The Constant Elasticity of Variance Option Pricing Model

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1 The Constant Elasticity of Variance Option Pricing Model John Randal A thesis submitted to the Victoria University of Wellington in partial fulfilment of the requirements for the degree of Master of Science in Statistics and Operations Research April, 1998

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3 Acknowledgements The author would like to thank his supervisors, Peter Thomson and Martin Lally, for their guidance and encouragement. Also, thanks to Credit Suisse First Boston NZ Limited for providing data, and to Edith Hodgen for valuable assistance with the preparation of this document. i

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5 Abstract The Constant Elasticity of Variance (CEV) Model was first presented in 1976 by John Cox and Stephen Ross as an extension to the famous Black-Scholes European Call Option Pricing Model of 1972/3. Unlike the Black-Scholes model, which is accessible to anyone with a pocket calculator and tables of the standard normal distribution, the CEV solution consists of a pair of infinite summations of gamma density and survivor functions. Its derivation rested on the risk-neutral pricing theory and the results of Feller. Moreover, descriptions of this model in journal and text-book literature frequently contained errors. One difficulty in implementation of the Black-Scholes model is that one of its arguments is an unobserved parameter σ, the share price volatility. Much research has concentrated on estimating this parameter with a general conclusion that it is better to imply this volatility from observed option prices, than to estimate it from stock price data. In the case of the CEV model there are two unobserved parameters, δ 2, with a relationship to the Black-Scholes parameter, and β, which defines the relationship between share price level and the variance of the instantaneous rate of return on the share. Attempts made to estimate this second parameter in the early 1980 s were not altogether satisfactory, perhaps condemning the CEV model to obscurity. A breakthrough was made in 1989 with a paper by Mark Schroder, who devised a method of evaluating the CEV option prices using the non-central Chi-Squared probability distribution, hence facilitating significantly simpler computation of the prices for those with suitable statistical software 1. 1 I use the statistical package SPLUS extensively in this thesis. iii

6 iv This thesis attempts to summarise the development of the CEV model, with comparisons made to the industry standard, the Black-Scholes model. The elegance of Schroder s method is also made clear. Joint estimation of the two parameters of the CEV model, δ and β, is attempted using both simulated data, and a sample of stocks traded on the Australian Stock Exchange. In this section, it appears that significant improvements can be made to earlier estimation methods. Finally, I would like to note that this thesis is primarily a statistical analysis of a financial topic. As a consequence of this, the focus of my analysis differs to that of articles in the financial journal literature, and that of finance texts. Furthermore, the literature on the CEV model is sparse, and in general theorems therein are stated without proof. Some theorems found in this thesis reflect the statistical nature of the analysis and are hence absent from the financial literature which I have surveyed and referenced. Throughout the thesis, I have attempted to make it clear when an idea or proof follows previously established material. Unattributed material is generally that which I have worked on with my supervisors guidance, but which is not found in the references I have used.

7 Contents 1 Introduction to Options The Call Option An example The Value of Call Options GBM and the Black-Scholes Model Introduction Share Price Evolution Simulation of Geometric Brownian Motion The Future Share Price S T Properties of C T - the Exercise Payoff Mean and Variance of C T given S t Simulation of C T The Black-Scholes Formula Properties of Call Price Prior to Maturity Best Predictor of a Future Black-Scholes Price Graphical Examination of C t Use of the Black-Scholes Model The CEV Model Introduction The CEV Share Price Solution Solution of the Forward Kolmogorov Equation v

8 vi CONTENTS Simulation of CEV Share Prices CEV Share Price Series Graphical Examination of S T Properties of C T - the Exercise Payoff The CEV Option Pricing Formula The CEV Solution Reconciling Various Forms of the CEV Solution Computing the Option Price The Absolute CEV Model Computing the General Model CEV Option Prices Behaviour of CEV Prices Use of the CEV Model Data Analysis Introduction Summary of Alternative Methods Estimating β from a share price series β Estimation Strategy The Variance of ˆβ Appraisal of the Estimation Technique Data Analysis Conclusions 141 A Definitions 143 B Proofs for Selected Results 147 B.1 Result B.2 Result B.3 Result B.4 Result

9 CONTENTS vii C Complete List of Shares 155 D SPLUS Code 157 D.1 GBM Simulation D.2 Inversion of the Black-Scholes Formula D.3 CEV Share Price Simulation D.4 Estimation of β Bibliography 164

10 viii CONTENTS

11 List of Figures 1.1 The lower bounds for call option value, and a possible form for the call price A realisation of GBM with initial value S t = $5, and parameters µ = 0.1, σ = 0.3 and 250 subintervals The daily returns for the series shown in Figure 2.1 with estimated and actual mean, and the estimated mean ± 2 standard deviation limits realisations of S T, a future geometric Brownian motion price, with initial value S t = $5, and parameters τ = 1, µ = 0.1 and σ = 0.3, and the theoretical lognormal distribution for S T The unbroken series is a realisation of GBM with initial value S t = $5 and parameters µ = 0.1, σ = 0.3, and 250 subintervals; the second series is the Black-Scholes prices for this share series added to the present value of the exercise price, with K = $5, and r = 0.06 and maturity at T = 1 year; finally the smooth broken line represents the present value of the exercise price ix

