The Constant Elasticity of Variance Option Pricing Model


 Amie Green
 3 years ago
 Views:
Transcription
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
2 2
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
4 ii
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 BlackScholes European Call Option Pricing Model of 1972/3. Unlike the BlackScholes 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 riskneutral pricing theory and the results of Feller. Moreover, descriptions of this model in journal and textbook literature frequently contained errors. One difficulty in implementation of the BlackScholes 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 BlackScholes 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 noncentral ChiSquared 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 BlackScholes 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 BlackScholes 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 BlackScholes Formula Properties of Call Price Prior to Maturity Best Predictor of a Future BlackScholes Price Graphical Examination of C t Use of the BlackScholes 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 BlackScholes 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 BlackScholes 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 BlackScholes 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 BlackScholes 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 BlackScholes 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 BlackScholes 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 BlackScholes implied volatilities for Absolute CEV option prices with S t = $5, τ = 1 year, σ = 0.3, r = 0.06 and $4 K $ Outofthemoney 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 = Atthemoney 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 = Inthemoney 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 loglikelihood 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 crosssection of the loglikelihood 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 pvalues 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 pvalues 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 riskfree rate, which is payable continuously on an asset whose future value is certain. Definition 1.6. A European option is inthemoney if the current share price S t exceeds K, atthemoney if the current share price is equal to K, and outofthemoney 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 riskfree 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 nonnegative 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 riskfree 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 riskfree 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 riskfree 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 BlackScholes Model 9
28 10 CHAPTER 2. GBM AND THE BLACKSCHOLES MODEL 2.1 Introduction Analysis of share prices over time would suggest that they are discretetime, discretevariable 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 continuoustime, continuousvariable 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 continuoustime, continuousvariable 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 BlackScholes 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 BlackScholes model. 2.2 Share Price Evolution Derivation of the BlackScholes 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 BLACKSCHOLES 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 BLACKSCHOLES 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 nonsystematic 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 BlackScholes 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 BLACKSCHOLES 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 equispaced 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 BLACKSCHOLES MODEL Logreturns 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 nonzero 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 equiwidth 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 BLACKSCHOLES 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 riskfree 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 BLACKSCHOLES 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 BLACKSCHOLES 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 BLACKSCHOLES 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 BlackScholes Formula Black and Scholes first presented the BlackScholes 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 BLACKSCHOLES 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 shortselling 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 riskfree rate. Proof. Consider forming a selffinancing 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 BLACKSCHOLES 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 BLACKSCHOLES 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 riskfree rate r, and go long (short) in the portfolio of stock and option and earn arbitrage
The BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationEuropean Options Pricing Using Monte Carlo Simulation
European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical
More informationBlackScholes Equation for Option Pricing
BlackScholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there
More informationARBITRAGEFREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGEFREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic
More informationValuing Stock Options: The BlackScholesMerton Model. Chapter 13
Valuing Stock Options: The BlackScholesMerton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The BlackScholesMerton Random Walk Assumption
More informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
More informationLecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing
Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common
More informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationCall Price as a Function of the Stock Price
Call Price as a Function of the Stock Price Intuitively, the call price should be an increasing function of the stock price. This relationship allows one to develop a theory of option pricing, derived
More informationDoes BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial
More informationInstitute of Actuaries of India Subject CT3 Probability and Mathematical Statistics
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in
More informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationNotes on BlackScholes Option Pricing Formula
. Notes on BlackScholes Option Pricing Formula by DeXing Guan March 2006 These notes are a brief introduction to the BlackScholes formula, which prices the European call options. The essential reading
More informationCS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options
CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common
More informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 11. The BlackScholes Model: Hull, Ch. 13.
Week 11 The BlackScholes Model: Hull, Ch. 13. 1 The BlackScholes Model Objective: To show how the BlackScholes formula is derived and how it can be used to value options. 2 The BlackScholes Model 1.
