TRUE,Ity=4,lwd=4) temp<-legend(“top”,legend=c(“Call”,”Put”),Ity=1, lwd=5,col=c(“blue”,”red”)

##Black&Scholes Model##

require(fImport)

S<-yahooSeries(“SRS.MI”, from=”2011-05-20”, to=”2012-05-12”)

Head(S)

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Close<-S[, “SRS.MI.Close”]

Require(quantmod)

chartSeries(Close, theme=”white”)

X<-returns(Close)

Delta<-1/252

sigma.hat<-sqrt(var(X)/Delta)[1,1]

sigma.hat

##Call Option (Saras)##

call.price<-function(x=1, t=0, T=1, r=1, sigma=1,K=1){

d2<-(log(X/K) + (r-0.5*sigma^2)*(T-t)/sigma*sqrt(T-t))

d1<-d2 + sigma*sqrt(T-t)

x*pnorm(d1)-K*exp(-r*(T-t))*pnorm(d2)}

S0<-0.75

K<-0.72

T<-20/252

r<-0.005

sigma<-0.5422024

C<-call.price(x=S0, t=0, T=T, r=r, K=K, sigma=sigma)

C

## Put Option (Saras)##

put.price<-function(x=1, t=0, T=1, r=1, sigma=1,K=1){

d2<-(log(X/K) + (r-0.5*sigma^2)*(T-t))/(sigma*sqrt(T-t))

d1<-d2 + sigma*sqrt(T-t)

K*exp(-r*(T-t))*pnorm(-d2) –x*pnorm(-d1)}

S0<-0.75

K<-0.72

T<-20/252

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r<-0.005

sigma<-0.5422024

p<-put.price(x=S0, t=0, T=T, r=r, K=K, sigma=sigma)

p

##Put-Call Parity##

C – S0 + K * exp(-r * T)

##Implied Volatility##

#call

p<-0.039

Delta<-1/252

T<-20*Delta

S0<-0.75

K<-0.72

r<-0.005

sigma.imp<-GBSVolatility(p,”c”,S=S0, X=K, Time=T, r=r, b=r)

sigma.imp

#put

p<-0.029

sigma.imp<-GBSVolatility(p, “c”, S=S0, X=K, Time=T, r = r, b = r)

sigma.imp

##Plot B&S##

library(tseries)

S <- get.hist.quote(“SRS.MI”, start=”2011-05-20”, end=”2012-05-20”)

chartSeries(S,TA=c(addVo(),addBBands(), addATR()), theme=”white”)

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##geometric Brwnian motion##

getSymbols(“SRS.MI”)

plot(SRS.MI)

str(SRS.MI)

S<-SRS.MI[, “SRS.MI.Close”]

X<-diff(log(S))

plot(X)

plot(density(X, na.rm=TRUE))

qqnorm(na.omit(X))

qqline(na.omit(X))

##Log-Returns##

getSymbols(“SRS.MI”)

plot(SRS.MI)

str(SRS.MI)

##Volatility Smile##

S<- (SRS.MI[, “SRS.MI.Close”]

X<-returns(Close)

Delta<-1/252

sigma.hat<-sqrt(var(X)/Delta)

sigma.hat

Pt<-c(0.0435, 0.0415, 0.032, 0.0205, 0.008, 0.003, 0.0005, 0.0005)

K<-c(0.74, 0.76, 0.78, 0.8, 0.85, 0.9, 0.95, 1, 1.05)

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S0<-0.75

nP<-length(Pt)

T<-20*Delta

R<-0.005

smile<- sapply(1:nP, function(i) GBSVolatiltiy(Pt[i], “c”, S=S0, X=K[i], Time=T, r=r, b=r))

vals<-c(smile, sigma.hat)

plot(K, smile, type =”l”, lty=3, col=”orange”)

axis(2, sigma.hat, expression(hat(sigma)), col=”blue”)

 

##MONTE CARLO Method##

S0<-0.75

K<-0.72

r<-0.005

Delta<-1/252

T<-20*Delta

sigma<-0.5422024

MCPrice<-function(x=1, t=0, T=1, r=1, sigma=1, M=1000, f){

h<-function(m){

u<- rnorm(m/2)

tmp<-c(x*exp((r-0.5*sigma^2)*(T-t)+sigma*sqrt(T-t)*u), x*exp((r-0.5*sigma^2)*(T-t)+sigma*sqrt(T-t)*(-u)))

mean(sapply(tmp, function(xx) f(xx)))

}

p<-h(M)

p*exp(-r*(T-t))

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}

##CALL##

f<-function(x) max(0, x-K)

set.seed(123)

