The Constant Elasticity of Variance Option Pricing Model

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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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Acknowledgements

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

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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 in-finite 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 con-tained 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, δ2, with a relationship to the

Black-Scholes parameter, andβ, 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 software1.

1I use the statistical package SPLUS extensively in this thesis.

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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 Ex-change. 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.

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Contents

1 Introduction to Options 1

1.1 The Call Option . . . 2

1.1.1 An example . . . 2

1.2 The Value of Call Options . . . 2

2 GBM and the Black-Scholes Model 9 2.1 Introduction . . . 10

2.2 Share Price Evolution . . . 10

2.2.1 Simulation of Geometric Brownian Motion . . . 15

2.2.2 The Future Share PriceST . . . 18

2.3 Properties ofCT - the Exercise Payoff . . . 20

2.3.1 Mean and Variance ofCT given St . . . 23

2.3.2 Simulation of CT . . . 25

2.4 The Black-Scholes Formula . . . 25

2.4.1 Properties of Call Price Prior to Maturity . . . 32

2.4.2 Best Predictor of a Future Black-Scholes Price . . . 34

2.4.3 Graphical Examination of Ct. . . 37

2.5 Use of the Black-Scholes Model . . . 42

3 The CEV Model 47 3.1 Introduction . . . 48

3.2 The CEV Share Price Solution . . . 50

3.2.1 Solution of the Forward Kolmogorov Equation . . . 54 v

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3.2.2 Simulation of CEV Share Prices . . . 62

3.2.3 CEV Share Price Series . . . 66

3.2.4 Graphical Examination of ST . . . 71

3.3 Properties of CT - the Exercise Payoff . . . 77

3.4 The CEV Option Pricing Formula . . . 78

3.4.1 The CEV Solution . . . 81

3.4.2 Reconciling Various Forms of the CEV Solution . . . . 84

3.5 Computing the Option Price . . . 85

3.5.1 The Absolute CEV Model . . . 85

3.5.2 Computing the General Model . . . 86

3.5.3 CEV Option Prices . . . 91

3.5.4 Behaviour of CEV Prices . . . 96

3.6 Use of the CEV Model . . . 98

4 Data Analysis 105 4.1 Introduction . . . 106

4.1.1 Summary of Alternative Methods . . . 107

4.2 Estimating β from a share price series . . . 113

4.2.1 β Estimation Strategy . . . 114

4.2.2 The Variance of ˆβ . . . 120

4.3 Appraisal of the Estimation Technique . . . 122

4.4 Data Analysis . . . 129

5 Conclusions 141 A Definitions 143 B Proofs for Selected Results 147 B.1 Result 2.2 . . . 147

B.2 Result 3.2 . . . 149

B.3 Result 3.4 . . . 150

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C Complete List of Shares 155

D SPLUS Code 157

D.1 GBM Simulation . . . 157

D.2 Inversion of the Black-Scholes Formula . . . 158

D.3 CEV Share Price Simulation . . . 160

D.4 Estimation of β . . . 161

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List of Figures

1.1 The lower bounds for call option value, and a possible form for the call price. . . 5

2.1 A realisation of GBM with initial valueSt = $5, and parame-ters µ= 0.1,σ = 0.3 and 250 subintervals. . . 16 2.2 The daily returns for the series shown in Figure 2.1 with

esti-mated and actual mean, and the estiesti-mated mean±2 standard deviation limits. . . 18 2.3 5000 realisations of ST, a future geometric Brownian motion

price, with initial value St = $5, and parameters τ = 1, µ = 0.1 andσ = 0.3, and the theoretical lognormal distribution for

ST. . . 20 2.4 The unbroken series is a realisation of GBM with initial value

St = $5 and parameters µ = 0.1, σ = 0.3, and 250 subin-tervals; 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; fi-nally the smooth broken line represents the present value of the exercise price. . . 39

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2.5 The unbroken series is a realisation of GBM with initial value

St = $5 and parameters µ = 0.1, σ = 0.3, and 250 subin-tervals; 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; fi-nally the smooth broken line represents the present value of the exercise price. . . 40 2.6 The distribution of call prices obtained using the Black-Scholes

formula on a sample of share prices St+s, where s = 1 year,

St = $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. . . 42 2.7 The distribution of call prices obtained using the Black-Scholes

formula on a sample of share prices St+s, where s = 1 year,

St = $5, µ = 0.1, σ = 0.1, K = $5, r = 0.06 and time to maturity τs ranges from 2 years to half a month. Also the theoretical density function for these prices. . . 45 2.8 The distribution of call prices obtained using the Black-Scholes

formula on a sample of share prices St+s, where s = 1 year,

St = $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 St+s. . . 46 3.1 The relationship between β and P(ST = 0), with St = $5,

τ = 1 year, µ= 0.10, σ =δStβ/2−1 = 0.3 in the solid curve and

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3.2 Firstly, a typical realisation of GBM, withSt = $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. . . 69 3.3 Firstly, a typical realisation of share price, withSt= $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. . . 70 3.4 5000 realisations of ST, a future CEV price with β = −1,

with St = $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. . . 74 3.5 The empirical cumulative distribution function of the 5000

realisations of ST shown in Figure 3.4, a future CEV price with β = 1, with parameters St = $5, τ = 1, µ = 0.1 and

σ = 0.3, and the theoretical distribution function for these prices. . . 75 3.6 The standard deviation of a future CEV share price ST, with

St = $5, τ = 1 year, µ= 0.1 and σ = 0.3, for −2≤β ≤2. . . . 76 3.7 A realisation of a CEV share price with initial value St = $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, withK = $5, andr= 0.06 and time to maturity indicated on the horizontal scale; finally the present value of the exercise price itself. . . . 92

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3.8 Black-Scholes implied volatilities for Absolute CEV option prices with St = $5, τ = 1 year, σ = 0.3, r = 0.06 and $4K $6. . . 99 3.9 Out-of-the-money CEV option prices, with β between -2 and

6, σ between 0.4 and 2, and additional parameters St = $50,

K = $55,τ = 0.5 years, and r= 0.06. . . 101 3.10 At-the-money CEV option prices, with β between -2 and 6,

σ between 0.4 and 2, and additional parameters St = $50,

K = $50,τ = 0.5 years, and r= 0.06. . . 102 3.11 In-the-money CEV option prices, with β between -2 and 6,