12 x LIST OF FIGURES 2.5 The unbroken series is a realisation of GBM with initial value S t = $5 and parameters µ = 0.1, σ = 0.3, and 250 subintervals; the second series is the Black-Scholes prices for this share series added to the present value of the exercise price, with K = $5, and r = 0.06 and maturity at T = 1 year; finally the smooth broken line represents the present value of the exercise price The distribution of call prices obtained using the Black-Scholes formula on a sample of share prices S t+s, where s = 1 year, S t = $5, µ = 0.1, σ = 0.1, K = $5, r = 0.06 and time to maturity τ s of 5 months. Also shown is the theoretical density function The distribution of call prices obtained using the Black-Scholes formula on a sample of share prices S t+s, where s = 1 year, S t = $5, µ = 0.1, σ = 0.1, K = $5, r = 0.06 and time to maturity τ s ranges from 2 years to half a month. Also the theoretical density function for these prices The distribution of call prices obtained using the Black-Scholes formula on a sample of share prices S t+s, where s = 1 year, S t = $5, µ = 0.1, σ = 0.1, K = $5, r = 0.06 and time to maturity τ s ranges from 2 to 75 years. Also the (solid) theoretical density function for these prices, and the (dotted) lognormal density function of S t+s The relationship between β and P (S T = 0), with S t = $5, τ = 1 year, µ = 0.10, σ = δs β/2 1 t = 0.3 in the solid curve and σ = 0.28 in the broken curve

13 LIST OF FIGURES xi 3.2 Firstly, a typical realisation of GBM, with S t = $5, τ = 1 year, µ = 0.10, σ = 0.3 and n = 250 subintervals; secondly, the daily returns for this series with estimated mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against the estimated standard deviation of the daily returns Firstly, a typical realisation of share price, with S t = $5, τ = 1 year, µ = 0.10, σ = 0.3 and n = 250 subintervals and CEV parameter β = 1; secondly, the daily returns for this series with estimated mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against the estimated standard deviation of the daily returns realisations of S T, a future CEV price with β = 1, with S t = $5, τ = 1 year, µ = 0.1 and σ = 0.3, the (solid) theoretical density for these prices, and the lognormal density function with the same parameters The empirical cumulative distribution function of the 5000 realisations of S T shown in Figure 3.4, a future CEV price with β = 1, with parameters S t = $5, τ = 1, µ = 0.1 and σ = 0.3, and the theoretical distribution function for these prices The standard deviation of a future CEV share price S T, with S t = $5, τ = 1 year, µ = 0.1 and σ = 0.3, for 2 β A realisation of a CEV share price with initial value S t = $5, and parameters β = 1, τ = 1, µ = 0.1, σ = 0.3, and 250 subintervals; also the CEV option prices for this share series added to the present value of the exercise price, with K = $5, and r = 0.06 and time to maturity indicated on the horizontal scale; finally the present value of the exercise price itself

14 xii LIST OF FIGURES 3.8 Black-Scholes implied volatilities for Absolute CEV option prices with S t = $5, τ = 1 year, σ = 0.3, r = 0.06 and $4 K $ Out-of-the-money CEV option prices, with β between -2 and 6, σ between 0.4 and 2, and additional parameters S t = $50, K = $55, τ = 0.5 years, and r = At-the-money CEV option prices, with β between -2 and 6, σ between 0.4 and 2, and additional parameters S t = $50, K = $50, τ = 0.5 years, and r = In-the-money CEV option prices, with β between -2 and 6, σ between 0.4 and 2, and additional parameters S t = $50, K = $45, τ = 0.5 years, and r = The log-likelihood surface, l(β, δ), for a simulated series with S t = $5, τ = 3 years, µ = 0.1, σ = 0.3, n = 250 subintervals per year, and β = The cross-section of the log-likelihood surface in Figure 4.1, l(β, ˆδ) for a simulated series with St = $5, τ = 3 years, µ = 0.1, σ = 0.3, n = 250 subintervals per year, and β = 0. In addition, the line β = ˆβ which identifies the maximum e n, given by Equation (4.14), for the simulated series examined previously with S t = $5, τ = 3 years, µ = 0.1, σ = 0.3, n = 250 subintervals per year, and β = Estimates of β, resulting from CEV share price simulation, used for the figures in Table 4.2. Superimposed on these are the density function of an N( β, 1) random variable over the range of the estimates, where β is the sample average of the β estimates Estimates of s( ˆβ), the standard deviation of ˆβ, from Table 4.2 against the true β

15 LIST OF FIGURES xiii 4.6 Sample values e n, defined in Equation (4.14), for the share price of AMC, with superimposed N(0, 1) density function The BHP share series; in addition, the daily returns for this series with moving mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against standard deviation of the daily returns The MIM share series; in addition, the daily returns for this series with moving mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against standard deviation of the daily returns The BOR share series; in addition, the daily returns for this series with moving mean, and mean ± 2 standard deviation series; thirdly, a plot of the share price level against standard deviation of the daily returns

16 xiv LIST OF FIGURES

17 List of Tables 3.1 A section of MacBeth and Merville s Table 1, of CEV option prices, calculated for the parameters shown, with additional parameters: S t = $50 and r = Beckers δ values (rounded to 4 d.p.) for the Square Root CEV process, found using Equation (3.45), with additional parameters S t = $40, and µ = log(1.05) A section of Beckers Table II, showing Square Root CEV prices, with additional parameters S t = $40, and r = log(1.05) Beckers δ values (rounded to 4 d.p.) for the Absolute CEV process, with additional parameters S t = $40, and µ = log(1.05) MacBeth and Merville s β estimates for six stocks Summary of the β estimates for simulated series, obtained using the maximum likelihood procedure described above, with S t = $5, µ = 0.1, σ = 0.3 and where 3 years data is used p-values for hypothesis test H 0 : β = i, i = 2, 1,..., 6 using the simulated data summarised in Table β estimates for the 44 ASX share series, with estimates of s( ˆβ), the standard deviation of ˆβ, and the resulting confidence intervals obtained by simulation p-values for the test of normality of e n for the 44 ASX share series, where e n is given by Equation (4.14) C.1 ASX Company Codes xv