More information第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model
1 第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model Outline 有 关 股 价 的 假 设 The BS Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American
More informationBlackScholesMerton approach merits and shortcomings
BlackScholesMerton approach merits and shortcomings Emilia Matei 1005056 EC372 Term Paper. Topic 3 1. Introduction The BlackScholes and Merton method of modelling derivatives prices was first introduced
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More informationBlackScholes Option Pricing Model
BlackScholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,
More informationOption pricing. Vinod Kothari
Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate
More informationFour Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com
Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive the
More informationThe BlackScholes pricing formulas
The BlackScholes pricing formulas Moty Katzman September 19, 2014 The BlackScholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationDerivation and Comparative Statics of the BlackScholes Call and Put Option Pricing Formulas
Derivation and Comparative Statics of the BlackScholes Call and Put Option Pricing Formulas James R. Garven Latest Revision: 27 April, 2015 Abstract This paper provides an alternative derivation of the
More informationLecture Notes: Basic Concepts in Option Pricing  The Black and Scholes Model
Brunel University Msc., EC5504, Financial Engineering Prof Menelaos Karanasos Lecture Notes: Basic Concepts in Option Pricing  The Black and Scholes Model Recall that the price of an option is equal to
More informationHedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies
Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative
More informationτ θ What is the proper price at time t =0of this option?
Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min
More informationOn MarketMaking and DeltaHedging
On MarketMaking and DeltaHedging 1 Market Makers 2 MarketMaking and BondPricing On MarketMaking and DeltaHedging 1 Market Makers 2 MarketMaking and BondPricing What to market makers do? Provide
More informationTHE BLACKSCHOLES MODEL AND EXTENSIONS
THE BLACSCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the BlackScholes pricing model of a European option by calculating the expected value of the option. We will assume that
More informationPricing of an Exotic Forward Contract
Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 111 Nojihigashi, Kusatsu, Shiga 5258577, Japan Email: {akahori,
More informationReview of Basic Options Concepts and Terminology
Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some
More informationMertonBlackScholes model for option pricing. Peter Denteneer. 22 oktober 2009
MertonBlackScholes model for option pricing Instituut{Lorentz voor Theoretische Natuurkunde, LION, Universiteit Leiden 22 oktober 2009 With inspiration from: J. Tinbergen, T.C. Koopmans, E. Majorana,
More informationOption Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013
Option Pricing Chapter 4 Including dividends in the BS model Stefan Ankirchner University of Bonn last update: 6th November 2013 Stefan Ankirchner Option Pricing 1 Dividend payments So far: we assumed
More informationInvesco Great Wall Fund Management Co. Shenzhen: June 14, 2008
: A Stern School of Business New York University Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008 Outline 1 2 3 4 5 6 se notes review the principles underlying option pricing and some of
More informationIntroduction to Options. Derivatives
Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived
More informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of inthemoney options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationThe BlackScholes Formula
ECO30004 OPTIONS AND FUTURES SPRING 2008 The BlackScholes Formula The BlackScholes Formula We next examine the BlackScholes formula for European options. The BlackScholes formula are complex as they
More informationJournal of Financial and Economic Practice
Analyzing Investment Data Using Conditional Probabilities: The Implications for Investment Forecasts, Stock Option Pricing, Risk Premia, and CAPM Beta Calculations By Richard R. Joss, Ph.D. Resource Actuary,
More informationComputational Finance Options
1 Options 1 1 Options Computational Finance Options An option gives the holder of the option the right, but not the obligation to do something. Conversely, if you sell an option, you may be obliged to
More informationLecture 12: The BlackScholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The BlackScholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The BlackScholesMerton Model
More informationA SimulationBased lntroduction Using Excel