σ between 0.4 and 2, and additional parameters St = $50,

K = $45,τ = 0.5 years, and r= 0.06. . . 103 4.1 The log-likelihood surface, ¯l(β, δ), for a simulated series with

St = $5, τ = 3 years, µ= 0.1, σ = 0.3, n = 250 subintervals per year, and β = 0. . . 119 4.2 The cross-section of the log-likelihood surface in Figure 4.1,

¯

l(β,ˆδ) for a simulated series withSt = $5,τ = 3 years,µ= 0.1,

σ = 0.3,n = 250 subintervals per year, andβ = 0. In addition, the line β = ˆβ which identifies the maximum. . . 119 4.3 en, given by Equation (4.14), for the simulated series examined

previously with St = $5, τ = 3 years, µ = 0.1, σ = 0.3,

n = 250 subintervals per year, and β = 0. . . 123 4.4 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 theβ

estimates. . . 124 4.5 Estimates ofs( ˆβ), the standard deviation of ˆβ, from Table 4.2

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4.6 Sample values en, defined in Equation (4.14), for the share price of AMC, with superimposedN(0,1) density function. . 133 4.7 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. . . 137 4.8 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. . . 138 4.9 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. . . 139

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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: St= $50 and r = 0.06. . . 93 3.2 Beckers’ δ values (rounded to 4 d.p.) for the Square Root

CEV process, found using Equation (3.45), with additional parametersSt = $40, andµ= log(1.05). . . 94 3.3 A section of Beckers’ Table II, showing Square Root CEV

prices, with additional parameters St= $40, and r= log(1.05). 95 3.4 Beckers’ δ values (rounded to 4 d.p.) for the Absolute CEV

process, with additional parametersSt= $40, andµ= log(1.05). 96 4.1 MacBeth and Merville’sβ estimates for six stocks. . . 110 4.2 Summary of theβ estimates for simulated series, obtained

us-ing the maximum likelihood procedure described above, with

St = $5, µ= 0.1, σ = 0.3 and where 3 years’ data is used. . . . 126 4.3 p-values for hypothesis test H0 : β = i, i = −2,−1, . . . ,6

using the simulated data summarised in Table 4.2. . . 127 4.4 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. . . 130 4.5 p-values for the test of normality of en for the 44 ASX share

series, whereen is given by Equation (4.14). . . 132 C.1 ASX Company Codes . . . 155

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Chapter 1

Introduction to Options

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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

willexercisethe option. The price they pay is called theexercise price. If the

option isEuropeanthe holder may exercise the option only on theexerciseor

maturitydate of the option. Americanoptions 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.

1.1.1

An example

In New Zealand, it is possible to buy call options on the shares of Brier-ley Investments Limited (BIL). These options are traded along with options on other underlying assets on the New Zealand Futures and Options Ex-change (NZFOE), information about which can be found on the internet at

http://www.nzfoe.co.nz/. 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 1998. If Brierley shares trade above $1.20 between now and August, Investor X may choose to exercise the option and receive the thou-sand 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

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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 St be the price at time t of a share paying no dividends

over the time interval [t, T].

Definition 1.2. Let Ct 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 St 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 St 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 valueST, in exchange for a cash paymentK. The payoff of the option at maturity can be represented mathematically by the equation:

Payoff =CT =

(

ST −K ST > K 0 ST ≤K

(1.1) or equivalently CT = max(0, ST −K) = (ST −K)+.

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Prior to maturity, at timet, 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 St 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).

Ct ≥max(0, St−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 Ct ≥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:

VTA=CT = max(0, ST −K)

VTB =ST −K respectively, and so we see VA

T ≥ VTB, and hence it follows that VtA ≥ VtB. If the latter relationship were not true, arbitrage profits could be earned by selling portfolio B and investing in portfolioA. Now VB

t is given by

VtB =St−Ke−rτ and hence

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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 price1.

Share Price

Option Price

0

0 K e−rτ

Ct St−K e−rτ

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:

• St, 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.

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In addition, the option price will depend on the stochastic properties ofST. 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 priceST, and in particular:

• σ, the share price volatility.

This parameter, which will be defined more formally later, represents the uncertainty of future share prices. As this parameter increases, the risk associated with a future share price increases. The option price may be thought of as the sum of the lower bound and a premium, where the premium is monotonically increasing inσ.

Treating St 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 rater diminishes:

lim τ→∞Ke

−rτ = 0,

and so the lower boundCt=St−Ke−rτ in Figure 1.1 is translated to the left as the y intercept term decreases, and so the option price Ct corresponding to any particularSt must increase to compensate.

As the exercise payment increases, there is an opposite effect on the lower bound Ct = St − Ke−rτ. In this case the y intercept term will increase, thus translating the lower bound to the right, and allowing Ct to decrease. Increasing the exercise price therefore decreases the value of a European option.

As the risk-free raterincreases, the present value of the exercise payment

Ke−rτ will decrease as it did when τ was increased, and hence the option value will increase.

At exercise, the option delivers the payoff (ST −K)+. ST is of course a random variable, since at the current time t we do not know for sure what

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the share price at T will be. In order to make any statements about the probabilistic properties of ST, assumptions must be made on how the share price evolves through time.

The evolution of a general process Xt may be described by the following stochastic differential equation (SDE):

dXt=µ(Xt, t)dt+σ(Xt, t)dBt (1.3) wheredXt may be interpreted as the change in Xtover the period [t, t+dt],

µ(Xt, t) andσ(Xt, t) are functions ofXtandt, and{Bt}is Brownian motion, with initial condition B0 = 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:

dSt =µStdt+σStdBt (t >0)

where µ and σ are constant, and B0 = 0.

Definition 1.8. A share price that follows the Constant Elasticity of Vari-ance (CEV) model is a solution to the following SDE:

dSt=µStdt+δS

β

2

t dBt (t >0) (1.4)

where µ, δ and β are constant, and B0 = 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 casesβ = 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,

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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.

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Chapter 2

GBM and the Black-Scholes

Model

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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 pro-cesses. 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 rele-vant 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(St+s< s|Su,0≤u≤t) =P(St+s < s|St).

This property is consistent with weak form efficiency in the share market, since it implies that all information in pricesS0, S1, . . . , St−1 is encapsulated

in the present priceSt.