18 xvi LIST OF TABLES

19 Chapter 1 Introduction to Options 1

20 2 CHAPTER 1. INTRODUCTION TO OPTIONS 1.1 The Call Option A call option gives the holder the opportunity, but not the obligation, to purchase a unit of its underlying asset some time in the future, at a price decided now. If the holder decides to buy the unit of underlying asset, they will exercise the option. The price they pay is called the exercise price. If the option is European the holder may exercise the option only on the exercise or maturity date of the option. American options may be exercised at any time up to and including the maturity date. An option will be either American or European, and it will have the fundamental characteristics: underlying asset, exercise price and maturity date An example In New Zealand, it is possible to buy call options on the shares of Brierley Investments Limited (BIL). These options are traded along with options on other underlying assets on the New Zealand Futures and Options Exchange (NZFOE), information about which can be found on the internet at For example, Investor X could purchase a BIL option traded on the NZFOE, maturing in August 1998, with an exercise price of $1.20. All share options on the NZFOE are American, and so this option allows Investor X to buy 1000 BIL shares for $1200 any time between now and August If Brierley shares trade above $1.20 between now and August, Investor X may choose to exercise the option and receive the thousand shares. Alternatively, X might exercise the option and immediately sell the shares at the current share price (which would be greater then $1.20) for a profit. If the BIL share price does not exceed $1.20 before August, Investor X can (and should) let the option lapse, at no further cost. 1.2 The Value of Call Options Although many traded options are American, and upon exercise yield many shares, for the remainder of this project I will consider only European call

21 1.2. THE VALUE OF CALL OPTIONS 3 options, whose underlying asset is a single share. A call option is a derivative asset, whose value is based on the value of another asset, namely the underlying stock. Since exercise of the call option in the future delivers this share, the price of the option will obviously reflect the present worth of this share and the chances of it being higher than the exercise price on the maturity date. Let me present the following definitions: Definition 1.1. Let S t be the price at time t of a share paying no dividends over the time interval [t, T ]. Definition 1.2. Let C t be the price at time t of a European option. Definition 1.3. Let K be the exercise price of a European option, payable on exercise, for a single share with price S t at time t. Definition 1.4. Let T be the maturity date of the European option, and τ = T t the time until maturity from time t. Definition 1.5. Let r be the risk-free rate, which is payable continuously on an asset whose future value is certain. Definition 1.6. A European option is in-the-money if the current share price S t exceeds K, at-the-money if the current share price is equal to K, and out-of-the-money if the current share price is less than K. On the maturity date T, the European option either expires worthless, or delivers a single share, with value S T, in exchange for a cash payment K. The payoff of the option at maturity can be represented mathematically by the equation: Payoff = C T = { S T K S T > K 0 S T K (1.1) or equivalently C T = max(0, S T K) = (S T K) +.

22 4 CHAPTER 1. INTRODUCTION TO OPTIONS Prior to maturity, at time t, the option will have value not less than zero, since the option cannot yield a negative payoff, and also not less than the current share price S t less the present value of the exercise price discounted at the risk-free rate Ke rτ. This is established in the following theorem, whose proof is standard and can be found in Hull (1997). Theorem 1.1 (Lower Bounds for Call Option Value). C t max(0, S t Ke rτ ) Proof. Since the option yields a non-negative payoff in the future, the price paid now for the opportunity to receive those payoffs must not be less than zero, Hence C t 0. Consider now two portfolios: Portfolio A, consisting of a single option; Portfolio B, consisting of a share and a liability of K, payable at T. At T, the values of portfolios A and B are: V A T = C T = max(0, S T K) V B T = S T K respectively, and so we see V A T V T B A, and hence it follows that Vt V B t. If the latter relationship were not true, arbitrage profits could be earned by selling portfolio B and investing in portfolio A. Now V B t is given by V B t = S t Ke rτ and hence C t S t Ke rτ. (1.2)

23 1.2. THE VALUE OF CALL OPTIONS 5 Option prices given by any option pricing model should obey the bounds given in Equation (1.2), and will have the same general appearance as the function shown in Figure 1.1. Properties of option prices appear in Merton (1973), and in particular, he proves that call prices must be convex in the share price 1. Option Price C t S t Ke rτ 0 0 Ke rτ Share Price Figure 1.1: The lower bounds for call option value, and a possible form for the call price. From the relationship given in Theorem 1.1 it is immediately apparent that the option price must depend on at least four factors: S t, the current share price; τ, the time to maturity of the option; K, the exercise price of the option; and r, the continuously compounding risk-free rate. 1 Merton (1973), Theorem 10, page 150.

24 6 CHAPTER 1. INTRODUCTION TO OPTIONS In addition, the option price will depend on the stochastic properties of S T. The fact that the call price is not equal to the lower bound given in Theorem 1.1 is due to the random nature of the future share price S T, and in particular: σ, the share price volatility. This parameter, which will be defined more formally later, represents the uncertainty of future share prices. associated with a future share price increases. As this parameter increases, the risk The option price may be thought of as the sum of the lower bound and a premium, where the premium is monotonically increasing in σ. Treating S t as fixed, and considering each of τ, K and r in turn, ceteris paribus, we can anticipate the effect a change in each factor might have on the current option price, using the lower bound for the option price given in Theorem 1.1. As the time to maturity τ increases, the present value of the exercise payment at the risk-free rate r diminishes: lim τ Ke rτ = 0, and so the lower bound C t = S t Ke rτ in Figure 1.1 is translated to the left as the y intercept term decreases, and so the option price C t corresponding to any particular S t must increase to compensate. As the exercise payment increases, there is an opposite effect on the lower bound C t = S t Ke rτ. In this case the y intercept term will increase, thus translating the lower bound to the right, and allowing C t to decrease. Increasing the exercise price therefore decreases the value of a European option. Ke rτ As the risk-free rate r increases, the present value of the exercise payment will decrease as it did when τ was increased, and hence the option value will increase. At exercise, the option delivers the payoff (S T K) +. S T is of course a random variable, since at the current time t we do not know for sure what