Quantitative Finance A SimulationBased lntroduction Using Excel Matt Davison University of Western Ontario London, Canada CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint
More informationA Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model
Applied Mathematical Sciences, vol 8, 14, no 143, 7157135 HIKARI Ltd, wwwmhikaricom http://dxdoiorg/11988/ams144644 A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting
More informationBlack Scholes Merton Approach To Modelling Financial Derivatives Prices Tomas Sinkariovas 0802869. Words: 3441
Black Scholes Merton Approach To Modelling Financial Derivatives Prices Tomas Sinkariovas 0802869 Words: 3441 1 1. Introduction In this paper I present Black, Scholes (1973) and Merton (1973) (BSM) general
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the
More informationLECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS
LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS Robert S. Pindyck Massachusetts Institute of Technology Cambridge, MA 02142 Robert Pindyck (MIT) LECTURES ON REAL OPTIONS PART II August, 2008 1 / 50
More informationEXP 481  Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0
EXP 481  Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Blackcholes model PutCall Parity Implied Volatility Options: Definitions A call option gives the buyer the
More informationThe Binomial Option Pricing Model André Farber
1 Solvay Business School Université Libre de Bruxelles The Binomial Option Pricing Model André Farber January 2002 Consider a nondividend paying stock whose price is initially S 0. Divide time into small
More informationOption Pricing. Chapter 11 Options on Futures. Stefan Ankirchner. University of Bonn. last update: 13/01/2014 at 14:25
Option Pricing Chapter 11 Options on Futures Stefan Ankirchner University of Bonn last update: 13/01/2014 at 14:25 Stefan Ankirchner Option Pricing 1 Agenda Forward contracts Definition Determining forward
More informationConstant Elasticity of Variance (CEV) Option Pricing Model:Integration and Detailed Derivation
Constant Elasticity of Variance (CEV) Option Pricing Model:Integration and Detailed Derivation YingLin Hsu Department of Applied Mathematics National Chung Hsing University Coauthors: T. I. Lin and C.
More informationUnderstanding Options and Their Role in Hedging via the Greeks
Understanding Options and Their Role in Hedging via the Greeks Bradley J. Wogsland Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 379961200 Options are priced assuming that
More informationLecture 6: Option Pricing Using a Onestep Binomial Tree. Friday, September 14, 12
Lecture 6: Option Pricing Using a Onestep Binomial Tree An oversimplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationOptions: Valuation and (No) Arbitrage
Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial
More informationThe Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models
780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond marketmaker would deltahedge, we first need to specify how bonds behave. Suppose we try to model a zerocoupon
More informationConvenient Conventions
C: call value. P : put value. X: strike price. S: stock price. D: dividend. Convenient Conventions c 2015 Prof. YuhDauh Lyuu, National Taiwan University Page 168 Payoff, Mathematically Speaking The payoff
More informationCAPM Option Pricing. Sven Husmann a, Neda Todorova b
CAPM Option Pricing Sven Husmann a, Neda Todorova b a Department of Business Administration, European University Viadrina, Große Scharrnstraße 59, D15230 Frankfurt (Oder, Germany, Email: husmann@europauni.de,
More informationJungSoon Hyun and YoungHee Kim
J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL JungSoon Hyun and YoungHee Kim Abstract. We present two approaches of the stochastic interest
More informationBINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract
BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple
More informationLecture 4: The BlackScholes model
OPTIONS and FUTURES Lecture 4: The BlackScholes model Philip H. Dybvig Washington University in Saint Louis BlackScholes option pricing model Lognormal price process Call price Put price Using BlackScholes
More informationChapter 2: Binomial Methods and the BlackScholes Formula
Chapter 2: Binomial Methods and the BlackScholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a calloption C t = C(t), where the
More informationBlackScholes and the Volatility Surface
IEOR E4707: Financial Engineering: ContinuousTime Models Fall 2009 c 2009 by Martin Haugh BlackScholes and the Volatility Surface When we studied discretetime models we used martingale pricing to derive
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationLecture. S t = S t δ[s t ].