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 assump-tions, 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 assump-tion that the future share price has a lognormal distribuassump-tion. This condiassump-tion

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is met if share price follows GBM, a process which is defined in Definition 1.7, and is a solution to the following SDE:

dSt=µStdt+σStdBt (2.1) with initial condition B0 = 0, and where St is the share price at time t, µ is the continuously compounding expected growth rate of St,σ is the standard deviation of the instantaneous return on St, and {Bt} is Brownian motion, with E(dBt) = 0 and Var(dBt) =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), withβ = 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 Itˆo’s Differentiation Lemma.

Result 2.1 (Itˆo’s Differentiation Lemma). Suppose that f(x, t) and its

partial derivatives fx, fxx and ft are continuous. If Xt is given by

dXt=µ(Xt, t)dt+σ(Xt, t)dBt

then Yt=f(Xt, t) has stochastic differential:

dYt= (µ(Xt, t)fx+ft+12σ2(Xt, t)fxx)dt+σ(Xt, t)fxdBt =fxdXt+ftdt+ 21σ2(Xt, t)fxxdt.

Hull (1997) gives a sketch proof of this result using a Taylor series ex-pansion. Using Itˆo’s Lemma, we can now prove the following well known theorems.

Theorem 2.1. Conditional on its value at time t, a share price ST that

follows GBM will have a lognormal distribution at T > t, with parameters

E(lnST) = lnSt + (µ− 12σ2)τ and Var(lnST) = σ2τ, where τ = T − t.

Moreover, the form for ST given St will be:

ST =Ste(µ−

1 2σ2)τ+σ

τ Z

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Proof. Consider the transformation Yt = lnSt. Referring to the SDE (2.1), and to Result 2.1, we see that:

Xt =St

µ(Xt, t) =µSt

σ(Xt, t) =σSt

f(Xt, t) = lnSt and so applying Itˆo’s Lemma

dlnSt=

∂lnSt

∂t dt+ ∂lnSt

∂St

dSt+ 12(σSt)2

∂2lnS

t

∂St2

dt

= 0 dt+ 1

St

(µStdt+σStdBt)− 12σ2St2 1

S2

t

dt

µ12σ2¢

dt+σdBt (2.2)

Thus, integrating both sides fromt to T yields: lnST −lnSt=

¡

µ12σ2¢

(T t) +σ

Z T

t

dBt =¡

µ12σ2¢

τ +σ(BT −Bt)

From the properties of Brownian motion, in particular its independent incre-ments:

BT −Bt ∼BT−t−B0 =Bτ

since B0 = 0, and where ∼ denotes “is distributed as”. Hence:

lnST ∼lnSt+

¡

µ 1 2σ

τ+σBτ. (2.3) Conditioned on St, it is clear that the only random variable in the RHS of the above equation is the Brownian motion term Bτ. Using the fact that

Bt ∼ N(0, t) for any t >0, we obtain the mean and variance of lnST given

St:

E(lnST|St) = E(lnSt+

¡

µ 1 2σ

τ+σBτ|St) = lnSt+

¡

µ 12σ2¢

τ +σE(Bτ) = lnSt+

¡

µ 12σ2¢

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Var(lnST|St) = Var(lnSt+

¡

µ12σ2¢

τ +σBτ|St) =σ2Var(Bτ)

=σ2τ

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 b2Var(X). Combining this with the mean and variance above

it is clear that

lnST|St ∼N

¡

lnSt+

¡

µ12σ2¢

τ, σ2τ¢

(2.4) i.e. ST is lognormal with parameters lnSt+

¡

µ 12σ2¢

τ and σ2τ. Moreover,

since Z = Bτ

τ ∼N(0,1), it follows directly from Equation (2.3) that

ST =Ste(µ−

1 2σ2)τ+σ

τ Z (2.5)

as required.

Theorem 2.2 (Moments of a Share Price following GBM). The mean

and variance of the share price ST conditional on an earlier share price St,

0t < T areSteµτ and St2e2µτ(eσ

2τ

−1) respectively.

Proof. Note first that the moment generating function of anN(0,1) variable

is E(esZ) = e12s2

. This can be used to find the moments of the random variable of interest ST|St.

E(ST|St) =E

µ

Ste(µ−

1 2σ2)τ+σ

√ τ Z ¯ ¯ ¯ ¯ St ¶

=Ste(µ−

1 2σ2)τE

³

eσ√τ Z´

=Ste(µ−

1

2σ2)τe12σ2τ

=Steµτ (2.6)

E(ST2|St) =E

µ

St2e2(µ−12σ2)τ+2σ

√ τ Z ¯ ¯ ¯ ¯ St ¶

=St2e2(µ−

1 2σ2)τE

³

e2σ√τ Z´

=St2e2(µ−21σ2)τe2σ2τ

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Forming the variance using the relationship Var(X) = E(X2)E(X)2, we

obtain

Var(ST|St) =St2e2µτ(eσ

2

−1) (2.7)

as required.

Note that if share price evolution is a deterministic process, i.e. σ = 0, then ST equals the mean of the stochastic process: Steµτ. Thus using either the expected value, or the deterministic case, the parameterµcan 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:

µj =r+ MRP

Cov(Rj, Rm)

σ2

m

where MRP is the market risk premium, Rj is the return on asset j, µj is the expected value of that return, Rm is the return on the market portfolio, and σ2

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 µj. The size of this effect is determined by the measure of systematic risk:

Cov(Rj, Rm)

σ2

m

which compares the risk associated with the particular asset to the risk as-sociated 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:

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whereRj andRmare as above,αj andbjare the regression model parameters, and ²j is a stochastic error term, with variance σ²2j. Applying the variance

operator to all terms in the equation above yields the relationship:

σj2 =b2jσm2 +σ2²j.

The variance of the market return, σ2

m measures the systematic risk in the system, whereas σ2

²j measures the non-systematic risk. Hence the parameter

σ reflects a combination of these, but not investor risk attitudes.