25 1.2. THE VALUE OF CALL OPTIONS 7 the share price at T will be. In order to make any statements about the probabilistic properties of S T, assumptions must be made on how the share price evolves through time. The evolution of a general process X t may be described by the following stochastic differential equation (SDE): dx t = µ(x t, t)dt + σ(x t, t)db t (1.3) where dx t may be interpreted as the change in X t over the period [t, t + dt], µ(x t, t) and σ(x t, t) are functions of X t and t, and {B t } is Brownian motion, with initial condition B 0 = 0. This process is too general for our purposes, and I will restrict attention to two specific cases, geometric Brownian motion (GBM), and Constant Elasticity of Variance (CEV) evolution. Definition 1.7. A share price that follows geometric Brownian motion (GBM) is a solution to the following SDE: ds t = µs t dt + σs t db t (t > 0) where µ and σ are constant, and B 0 = 0. Definition 1.8. A share price that follows the Constant Elasticity of Variance (CEV) model is a solution to the following SDE: ds t = µs t dt + δs β 2 t db t (t > 0) (1.4) where µ, δ and β are constant, and B 0 = 0. It is clear from Definitions 1.7 and 1.8 that GBM is a special case of the CEV model, and corresponds to the case when β = 2. The solution to Equation (1.4) has very different behaviour for the three cases β = 2, β < 2, and β > 2. In the first case, the solution to the SDE above is geometric Brownian motion, which is a well known and widely studied process. This process, and the price of a call option over a share price following GBM,

26 8 CHAPTER 1. INTRODUCTION TO OPTIONS are described in Chapter 2. When β < 2, the share price is the original Constant Elasticity of Variance process, and is considered, again with its companion option prices, in Chapter 3. The third case, corresponding to β > 2 is mentioned briefly at the end of Chapter 3, and is applied in Chapter 4 along with both other cases.

27 Chapter 2 GBM and the Black-Scholes Model 9

28 10 CHAPTER 2. GBM AND THE BLACK-SCHOLES MODEL 2.1 Introduction Analysis of share prices over time would suggest that they are discrete-time, discrete-variable processes. This means that they change at discrete time points, and may take on only discrete values. In practice, however, share prices are generally modelled using continuous-time, continuous-variable processes. The geometric Brownian motion (GBM) process examined in this chapter, and the Constant Elasticity of Variance (CEV) process considered in the following chapter are continuous-time, continuous-variable processes. In addition, both these models are Markovian, meaning that the only relevant information regarding the future of the process is the present value, and that the past is irrelevant. This can be expressed mathematically as follows: P (S t+s < s S u, 0 u t) = P (S t+s < s S t ). This property is consistent with weak form efficiency in the share market, since it implies that all information in prices S 0, S 1,..., S t 1 is encapsulated in the present price S t. The Black-Scholes Model was first presented in an empirical paper by Black & Scholes (1972), which was quickly followed by its derivation in Black & Scholes (1973). This model is based on a number of restrictive assumptions, one of which is that the price of the underlying asset has a lognormal distribution at the end of any finite (forward) interval, conditional on its value at some initial starting point. It will be shown that if we assume that the share price follows GBM, then this condition will be met. In this chapter I will deal initially with properties of geometric Brownian motion, and then examine the Black-Scholes model. 2.2 Share Price Evolution Derivation of the Black-Scholes option pricing equation requires the assumption that the future share price has a lognormal distribution. This condition

29 2.2. SHARE PRICE EVOLUTION 11 is met if share price follows GBM, a process which is defined in Definition 1.7, and is a solution to the following SDE: ds t = µs t dt + σs t db t (2.1) with initial condition B 0 = 0, and where S t is the share price at time t, µ is the continuously compounding expected growth rate of S t, σ is the standard deviation of the instantaneous return on S t, and {B t } is Brownian motion, with E(dB t ) = 0 and Var(dB t ) = dt. Note that both µ and σ are constants, independent of time and the current share price, and that Equation (2.1) is a special case of Equation (1.4), with β = 2 and δ = σ. I show below that the solution of the SDE above has a lognormal distribution. To prove this it is necessary to use a result from stochastic calculus called Itô s Differentiation Lemma. Result 2.1 (Itô s Differentiation Lemma). Suppose that f(x, t) and its partial derivatives f x, f xx and f t are continuous. If X t is given by dx t = µ(x t, t)dt + σ(x t, t)db t then Y t = f(x t, t) has stochastic differential: dy t = (µ(x t, t)f x + f t σ2 (X t, t)f xx )dt + σ(x t, t)f x db t = f x dx t + f t dt σ2 (X t, t)f xx dt. Hull (1997) gives a sketch proof of this result using a Taylor series expansion. Using Itô s Lemma, we can now prove the following well known theorems. Theorem 2.1. Conditional on its value at time t, a share price S T that follows GBM will have a lognormal distribution at T > t, with parameters E(ln S T ) = ln S t + (µ 1 2 σ2 )τ and Var(ln S T ) = σ 2 τ, where τ = T t. Moreover, the form for S T given S t will be: S T = S t e (µ 1 2 σ2 )τ+σ τz where Z is a standard normal variable with mean zero and unit variance.