M<-1000

MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)

M<-10000

MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)

M<-100000

MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)

M<-1000000

MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)

 

##PUT##

f<- function(x) max(0,K-x)

M<-1000

MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)

M<-10000

MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)

M<-100000

MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)

M<- 1000000

MCPrice(x=S0, t=0, T=T, r=r, sigma=sigma.hat, M=M, f=f)

 

##Speed of Convergency##

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S0<-0.75

K<- 0.72

r<-0.005

Delta<- 1/252

sigma.hat<-0.5422024

f<- function(x) max(0,x-K)

set.seed(123)

m<-c(10, 50, 100, 150, 200, 250,500, 1000)

p1<-NULL

err<-NULL

nM<-length(m)

repl<-100

mat<-matrix(, repl, nM)

for(k in 1:nM){

tmp<-numeric(repl)

fot(i in 1:repl) tmp[i] <- MCPrice(x=S0, t=0, T=T, r=r, sigma, M=m[k], f=f)

mat[, k] <- tmp

p1<-c(p1, mean(tmp))

err<-c(err, sd(tmp))

}

colnames(mat)<-m

p0<-GBSOption(TypeFlag=”c”, S=S0, X=K, Time=T, r=r, b=r, sigma=sigma)

minP<-min(p1-err)

maxP<-max(p1 + err)

plot(m, p1, type=”n”, ylim=c(minP, maxP), axes=F, ylab=”MC price”, xlab0 “MC replications”, main= “Speed of Convergency – Saras S.p.a.”)

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lines(m, p1 + err, col= “blue”)

lines(m, p1-err, col= “blue”)

axis(2, p0, “B&S price”)

axis(1,m)

boxplot(mat, add=TRUE, at=m, boxwex=15, col= “orange”, axes=F)

points(m, p1, col= “blue”, lwd=3, lty=3)

 

##Greeks  Analysis##  

require(fOptions)

r<- 0.005

T<-20/252

K<-0.72

sigma<- 0.5422024

S0<- 0.75

GBSCharacteristics(TypeFlag= “c”, S=S0, X=K, Time=T, r=r, b=r, sigma=sigma)

 

 

##Lévy process##

require(fImport)

data <- yahooSeries("ENI.MI", from="2009-01-01", to="2009-12-31")

S <- data[, "ENI.MI.Close"] > X <- returns(S) > require(quantmod) >

lineChart(S,layout=NULL,theme="white")

lineChart(X,layout=NULL,theme="white")

sigma.hat <- sqrt( var(X)/deltat(S) )

alpha.hat <- mean(X)/deltat(S)

mu.hat <- alpha.hat + 0.5 * sigma.hat^2

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plot(density(X))

d <- function(x) dnorm(x, mean=mu.hat, sd=sigma.hat)

curve(d, -.2, .2, col="red",add=TRUE, n=500)

par(mfrow=c(2,2))

par(mar=c(3,3,3,1))

grid <- NULL

library(fBasics)

nFit(X)

nigFit(X,trace=FALSE)

hypFit(X,trace=FALSE)

ghFit(X,trace=FALSE)

##FFT##

FFTcall.price <- function(phi, S0, K, r, T, alpha = 1, N = 2^12, eta = 0.25) {

m <- r - log(phi(-(0+1i)))

phi.tilde <- function(u) (phi(u) * exp((0+1i) * u * m))^T

psi <- function(v) exp(-r * T) * phi.tilde((v - (alpha +

1) * (0+1i)))/(alpha^2 + alpha - v^2 + (0+1i) * (2 *

alpha + 1) * v)

lambda <- (2 * pi)/(N * eta)

b <- 1/2 * N * lambda

ku <- -b + lambda * (0:(N - 1))

v <- eta * (0:(N - 1))

tmp <- exp((0+1i) * b * v) * psi(v) * eta * (3 + (-1)^(1:N) –

((1:N) - 1 == 0))/3

ft <- fft(tmp)

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res <- exp(-alpha * ku) * ft/pi

inter <- spline(ku, Re(res), xout = log(K/S0))

return(inter$y * S0)

}

phiBS <- function(u) exp((0+1i) * u * (mu - 0.5 * sigma^2) - 0.5 * sigma^2 * u^2)

S0<- 0.75

K<-0.72

R<-0.005

T<-20/252

Sigma<-0.5422024

mu <- 1

require(fOptions)

GBSOption(TypeFlag = "c", S = S0, X = K, Time = T, r = r, b = r, sigma = sigma)

FFTcall.price(phiBS, S0 = S0, K = K, r = r, T = T) }