2.2.1

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:

ST =Ste(µ−

1 2σ2)τ+σ

√ τ Z

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 =t0 < t1 <· · ·< ti <· · ·< tn−1 < tn=T

where the intervals are not necessarily of equal length. Equation (2.5) can also be written to give the share price at time ti conditional on the share price at ti−1:

Sti =Sti−1e

(µ−1

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hence simulatingn realisations ofZ, the series can be constructed as follows:

Sti =St

i

Y

j=1

Stj

Stj−1

=St i

Y

j=1

e(µ−12σ2)(tj−tj−1)+σ

tj−tj−1Zj

=Stexp

à i

X

j=1

12σ2)(tj −tj−1) +σ

p

tj −tj−1Zj

!

=Ste(µ−

1

2σ2)(ti−t)exp

Ã

σ

i

X

j=1

p

tj −tj−1Zj

!

. (2.8)

A program which simulates GBM using Equation (2.8) is given in Appendix D.1.

Time - years

Share Price

0.0 0.2 0.4 0.6 0.8 1.0

4.5

5.0

5.5

6.0

Figure 2.1: A realisation of GBM with initial valueSt= $5, and parameters

µ= 0.1, σ = 0.3 and 250 subintervals.

Figure 2.1 shows a single realisation of geometric Brownian motion with

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ST = $5.12, which is smaller than its expected value E(ST|St) = $5e0.1 = $5.53 but well within a single standard deviations(ST|St) =Steµ

eσ2

−1 = $1.696 of it.

The daily returns for the series {St}are defined as:

Rt = lnSt+∆t−lnSt

where ∆t = 2501 years is approximately one trading day. Whilst the series

{St} is clearly not a stationary process, Equation (2.2) indicates that the daily returns should be stationary, with a mean (µ 12σ2)∆t and variance

σ2t. A plot of the daily returns can be seen in Figure 2.2, with an estimate

of the mean level and standard deviation shown1. 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 ti using weights from the bisquare function:

B(x) =

(

(1x2)2 |x|<1

0 |x| ≥1

where x depends on the time ti 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

2σ2)∆t which is negligible for the parameters chosen2. We see that the

es-timated mean does indeed oscillate about the actual mean level, and that

1The 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.

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Time - years

Log-returns

0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.0

0.02

0.04

0.06

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 au-tocorrelation 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.

2.2.2

The Future Share Price

S

T

As discussed in Chapter 1, the value of a European option att will certainly depend on the share price at the future exercise timeT. 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 knownµ and σ, then N realisations of the share price at T can be obtained

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using the second program seen in Appendix D.1. Simulation of the quantity

ST is a useful way of deciding what properties the share price will have atT, 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 sim-ulation 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 St = $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(lnST|St) = ln 5 + (0.1 + 120.32) and Var(lnST|St) = 0.32. 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 fre-quency 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) respec-tively. These estimates are given by the sample mean and sample variance of 5.5397 and 2.8756 respectively, compared to theoretical values of 5.5259 and 2.8756. 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.

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2 4 6 8 10 12 14

0.0

0.05

0.10

0.15

0.20

0.25

Share Price

Relative Frequency

Figure 2.3: 5000 realisations of ST, a future geometric Brownian motion price, with initial valueSt = $5, and parametersτ = 1,µ= 0.1 and σ= 0.3, and the theoretical lognormal distribution for ST.

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:

CT = max(ST −K,0)

= (ST −K)+ (2.9) whereK 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

Pt=e−λτE(CT|St) (2.10) where Pt would be the price paid now for the payoff CT, which would be received at time T. This equation features two rates particular to the risk

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preferences of investors in aggregate: µ, the continuously compounded ex-pected 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 transformationYT =ST−K. It is clearYT has mean E(ST)−K, variance Var(ST), andYT+K has a lognormal distribution. Note that YT itself does not have a lognormal distribution, since the lognormal distribution is defined on the range [0,) whereasYT is defined on [−K,∞). However, the shape of the distribution of YT will be identical to that of ST, since we have the relationship

P(YT < y) =P(ST −K < y) = P(ST < y+K).

Next consider CT = YT+ = max(0, YT). It is clear that P(CT = 0) =

P(ST ≤K) and so CT will not have a continuous distribution function. Theorem 2.3 (Distribution of CT given St). GivenSt,CT = (ST−K)+

has a mixed distribution

P(CT ≤c|St) =

(

0 c <0

FST|St(K+c, τ) c≥0 with density:

fCT|St(c, τ) =FST|St(K, τ) δ(c) +

(

0 c < 0

fST|St(K+c, τ) c >0

where FST|St and fST|St are the cumulative distribution and density functions

for the share price ST, and δ(c) is the Dirac delta function.

Proof. SinceST givenSthas a lognormal distribution with parameters lnSt+

12σ2)τ and σ2τ, givenSt:

P(ST −K ≤c|St) =P(ST ≤K+c|St) =P(lnST ≤ln(K+c)|St) = Φ

µln(K +c)

−lnSt−(µ−12σ2)τ

σ√τ

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where Φ(x) = Rx

−∞φ(z)dz is the standard normal cumulative distribution function, andφ(z) is the standard normal probability density function. Thus:

FCT|St(c, τ) =P(CT ≤c|St)

=

(

0 c <0

P(ST −K ≤c|St) c≥0 =

(

0 c <0

Φ³ln(K+c)−lnSt−(µ−12σ2)τ

σ√τ

´

c0 (2.11)

fCT|St(c, τ) =

      

0 c <0

P(CT = 0|St)δ(c) c= 0 ∂

∂cΦ

³ln(K+c)

−lnSt−(µ−12σ2)τ

σ√τ

´

c >0

=       

0 c <0

P(ST ≤K|St)δ(c) c= 0

1 (K+c)σ√τφ

³ln(K+c)

−lnSt−(µ−12σ2)τ

σ√τ

´

c >0

=     

0 c < 0

FST|St(K, τ)δ(c) c= 0

fST|St(K+c, τ) c > 0

(2.12)

whereFST|St andfST|St are the cumulative distribution and density functions

for the terminal share price ST given by:

FST|St(s, τ) =P(ST < s|St)

= Φ

µ

lnslnSt−(µ− 12σ2)τ

σ√τ

(2.13)

fST|St(s, τ) =

∂FST|St(s, τ)

∂s

= 1

sσ√τφ

µlns

−lnSt−(µ−12σ2)τ

σ√τ

(2.14) and δ(c) is the Dirac delta function defined by:

Z x

−∞

δ(c)dc=

(

0 x <0 1 x0.