30 12 CHAPTER 2. GBM AND THE BLACK-SCHOLES MODEL Proof. Consider the transformation Y t = ln S t. Referring to the SDE (2.1), and to Result 2.1, we see that: and so applying Itô s Lemma d ln S t = ln S t t X t = S t µ(x t, t) = µs t σ(x t, t) = σs t f(x t, t) = ln S t dt + ln S t S t ds t + 1(σS 2 t) 2 2 ln S t 2 S dt t = 0 dt + 1 S t (µs t dt + σs t db t ) 1 2 σ2 S 2 t = ( µ 1 2 σ2) dt + σdb t (2.2) Thus, integrating both sides from t to T yields: ln S T ln S t = ( µ 1 2 σ2) (T t) + σ T = ( µ 1 2 σ2) τ + σ(b T B t ) From the properties of Brownian motion, in particular its independent increments: B T B t B T t B 0 = B τ t 1 S 2 t db t since B 0 = 0, and where denotes is distributed as. Hence: ln S T ln S t + ( µ 1 2 σ2) τ + σb τ. (2.3) Conditioned on S t, it is clear that the only random variable in the RHS of the above equation is the Brownian motion term B τ. Using the fact that B t N(0, t) for any t > 0, we obtain the mean and variance of ln S T given S t : E(ln S T S t ) = E(ln S t + ( µ 1 2 σ2) τ + σb τ S t ) = ln S t + ( µ 1 2 σ2) τ + σe(b τ ) = ln S t + ( µ 1 2 σ2) τ dt

31 2.2. SHARE PRICE EVOLUTION 13 Var(ln S T S t ) = Var(ln S t + ( µ 1 2 σ2) τ + σb τ S t ) = σ 2 Var(B τ ) = σ 2 τ In addition, if X is a normal random variable, then the linear combination a + bx, where a and b are both constant, is also normal with mean ae(x) and variance b 2 Var(X). Combining this with the mean and variance above it is clear that ln S T S t N ( ln S t + ( µ 1 2 σ2) τ, σ 2 τ ) (2.4) i.e. S T is lognormal with parameters ln S t + ( µ 1 2 σ2) τ and σ 2 τ. Moreover, since Z = Bτ τ N(0, 1), it follows directly from Equation (2.3) that S T = S t e (µ 1 2 σ2 )τ+σ τz (2.5) as required. Theorem 2.2 (Moments of a Share Price following GBM). The mean and variance of the share price S T conditional on an earlier share price S t, 0 t < T are S t e µτ and S 2 t e 2µτ (e σ2τ 1) respectively. Proof. Note first that the moment generating function of an N(0, 1) variable is E(e sz ) = e 1 2 s2. This can be used to find the moments of the random variable of interest S T S t. ( ) E(S T S t ) = E S t e (µ 1 2 σ2 )τ+σ τz S t ( ) = S t e (µ 1 2 σ2 )τ E e σ τz = S t e (µ 1 2 σ2 )τ e 1 2 σ2 τ = S t e µτ (2.6) ( ) E(ST 2 S t ) = E St 2 e 2(µ 1 2 σ2 )τ+2σ τz S t ( ) = St 2 e 2(µ 1 2 σ2 )τ E e 2σ τz = S 2 t e 2(µ 1 2 σ2 )τ e 2σ2 τ = S 2 t e (2µ+σ2 )τ

32 14 CHAPTER 2. GBM AND THE BLACK-SCHOLES MODEL Forming the variance using the relationship Var(X) = E(X 2 ) E(X) 2, we obtain as required. Var(S T S t ) = S 2 t e 2µτ (e σ2 1) (2.7) Note that if share price evolution is a deterministic process, i.e. σ = 0, then S T equals the mean of the stochastic process: S t e µτ. Thus using either the expected value, or the deterministic case, the parameter µ can be thought of as the continuously compounding expected rate of return on the share per unit time, and may be modelled for a particular stock using the Capital Asset Pricing Model (CAPM), which features both a market risk premium, and a measure of the riskiness of the particular firm compared to the market. The CAPM describes the expected return on a particular asset: µ j = r + MRP Cov(R j, R m ) σ 2 m where MRP is the market risk premium, R j is the return on asset j, µ j is the expected value of that return, R m is the return on the market portfolio, and σ 2 m is the variance of the market return. Aggregate investor attitudes towards risk affect the size of the market risk premium, which has a direct effect on the size of µ j. The size of this effect is determined by the measure of systematic risk: Cov(R j, R m ) σm 2 which compares the risk associated with the particular asset to the risk associated with the market as a whole. The parameter µ is the only component of the GBM model that reflects investor risk attitudes. The return on an asset j can be modelled using the standard univariate linear regression model: R j = α j + b j R m + ɛ j

33 2.2. SHARE PRICE EVOLUTION 15 where R j and R m are as above, α j and b j are the regression model parameters, and ɛ j is a stochastic error term, with variance σ 2 ɛ j. Applying the variance operator to all terms in the equation above yields the relationship: σ 2 j = b 2 jσ 2 m + σ 2 ɛ j. The variance of the market return, σ 2 m measures the systematic risk in the system, whereas σ 2 ɛ j measures the non-systematic risk. Hence the parameter σ reflects a combination of these, but not investor risk attitudes Simulation of Geometric Brownian Motion Simulation of GBM series, or prices at a particular time in the future is a useful way of analysing the properties of GBM, and later, analysing the properties of Black-Scholes option prices. Because the solution to the SDE (2.1) is known, there is no need to use a numerical method to approximate the solution, but rather the solution can be simulated directly. The solution to the SDE was given in Equation (2.5) and is: S T = S t e (µ 1 2 σ2 )τ+σ τz where Z N(0, 1). Hence in order to simulate a single price at T, a single realisation of Z can be obtained, and the share price computed directly. In order to simulate an entire GBM series, we can divide the interval of interest [t, T ] into n subintervals defined by the times: t = t 0 < t 1 < < t i < < t n 1 < t n = T where the intervals are not necessarily of equal length. Equation (2.5) can also be written to give the share price at time t i conditional on the share price at t i 1 : S ti = S ti 1 e (µ 1 2 σ2 )(t i t i 1 )+σ t i t i 1 Z

34 16 CHAPTER 2. GBM AND THE BLACK-SCHOLES MODEL hence simulating n realisations of Z, the series can be constructed as follows: i S ti = S t S tj S j=1 tj 1 i = S t e (µ 1 2 σ2 )(t j t j 1 )+σ t j t j 1 Z j j=1 ( i = S t exp (µ 1 2 σ2 )(t j t j 1 ) + σ ) t j t j 1 Z j j=1 ( = S t e (µ 1 2 σ2 )(t i t) exp σ ) i tj t j 1 Z j. (2.8) j=1 A program which simulates GBM using Equation (2.8) is given in Appendix D.1. Share Price Time - years Figure 2.1: A realisation of GBM with initial value S t = $5, and parameters µ = 0.1, σ = 0.3 and 250 subintervals. Figure 2.1 shows a single realisation of geometric Brownian motion with S t = $5, τ = 1 year, µ = 0.1 and σ = 0.3. This particular realisation has

35 2.2. SHARE PRICE EVOLUTION 17 S T = $5.12, which is smaller than its expected value E(S T S t ) = $5e 0.1 = $5.53 but well within a single standard deviation s(s T S t ) = S t e µ e σ2 1 = $1.696 of it. The daily returns for the series {S t } are defined as: R t = ln S t+ t ln S t where t = 1 years is approximately one trading day. Whilst the series 250 {S t } is clearly not a stationary process, Equation (2.2) indicates that the daily returns should be stationary, with a mean (µ 1 2 σ2 ) t and variance σ 2 t. A plot of the daily returns can be seen in Figure 2.2, with an estimate of the mean level and standard deviation shown 1. These have been estimated using a Lowess filter with a smoothing window of 30 observations. The Lowess filter is a robust, centred moving average filter, and was derived by Cleveland (1979). This filter was designed to smooth scatterplots, but has application to the equi-spaced observations of a time series. It estimates the average value at t i using weights from the bisquare function: { (1 x 2 ) 2 x < 1 B(x) = 0 x 1 where x depends on the time t i and the smoothing window, and by making adjustments for outlying values. In this case I have used a smoothing window of 30 days, so that values outside this window are given zero weight, and hence do not affect the estimate. The estimates produced at the ends of the series are unreliable due to back- and forecasting of share price values. These estimates, obtained using estimated share price data, are not shown in the graph. Also shown in Figure 2.2 is the true mean for the daily returns of (µ 1 2 σ2 ) t which is negligible for the parameters chosen 2. We see that the estimated mean does indeed oscillate about the actual mean level, and that 1 The series actually shown are the estimated mean, and the estimated mean ± two estimated standard deviations, from which the estimated standard deviation itself can be recovered. 2 Nevertheless, over a year it compounds to a significant amount.

36 18 CHAPTER 2. GBM AND THE BLACK-SCHOLES MODEL Log-returns Time - years Figure 2.2: The daily returns for the series shown in Figure 2.1 with estimated and actual mean, and the estimated mean ± 2 standard deviation limits. the estimated standard deviation appears approximately constant. In fact, an autocorrelation plot for the daily return series shows no significant autocorrelation for non-zero lags. This is not surprising, since the series was generated in the first place using realisations of the standard normal variable that should indeed be independent of one another, and hence uncorrelated The Future Share Price S T As discussed in Chapter 1, the value of a European option at t will certainly depend on the share price at the future exercise time T. Since we are able to simulate geometric Brownian motion, we can also simulate the distribution of future share prices by generating many realisations of the same process. The value of this is largely illustrative, since many of the properties of GBM are well known. Suppose a particular share price truly follows GBM, with known µ and σ, then N realisations of the share price at T can be obtained

37 2.2. SHARE PRICE EVOLUTION 19 using the second program seen in Appendix D.1. Simulation of the quantity S T is a useful way of deciding what properties the share price will have at T, and hence what payoff (if any) the option is likely to deliver. In the case of GBM, the theoretical distribution is known, and in this case can be compared to the results of the simulation to appraise the simulation procedure, and help guide the eye. Figure 2.3 shows the result of a simulation of 5000 share prices τ = 1 year in the future, with additional parameters S t = $5, µ = 0.1 and σ = 0.3. The final value of the time series in Figure 2.1 has the same properties as each of the observations shown in the histogram. Superimposed on the observed distribution is the theoretical lognormal density curve with parameters E(ln S T S t ) = ln 5 + ( ) and Var(ln S T S t ) = It is clear from the graph that the fit is very good, particularly when the bars are small. This is expected since the height of frequency histogram bars is a Poisson random variable with mean and variance equal to the expected height of the bars. Hence, the smaller bars heights will have a small standard deviation and should be closer to the curve than the higher bars. The heights of the equi-width bars in the relative frequency histogram shown in Figure 2.3 are proportional to the Poisson heights in a frequency histogram, and so the variability comments hold. A useful means of gauging the success of the simulation procedure is to estimate the mean and variance of a sample of share prices, and compare these to the theoretical values given by Equations (2.6) and (2.7) respectively. These estimates are given by the sample mean and sample variance of and respectively, compared to theoretical values of and Note that while the means differ slightly, the variance estimate is accurate to within 4 decimal places, again testimony to the accuracy with which GBM can be simulated. Note that these estimates do not correspond to the parameters of the lognormal distribution, which are the mean and variance of the log share prices. The maximum likelihood method could be used to fit the best lognormal distribution to the sample values but this has not been done here.