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2.3.1

Mean and Variance of

C

T

given

S

t

The mean and variance ofCT givenStcan be found using the density function forCT given in Equation (2.12). An equivalent but simpler method is to note that CT = h(Z), where Z is a standard normal random variable. Then the expected value of any function,g, of CT can be found using the relationship

E{(gh)(Z)}=

Z ∞

−∞

(gh)(z)φ(z)dz

where gh is the composition ofg with h. This yields the following. Theorem 2.4 (Moments of CT). Given St, CT = (ST −K)+ has mean

E(CT|St) =SteµτΦ(gt)−KΦ(gt−σ√τ)

and mean square

E(CT2|St) = St2e(2µ+σ

2)τ

Φ(gt+σ√τ)−2SteµτKΦ(gt) +K2Φ(gt−σ√τ)

where

gt=

lnSt−lnK+ (µ+ 12σ2)τ

σ√τ

Proof. Note firstly that from Equation (2.5), it is clear thatST is a function

of a standard normal variable Z. Hence, we find that CT too is a function of

Z.

CT = (ST −K)+ =

(

0 ST ≤K

Ste(µ−

1 2σ2)τ+σ

τ ZK S T > K Note thatST ≤K implies

Z −lnSt+ lnK−(µ−

1 2σ

2)τ

σ√τ =−gt+σ

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and therefore the first moment can be determined thus: E(CT|St) =

Z ∞

−gt+σ√τ

³

Ste(µ−

1 2σ2)τ+σ

τ z

−K´φ(z)dz

=Steµτ

Z ∞

−gt+σ√τ

e−1 2(z−σ

τ)2

2π dz−K

Z ∞

−gt+σ√τ

φ(z)dz

=Steµτ

Z ∞

−gt

φ(y)dyKΦ(gt−σ√τ) =SteµτΦ(gt)−KΦ(gt−σ√τ).

The second moment gives us a means of calculating the variance ofCT given

St, and is found as follows: E(CT2|St) =

Z ∞

−gt+σ√τ

³

Ste(µ−

1 2σ2)τ+σ

√ τ z

−K´2φ(z)dz

=St2e(2µ+σ2)τ

Z ∞

−gt+σ√τ

e−1 2(z−2σ

√ τ)2

2π dz−2Ste

µτKΦ(g

t) +K2Φ(gt−σ√τ) =St2e(2µ+σ2)τ

Z ∞

−gt−σ√τ

φ(y)dy2SteµτKΦ(gt) +K2Φ(gt−σ√τ) =St2e(2µ+σ

2)τ

Φ(gt+σ√τ)−2SteµτKΦ(gt) +K2Φ(gt−σ√τ).

The variance can be formed as usual using Var(CT|St) = E(CT2|St)−E(CT|St)2. Thus, from Equation (2.10), an investor may value the option using the expected value, and the market rates µand λ, to give a price at t:

Pt=Ste(µ−λ)τΦ(gt)−Ke−λτΦ(gt−σ√τ). (2.15) wheregt 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 Pt will change. However, there is theory to show that the true value of Ct should be independent of risk premia on assets, thus any change inµis offset by an appropriate change inλ when the fair option price is calculated.

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2.3.2

Simulation of

C

T

A sample of terminal call values CT 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 ST 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 CT there is also good agreement between the theoretical and observed mean and variance. For an exercise price ofK = $5, the sample yields respective values of 0.9309 and 1.7714 for the estimated mean and variance, compared to theoretical values of 0.9247 and 1.7497.

It is also interesting to note the relative frequency of the event CT = 0, i.e. the proportion of occasions on which the option would not be exercised. The theoretical probability that the value of CT will be zero is given by the equation

P(CT = 0|St) =P(ST < K|St) = Φ(−gt+σ√τ)

which for the simulation equals 0.4273. Hence, for this particular simulation, the expected number of options that are not exercised is 2136. This can be compared to the sample estimate of 2109.

2.4

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 fol-lows directly:

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– the variance rate σ2 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 restric-tions, 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 St is the solution to the

SDE:

dSt =µ(St, t)Stdt+σ(St, t)StdBt (2.16)

then the price at timet of a call option over a share with priceStmust satisfy

the PDE:

1 2σ

2(S

t, t)St2

∂2C ∂S2 +rSt

∂C ∂S +

∂C

∂t −rCt= 0.

subject to the boundary condition

CT = (ST −K)+

where r is the continuously compounding risk-free rate.

Proof. Consider forming a self-financing portfolio att, ofatunits of the stock

with value St, and bt units of the call option with value Ct, where at and bt may be functions of both share price at t and time. This portfolio has value att:

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and over the period [t, t+dt] the change in portfolio value will be:

dVt =Vt+dt−Vt

=at+dtSt+dt+bt+dtCt+dt−atSt−btCt

= (at+dt−at)St+dt+atdSt+ (bt+dt−bt)Ct+dt+btdCt =atdSt+btdCt.

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

datSt+dt+dbtCt+dt = 0 as required.

The form for dSt is given by Equation (2.16), and dCt can be obtained from it using Itˆo’s Lemma. The option price can be considered a function of two variables: the current share price and time. Hence we can write

Ct =C(St, t) where St is a solution to the familiar SDE

dSt =µ(St, t)Stdt+σ(St, t)StdBt. Referring to Result 2.1, we see that

Xt=St µ(Xt, t) = µ(St, t)St σ(Xt, t) = σ(St, t)St f(Xt, t) =C(St, t). Hence, applying Itˆo’s Lemma we can determine the change in Ct over the period [t, t+dt]:

dCt=

∂C ∂tdt+

∂C ∂SdSt+

1

2(σ(St, t)St) 2∂2C

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The change in portfolio value over the interval [t, t+dt] thus becomes:

dVt =atdSt+btdCt =atdSt+bt

µ

∂C ∂tdt+

∂C ∂SdSt+

1 2σ

2(S

t, t)St2

∂2C

∂S2dt

=

µ

at+bt

∂C ∂S

dSt+bt

µ

∂C ∂t +

1 2σ

2(S

t, t)St2

∂2C

∂S2

dt

and the portfolio’s rate of return:

dVt

Vt

= atdSt+btdCt

atSt+btCt =

¡

at+bt∂C∂S

¢

dSt+bt

³

∂C ∂t +

1

2σ2(St, t)St2∂

2C

∂S2

´

dt atSt+btCt

=

¡

ft+ ∂C∂S

¢

dSt+

³

∂C ∂t +

1 2σ

2(S

t, t)St2∂

2C

∂S2

´

dt ftSt+Ct

whereft=at/bt.