38 20 CHAPTER 2. GBM AND THE BLACK-SCHOLES MODEL Relative Frequency Share Price Figure 2.3: 5000 realisations of S T, a future geometric Brownian motion price, with initial value S t = $5, and parameters τ = 1, µ = 0.1 and σ = 0.3, and the theoretical lognormal distribution for S T. 2.3 Properties of C T - the Exercise Payoff The properties of the call value at maturity are intimately linked to those of the share price by the equation: C T = max(s T K, 0) = (S T K) + (2.9) where K is the exercise price of the call option. It is this payoff that investors are interested in valuing. Such a future cash flow could be valued using the equation P t = e λτ E(C T S t ) (2.10) where P t would be the price paid now for the payoff C T, which would be received at time T. This equation features two rates particular to the risk

39 2.3. PROPERTIES OF C T - THE EXERCISE PAYOFF 21 preferences of investors in aggregate: µ, the continuously compounded expected rate of return which features in the expectation, and λ, the discount rate for future cash flows. It will be shown later that both µ and λ can be treated as if they were the risk-free rate r, for valuing this particular future cash flow. Consider the simple transformation Y T = S T K. It is clear Y T has mean E(S T ) K, variance Var(S T ), and Y T +K has a lognormal distribution. Note that Y T itself does not have a lognormal distribution, since the lognormal distribution is defined on the range [0, ) whereas Y T is defined on [ K, ). However, the shape of the distribution of Y T will be identical to that of S T, since we have the relationship P (Y T < y) = P (S T K < y) = P (S T < y + K). Next consider C T = Y + T = max(0, Y T ). It is clear that P (C T = 0) = P (S T K) and so C T will not have a continuous distribution function. Theorem 2.3 (Distribution of C T given S t ). Given S t, C T = (S T K) + has a mixed distribution with density: P (C T c S t ) = f CT S t (c, τ) = F ST S t (K, τ) δ(c) + { 0 c < 0 F ST S t (K + c, τ) c 0 { 0 c < 0 f ST S t (K + c, τ) c > 0 where F ST S t and f ST S t are the cumulative distribution and density functions for the share price S T, and δ(c) is the Dirac delta function. Proof. Since S T given S t has a lognormal distribution with parameters ln S t + (µ 1 2 σ2 )τ and σ 2 τ, given S t : P (S T K c S t ) = P (S T K + c S t ) = P (ln S T ln(k + c) S t ) ( ln(k + c) ln St (µ 1 ) 2 = Φ σ2 )τ σ τ

40 22 CHAPTER 2. GBM AND THE BLACK-SCHOLES MODEL where Φ(x) = x φ(z)dz is the standard normal cumulative distribution function, and φ(z) is the standard normal probability density function. Thus: F CT S t (c, τ) = P (C T c S t ) { 0 c < 0 = P (S T K c S t ) c 0 { 0 c < 0 = ( ) ln(k+c) ln St (µ Φ 1 2 σ2 )τ c 0 σ τ (2.11) 0 c < 0 f CT S t (c, τ) = P (C ( T = 0 S t ) δ(c) ) c = 0 Φ ln(k+c) ln St (µ 1 2 σ2 )τ c σ c > 0 τ 0 c < 0 = P (S T K S ( t ) δ(c) ) c = 0 1 (K+c)σ φ ln(k+c) ln St (µ 1 2 σ2 )τ τ σ c > 0 τ 0 c < 0 = F ST S t (K, τ) δ(c) c = 0 (2.12) f ST S t (K + c, τ) c > 0 where F ST S t and f ST S t are the cumulative distribution and density functions for the terminal share price S T given by: F ST S t (s, τ) = P (S T < s S t ) ( ln s ln St (µ 1 ) 2 = Φ σ2 )τ σ τ f ST S t (s, τ) = F S T S t (s, τ) s = 1 ( ln s ln sσ τ φ St (µ 1 ) 2 σ2 )τ σ τ and δ(c) is the Dirac delta function defined by: { x 0 x < 0 δ(c)dc = 1 x 0. (2.13) (2.14)

41 2.3. PROPERTIES OF C T - THE EXERCISE PAYOFF Mean and Variance of C T given S t The mean and variance of C T given S t can be found using the density function for C T given in Equation (2.12). An equivalent but simpler method is to note that C T = h(z), where Z is a standard normal random variable. Then the expected value of any function, g, of C T can be found using the relationship E{(g h)(z)} = (g h)(z)φ(z)dz where g h is the composition of g with h. This yields the following. Theorem 2.4 (Moments of C T ). Given S t, C T = (S T K) + has mean E(C T S t ) = S t e µτ Φ(g t ) KΦ(g t σ τ) and mean square E(C 2 T S t ) = S 2 t e (2µ+σ2 )τ Φ(g t + σ τ) 2S t e µτ KΦ(g t ) + K 2 Φ(g t σ τ) where g t = ln S t ln K + (µ σ2 )τ σ τ Proof. Note firstly that from Equation (2.5), it is clear that S T is a function of a standard normal variable Z. Hence, we find that C T too is a function of Z. Note that S T K implies C T = (S T K) + { 0 S T K = S t e (µ 1 2 σ2 )τ+σ τz K S T > K Z ln S t + ln K (µ 1 2 σ2 )τ σ τ = g t + σ τ

42 24 CHAPTER 2. GBM AND THE BLACK-SCHOLES MODEL and therefore the first moment can be determined thus: E(C T S t ) = g t+σ τ ( S t e (µ 1 2 σ2 )τ+σ ) τz K φ(z)dz = S t e µτ e 1 2 (z σ τ) 2 dz K φ(z)dz g t+σ τ 2π g t+σ τ = S t e µτ φ(y)dy KΦ(g t σ τ) g t = S t e µτ Φ(g t ) KΦ(g t σ τ). The second moment gives us a means of calculating the variance of C T given S t, and is found as follows: E(C 2 T S t ) = g t+σ τ = S 2 t e (2µ+σ2 )τ = S 2 t e (2µ+σ2 )τ ( S t e (µ 1 2 σ2 )τ+σ τz K) 2 φ(z)dz g t+σ τ g t σ τ e 1 2 (z 2σ τ) 2 2π dz 2S t e µτ KΦ(g t ) + K 2 Φ(g t σ τ) φ(y)dy 2S t e µτ KΦ(g t ) + K 2 Φ(g t σ τ) = S 2 t e (2µ+σ2 )τ Φ(g t + σ τ) 2S t e µτ KΦ(g t ) + K 2 Φ(g t σ τ). The variance can be formed as usual using Var(C T S t ) = E(CT 2 S t) E(C T S t ) 2. Thus, from Equation (2.10), an investor may value the option using the expected value, and the market rates µ and λ, to give a price at t: P t = S t e (µ λ)τ Φ(g t ) Ke λτ Φ(g t σ τ). (2.15) where g t is defined in Theorem 2.4 above. Note that as aggregate investor risk attitudes change, both µ and λ will change, and hence it appears that P t will change. However, there is theory to show that the true value of C t should be independent of risk premia on assets, thus any change in µ is offset by an appropriate change in λ when the fair option price is calculated.