Since the only stochastic elements are present in thedSt term, these can be eliminated by choice of ft to form a portfolio whose change in value over the period [t, t+dt] is deterministic. Setting the coefficient of dSt to zero gives

ft=−

∂C ∂S

and so Var(dVt

Vt ) = 0 and the rate of return on the portfolio becomes:

dVt Vt = ³ ∂C ∂t + 1 2σ

2(S

t, t)St2∂

2C

∂S2

´

dt

−∂C∂SSt+Ct

. (2.17)

By this choice offt, 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 rater, giving

dVt=rVtdt or dVt−rVtdt= 0.

If this were not the case, investors could borrow (lend) at the risk-free rate

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profits. Substituting terms fordVtand Vt in the above expression, yields the desired PDE:

µ

∂C ∂t +

1 2σ

2(S

t, t)St2

∂2C

∂S2

dtr

µ

−∂C∂SSt+Ct

dt = 0

1 2σ

2(S

t, t)St2

∂2C

∂S2 +rSt

∂C ∂S +

∂C

∂t −rCt = 0. (2.18)

The solutionCt must also satisfy the boundary condition

CT = (ST −K)+ since atT, the option yields the payoff CT.

Theorem 2.6 (The Black-Scholes PDE). The price of a European call

option,Ct, over a share whose price follows GBM, with timeτ until maturity,

must satisfy the partial differential equation (PDE):

1 2σ

2∂2C

∂S2 +rS

∂C ∂S +

∂C

∂t −rCt= 0 (2.19)

subject to the boundary condition

CT = (ST −K)+

where r is the continuously compounding risk-free rate.

Proof. The Black-Scholes PDE follows from Theorem 2.5, with

σ(St, t) =σ as does the boundary condition.

It is significant that neither of the PDEs (2.19) nor (2.18) feature µ, the only component of the respective models that describes investor risk preferences. Cox & Ross (1976) use this fact to solve this and other PDEs using a technique called risk-neutral valuation. Cox & Ross suggest that since the PDE does not feature risk preferences, then solution of the PDE

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can be achieved by assuming any convenient scenario for investor preferences. In particular, assuming that investors are risk-neutral provides an alternative method of deriving solutions like the Black-Scholes, and Constant Elasticity of Variance formulae, since both µ and λ in the Equation (2.10) can be set to r, and the price Pt evaluated. The general validity of this approach was established by Harrison & Kreps (1979).

The following theorem was first given by Black & Scholes (1972).

Theorem 2.7 (The Black-Scholes Formula). The solution to Equation

(2.19), subject to the boundary condition CT = (ST −K)+ is given by the

Black-Scholes Formula:

Ct=StΦ(ht)−Ke−rτΦ(ht−σ√τ)

where

ht=

lnSt−lnK+ (r+12σ2)τ

σ√τ

Proof. Evaluation of the partial derivatives ∂C∂S, ∂C∂t and ∂∂S2C2, where

Ct=StΦ(ht)−Ke−rτΦ(ht−σ√τ) (2.20) and substitution into the PDE indeed shows that the Black-Scholes formula is a solution. It is useful to note the relationship:

Stφ(ht)−Ke−rτφ(ht−σ√τ) = 0. From this it follows that the partial derivatives are:

∂C

∂S = Φ(ht)

∂2C

∂S2 =

φ(ht)

Stσ√τ and

∂C ∂t =−

Stσφ(ht)

2√τ −rKe

−rτΦ(h

t−σ√τ).

Substituting these derivatives into the PDE (2.19) shows thatCt as given by the Black-Scholes formula is indeed a solution of the PDE. Moreover,Ctalso satisfies the boundary condition, since setting t =T gives CT = (ST −K)+ as required.

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Result 2.2. Let the share price St be the solution to the SDE:

dSt =µStdt+σtStdBt

where σt is a deterministic function of time. Then the price of a call option

on a stock with share price St must satisfy the PDE:

1 2σ

2

tSt2

∂2C¯

∂S2 +rSt

∂C¯ ∂S +

∂C¯

∂t −rC¯t = 0

with boundary condition C¯T = (St−K)+, and the call price is given by the

Black-Scholes equation, with the substitution:

σ2 = ¯σ2 1 τ

Z T

t

σu2du.

Proof. The proof of this result is given in Appendix B.1.

Equation (2.15) gives the present value at time t, of the expected value of

CT, and features two rates µandλ, which depend on investor risk attitudes. Cox & Ross (1976) evaluate Pt with both µ and λ replaced by r giving a price which is identical to the Black-Scholes price. This method of valuing options is called the risk-neutral valuation method, and prices are given by:

Ct=e−rτE∗(CT|St)

where E∗ is the expectation operator taken in a risk-neutral world (so that

µ=λ=r). This is appropriate because the PDE of interest does not contain any parameters which reflect risk attitudes.

Note that Result 2.2 has interesting implications for the interpretation of the volatility parameter σ used in the Black-Scholes formula. Whilst σ2 is

defined as the instantaneous variance of the rate of return on the share price, it can be treated as if it is the average variance over the remaining life of the option, since the same pricing formula results. This framework gives a more intuitive meaning to the parameter σ.

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2.4.1

Properties of Call Price Prior to Maturity

Prior to maturity, the call price Ct and the share price St are linked by the Black-Scholes formula. This link can be used to derive the probability distribution of the option price at timet+s, wheret < t+s < T.

At maturity of the option, the relationship between the share and option prices was relatively simple:

CT = (ST −K)+

and yielded a distribution for CT that was very similar to that of ST. If

t < T, then the relationship between share and option price becomes the Black-Scholes formula:

Ct=StΦ(ht)−Ke−rτΦ(ht−σ√τ)

which yields a more complicated probability function for an option price at a time in the future, but prior to maturity.