43 2.4. THE BLACK-SCHOLES FORMULA Simulation of C T A sample of terminal call values C T can be obtained from a sample of share prices using the simple relationship (2.9). The distribution of such a sample can easily be recovered from the density function of S T shown in Figure 2.3 by repositioning the origin at K, and setting all observations less than K to zero. As before, in the case of C T there is also good agreement between the theoretical and observed mean and variance. For an exercise price of K = $5, the sample yields respective values of and for the estimated mean and variance, compared to theoretical values of and It is also interesting to note the relative frequency of the event C T = 0, i.e. the proportion of occasions on which the option would not be exercised. The theoretical probability that the value of C T will be zero is given by the equation P (C T = 0 S t ) = P (S T < K S t ) = Φ( g t + σ τ) which for the simulation equals Hence, for this particular simulation, the expected number of options that are not exercised is This can be compared to the sample estimate of The Black-Scholes Formula Black and Scholes first presented the Black-Scholes model in an empirical paper (Black & Scholes 1972) with the theoretical underpinnings following in Black & Scholes (1973). Their model considers pricing a European call option, over a stock traded in a market with the following properties: the instantaneous interest rate is known, and constant; the share price follows geometric Brownian motion, from which it follows directly:

44 26 CHAPTER 2. GBM AND THE BLACK-SCHOLES MODEL the variance rate σ 2 of the return on the stock is constant, and known; no dividends are paid on the share over the life of the option; there are no transaction costs, differential taxes, or short-selling restrictions, and it is possible to trade any fraction of the stock or option. Under these conditions Black & Scholes were able to obtain a price for the option that depends only on the current share price, the time to maturity, and on constants K, r, and σ that are assumed known. The partial differential equation (PDE) for the price of a call over a slightly more general share price process than GBM is established in the following well known theorem. Theorem 2.5 (The Call Price PDE). Suppose S t is the solution to the SDE: ds t = µ(s t, t)s t dt + σ(s t, t)s t db t (2.16) then the price at time t of a call option over a share with price S t must satisfy the PDE: 1 2 σ2 (S t, t)st 2 2 C S + rs C 2 t S + C t rc t = 0. subject to the boundary condition C T = (S T K) + where r is the continuously compounding risk-free rate. Proof. Consider forming a self-financing portfolio at t, of a t units of the stock with value S t, and b t units of the call option with value C t, where a t and b t may be functions of both share price at t and time. This portfolio has value at t: V t = a t S t + b t C t

45 2.4. THE BLACK-SCHOLES FORMULA 27 and over the period [t, t + dt] the change in portfolio value will be: dv t = V t+dt V t = a t+dt S t+dt + b t+dt C t+dt a t S t b t C t = (a t+dt a t )S t+dt + a t ds t + (b t+dt b t )C t+dt + b t dc t = a t ds t + b t dc t. The final step is justified by the assumption that the portfolio is self financing. This means that any change in the quantity of stock held is financed by a change in the quantity of the option held, and vice versa. This yields da t S t+dt + db t C t+dt = 0 as required. The form for ds t is given by Equation (2.16), and dc t can be obtained from it using Itô s Lemma. The option price can be considered a function of two variables: the current share price and time. Hence we can write C t = C(S t, t) where S t is a solution to the familiar SDE ds t = µ(s t, t)s t dt + σ(s t, t)s t db t. Referring to Result 2.1, we see that X t = S t µ(x t, t) = µ(s t, t)s t σ(x t, t) = σ(s t, t)s t f(x t, t) = C(S t, t). Hence, applying Itô s Lemma we can determine the change in C t over the period [t, t + dt]: dc t = C C dt + t S ds t + 1(σ(S 2 t, t)s t ) 2 2 C S dt. 2

46 28 CHAPTER 2. GBM AND THE BLACK-SCHOLES MODEL The change in portfolio value over the interval [t, t + dt] thus becomes: dv t = a t ds t + b t dc t ( C = a t ds t + b t t ( C = a t + b t S and the portfolio s rate of return: dv t V t = C dt + S ds t σ2 (S t, t)st 2 2 C ) ds t + b t ( C t σ2 (S t, t)s 2 t ) S dt 2 ) 2 C dt S 2 = a tds t + b t dc t a t S t + b t C t ( ) ( ) C C at + b t dst + b S t + 1 t 2 σ2 (S t, t)st 2 2 C dt S 2 = where f t = a t /b t. ( ft + C S ) dst + a t S t + b t C ( t ) C + 1 t 2 σ2 (S t, t)st 2 2 C dt S 2 f t S t + C t Since the only stochastic elements are present in the ds t term, these can be eliminated by choice of f t to form a portfolio whose change in value over the period [t, t + dt] is deterministic. Setting the coefficient of ds t to zero gives dv t V t = f t = C S and so Var( dvt V t ) = 0 and the rate of return on the portfolio becomes: ( ) C + 1 t 2 σ2 (S t, t)st 2 2 C dt S 2 C S S t + C t. (2.17) By this choice of f t, the number of units of stock held per unit of option held, the portfolio has no risk over [t, t + dt], and so by arbitrage theory its rate of return should be the risk free rate r, giving dv t = rv t dt or dv t rv t dt = 0. If this were not the case, investors could borrow (lend) at the risk-free rate r, and go long (short) in the portfolio of stock and option and earn arbitrage

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