Theorem 2.8 (Distribution of Black-Scholes Prices). Ct+sconditional

on St with s ∈ (0, τ), is given by the Black-Scholes formula, and has

proba-bility density function:

fCt+s|St(c, τ) =

 

0 c0

1

BSS−1(c;t+s)σ

√ sΦ(h∗

t+s)

φ³lnBSS−1(c;t+s)−lnSt−(µ−12σ2)s

σ√s

´

c >0

where BSS−1(c;t+s)is the inverse of the Black-Scholes function with respect

to S at time t+s, evaluated at c, and

h∗t+s = lnBS −1

S (c;t+s)−lnK+ (r+12σ2)(τ −s)

σ√τ s

Proof. First note thatCt+s >0, since the probability of a positive payout is

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temporarily the Black-Scholes priceCt+s =BS(St+s) whereBS is the

Black-Scholes formula with single argument St, consider now the case c >0:

FCt+s|St(c, τ) = P(Ct+s< c|St)

=P(BS(St+s)< c|St)

=P(St+s< BSS−1(c;t+s)|St) =P(lnSt+s <lnBSS−1(c;t+s)|St) = Φ

µlnBS−1

S (c;t+s)−lnSt−(µ− 12σ2)s

σ√s

since lnSt+s ∼N(lnSt+ (µ−12σ2)s, σ2s). This distribution function features the inverse of the Black-Scholes function with respect to St+s. Whilst the

form for this inverse cannot be written directly, the Black-Scholes model is a monotonic increasing function ofSt, and so computation of the single inverse value can be achieved using a numerical method such as the Newton-Raphson algorithm. This method is particularly useful here since the first derivative of the inverse with respect to c does have an explicit form which is easily evaluated. Differentiating this cumulative distribution function on the range

c > 0 gives the form for the density function on the same range. Noting that if y=BSS−1(c;t+s), then ∂y∂c can be found by using the chain rule for differentiation:

c=BS(y) 1 = ∂BS(y)

∂y ∂y ∂c ∂y

∂c =

1

Φ³lny−lnK+(r+12σ2)(τ−s)

σ√τ−s

´

since ∂C

∂S = Φ(ht). So, noting

∂BSS−1(c;t+s)

∂c =

1 Φ³lnBSS−1(c;t+s)−lnK+(r+

1

2σ2)(τ−s)

σ√τ−s

´ =

1 Φ(h∗

(52)

the derivative ofFCt+s|St(c, τ) with respect to c, with c >0 is:

fCt+s|St(c, τ) =

∂cFCt+s|St(c)

= 1

BSS−1(c;t+s)σ√sφ

µlnBS−1

S (c;t+s)−lnSt−(µ−12σ2)s

σ√s

1 Φ(h∗

t+s)

= 1

BSS−1(c;t+s)σ√sΦ(h∗ t+s)

φ

µ

lnBSS−1(c;t+s)lnSt−(µ− 12σ2)s

σ√s

giving finally the density function forCt+s:

fCt+s|St(c, τ) =

 

0 c0

1

BSS−1(c;t+s)σ

√ sΦ(h∗

t+s)

φ³lnBSS−1(c;t+s)−lnSt−(µ−12σ2)s

σ√s

´

c >0 (2.21)

Note that in order to evaluate this density function, the Black-Scholes function must be inverted with respect to St. This cannot be done analyti-cally and the equation

c=BS(S)

must be solved forS numerically. This can be done using a numerical method such as the Newton-Raphson method. Details of this method, and its imple-mentation in this case can be found in Appendix D.2.

2.4.2

Best Predictor of a Future Black-Scholes Price

Although the moments of the theoretical distribution look impossible to de-rive using the density function above, it is possible to dede-rive the mean price at a timet+s where 0< s < τ. It is known that the best predictor of this future price, in a mean square sense, is given by the expected value, condi-tional on the past and present values of the series. In order to calculate the expected value of CT, the fact that ST given St is a function of a standard normal random variable Z was utilised. The same procedure can be used here.

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Theorem 2.9. The expected value of Ct+s, given St, where s ∈ (0, τ), is

given by the equation:

E(Ct+s|St) =SteµsΦ

³

ht+ (µσ−√rτ)s

´

−Ke−r(τ−s)Φ³ht−σ√τ+ (µσ−√rτ)s

´

where ht is as in the Black-Scholes formula, and is given in Theorem 2.7.

Proof. The form ofSt+s, where 0 < s < τ, is given by Equation (2.5)

St+s =Ste(µ−

1 2σ2)s+σ

√ sZ.

Moreover Ct+s is given by the Black-Scholes formula (2.20) and so: E(Ct+s|St) = E(St+sΦ(ht+s)|St)−Ke−rτE(Φ(ht+s−σ

τ s)|St) (2.22) with ht+s a function of the random variable St+s.

Noting then that the Black-Scholes priceCt+s is a function ofSt+s which is in turn a function of Z, we conclude that Ct+s is itself a function of Z and its expected value can be computed using the standard normal density function φ(z) rather than the very complicated fCt+s|St(c, τ). I will evaluate

the two expectations in the above equation separately. The first expectation of interest is

E(St+sΦ(ht+s)|St) =

Z ∞

z=−∞

St+sΦ(ht+s)φ(z)dz

and can be written as two separate components, which I will simplify inde-pendently:

St+sφ(z) =Ste(µ−

1 2σ2)s+σ

√ sz1

2πe

−12z2

=Steµs 1

2πe

−12(−z+σ

√ s)2

=Steµsφ(y)

wherey=z+σ√s. Simplifying first lnSt+s, using the same transformation:

lnSt+s= lnSt+ (µ−12σ2)s+σ√sz = lnSt+ (µ+12σ2)s−σ√sy

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thusht+s becomes:

ht+s =

lnSt+s−lnK + (r+ 12σ2)(τ−s)

σ√τ s

= lnSt+ (µ+

1 2σ

2)sσsylnK+ (r+ 1 2σ

2)(τ s)

σ√τ s

=

r

τ τ s

µ

lnSt−lnK+ (r+ 12σ2)τ

σ√τ +

r)s σ√τ −y

r s τ ¶ = r τ τ s

µ

ht+

r)s σ√τ −y

r

s τ

Substituting these expressions into the first expectation on the right hand side of Equation (2.22), and making the change of variables z = y+σ√s

yields:

E(St+sΦ(ht+s)|St) =

Z ∞

y=−∞

Steµsφ(y)Φ

ht+ (µσ−√rτ)s −ypsτ

q

τ−s τ

dy

=SteµsΦ2

³

∞, ht+ (µσ−√rτ)s;

q

s τ

´

where

Φ2(x, y;ρ) =

Z x

−∞

φ(z)Φ

Ã

yρz

p

1ρ2

!

dz

is the cumulative distribution function for a bivariate normal random variable with zero means, unit variances, and correlation coefficientρ. This distribu-tion funcdistribu-tion is symmetric aboutxandy, and so the order of integration can be changed to give an alternative distribution function:

Φ2(x, y;ρ) =

Z y

−∞

φ(z)Φ

Ã

xρz

p

1ρ2

!

dz.

and in particular

Φ2(∞, y;ρ) =

Z y

−∞

φ(z)Φ()dz = Φ(y)

since Φ() = 1. This leads to the simplification of the first expectation: E(St+sΦ(ht+s)|St) =SteµsΦ

µ

ht+

r)s σ√τ

(55)

Now consider the second expectation in Equation (2.22). I will again simplify the component of the expression separately, but without making the

y transformation as done in the previous term:

ht+s−σ

τ s= lnSt+s−lnK + (r−

1 2σ

2)(τs)

σ√τ s

= lnSt+ (µ−

1 2σ

2)s+σszlnK+ (r 1 2σ

2)(τ s)

σ√τs

=

r

τ τ s

µ

lnSt−lnK+ (r− 12σ2)τ

σ√τ +

r)s σ√τ +z

r s τ ¶ = r τ τ s

µ

ht−σ√τ +

r)s σ√τ +z

r

s τ

Substituting this into the expectation and interchanging the order of inte-gration as above gives:

E(Φ(ht+s−σ√τ −s|St) =

Z ∞

z=−∞

φ(z)Φ

ht−σ√τ +(µσ−√rτ)s +zpsτ

q

τ−s τ

dz

= Φ2

³

∞, ht−σ√τ +(µσ−√rτ)s;−

q

s τ

´

= Φ³ht−σ√τ +(µσ−√rτ)s

´

.

Hence substituting the two expectations into Equation (2.22), we obtain the expected value ofCt+s:

E(Ct+s|St) =SteµsΦ

³

ht+ (µσ−√rτ)s

´

−Ke−r(τ−s)Φ³ht−σ√τ+ (µσ−√rτ)s

´

(2.23) as required.

This formula is intriguing, since it is so similar to the Black-Scholes formula itself, but with the firstSt+s replaced by E(St+s|St), and the correction term in the Φ terms.

2.4.3

Graphical Examination of

C

t

Just as the series {St} can be simulated, so too can the series {Ct}. This is done using a share price time series and the Black-Scholes formula. In

Figure

Figure 1.1: The lower bounds for call option value, and a possible form for the call price.

Figure 1.1:

The lower bounds for call option value, and a possible form for the call price. p.23
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

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 p.34
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:

A realisation of GBM with initial value S t = $5, and parameters µ = 0.1, σ = 0.3 and 250 subintervals. p.34
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.

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. p.36
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 .

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 . p.38
Figure 2.4: 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,

Figure 2.4:

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, p.57
Figure 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,

Figure 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, p.58
Figure 2.6: 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

Figure 2.6:

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 p.60
Figure 2.7: 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

Figure 2.7:

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 p.63
Figure 2.8: 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

Figure 2.8:

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 p.64
Figure 3.1: The relationship between β and P (S T = 0), with S t = $5, τ = 1 year, µ = 0.10, σ = δS t β/2−1 = 0.3 in the solid curve and σ = 0.28 in the broken curve.

Figure 3.1:

The relationship between β and P (S T = 0), with S t = $5, τ = 1 year, µ = 0.10, σ = δS t β/2−1 = 0.3 in the solid curve and σ = 0.28 in the broken curve. p.77
Figure 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

Figure 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 p.87
Figure 3.3: 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 seri

Figure 3.3:

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 seri p.88
Figure 3.4: 5000 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.

Figure 3.4:

5000 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. p.92
Figure 3.5: The empirical cumulative distribution function of the 5000 re- re-alisations 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 p

Figure 3.5:

The empirical cumulative distribution function of the 5000 re- re-alisations 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 p p.93
Figure 3.6: 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 ≤ 2.

Figure 3.6:

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 ≤ 2. p.94
Figure 3.7: 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

Figure 3.7:

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 p.110
Figure 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 ≤ $6.

Figure 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 ≤ $6. p.117
Figure 3.9: 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 = 0.06.

Figure 3.9:

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 = 0.06. p.119
Figure 3.11: 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 = 0.06.

Figure 3.11:

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 = 0.06. p.121
Figure 4.1: 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 = 0.

Figure 4.1:

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 = 0. p.137
Figure 4.2: The cross-section of the log-likelihood surface in Figure 4.1, ¯l(β, ˆ 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 = 0

Figure 4.2:

The cross-section of the log-likelihood surface in Figure 4.1, ¯l(β, ˆ 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 = 0 p.137
Figure 4.3: 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 = 0.

Figure 4.3:

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 = 0. p.141
Figure 4.5: Estimates of s( ˆ β), the standard deviation of ˆ β, from Table 4.2 against the true β.

Figure 4.5:

Estimates of s( ˆ β), the standard deviation of ˆ β, from Table 4.2 against the true β. p.146
Table 4.4: β estimates for the 44 ASX share series, with estimates of s( ˆ β), the standard deviation of ˆ β, and the resulting confidence intervals obtained by simulation.

Table 4.4:

β estimates for the 44 ASX share series, with estimates of s( ˆ β), the standard deviation of ˆ β, and the resulting confidence intervals obtained by simulation. p.148
Table 4.5: 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).

Table 4.5:

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). p.150
Figure 4.6 shows the sample values e n for the stock AMC, which has been randomly selected from the sample of stocks for which the test for normality was rejected

Figure 4.6

shows the sample values e n for the stock AMC, which has been randomly selected from the sample of stocks for which the test for normality was rejected p.151
Figure 4.7: 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.

Figure 4.7:

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. p.155
Figure 4.8: 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.

Figure 4.8:

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. p.156
Figure 4.9: 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.

Figure 4.9:

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. p.157

References

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