Lecture In real life the vast majority of all traded options are written on stocks having at least one dividend left before the date of expiration of the option. Thus the study of dividends is important
More informationLectures. Sergei Fedotov. 20912  Introduction to Financial Mathematics. No tutorials in the first week
Lectures Sergei Fedotov 20912  Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics
More informationLecture 11: The Greeks and Risk Management
Lecture 11: The Greeks and Risk Management This lecture studies market risk management from the perspective of an options trader. First, we show how to describe the risk characteristics of derivatives.
More informationFinite Differences Schemes for Pricing of European and American Options
Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the BlackScholes
More informationEC3070 FINANCIAL DERIVATIVES
BINOMIAL OPTION PRICING MODEL A OneStep Binomial Model The Binomial Option Pricing Model is a simple device that is used for determining the price c τ 0 that should be attributed initially to a call option
More informationLOGNORMAL MODEL FOR STOCK PRICES
LOGNORMAL MODEL FOR STOCK PRICES MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD 1. INTRODUCTION What follows is a simple but important model that will be the basis for a later study of stock prices as
More informationStocks paying discrete dividends: modelling and option pricing
Stocks paying discrete dividends: modelling and option pricing Ralf Korn 1 and L. C. G. Rogers 2 Abstract In the BlackScholes model, any dividends on stocks are paid continuously, but in reality dividends
More informationThe interest volatility surface
The interest volatility surface David Kohlberg Kandidatuppsats i matematisk statistik Bachelor Thesis in Mathematical Statistics Kandidatuppsats 2011:7 Matematisk statistik Juni 2011 www.math.su.se Matematisk
More informationFINANCIAL ECONOMICS OPTION PRICING
OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.
More informationINTERNATIONAL COMPARISON OF INTEREST RATE GUARANTEES IN LIFE INSURANCE
INTERNATIONAL COMPARISON OF INTEREST RATE GUARANTEES IN LIFE INSURANCE J. DAVID CUMMINS, KRISTIAN R. MILTERSEN, AND SVEINARNE PERSSON Abstract. Interest rate guarantees seem to be included in life insurance
More informationPRICING DIGITAL CALL OPTION IN THE HESTON STOCHASTIC VOLATILITY MODEL
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Number 3, September 003 PRICING DIGITAL CALL OPTION IN THE HESTON STOCHASTIC VOLATILITY MODEL VASILE L. LAZAR Dedicated to Professor Gheorghe Micula
More informationBinomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
More informationJournal Of Financial And Strategic Decisions Volume 10 Number 2 Summer 1997
Journal Of Financial And Strategic Decisions Volume 10 Number 2 Summer 1997 AN EMPIRICAL INVESTIGATION OF PUT OPTION PRICING: A SPECIFICATION TEST OF ATTHEMONEY OPTION IMPLIED VOLATILITY Hongshik Kim,
More informationDiusion processes. Olivier Scaillet. University of Geneva and Swiss Finance Institute
Diusion processes Olivier Scaillet University of Geneva and Swiss Finance Institute Outline 1 Brownian motion 2 Itô integral 3 Diusion processes 4 BlackScholes 5 Equity linked life insurance 6 Merton
More informationFIN 411  Investments Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices
FIN 411  Investments Option Pricing imple arbitrage relations s to call options Blackcholes model PutCall Parity Implied Volatility Options: Definitions A call option gives the buyer the right, but
More informationThe Valuation of Currency Options
The Valuation of Currency Options Nahum Biger and John Hull Both Nahum Biger and John Hull are Associate Professors of Finance in the Faculty of Administrative Studies, York University, Canada. Introduction
More informationOther variables as arguments besides S. Want those other variables to be observables.
Valuation of options before expiration Need to distinguish between American and European options. Consider European options with time t until expiration. Value now of receiving c T at expiration? (Value
More informationInstitutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)
Copyright 2003 Pearson Education, Inc. Slide 081 Institutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared
More informationFinancial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan
Financial Modeling An introduction to financial modelling and financial options Conall O Sullivan Banking and Finance UCD Smurfit School of Business 31 May / UCD Maths Summer School Outline Introduction
More informationOscillatory Reduction in Option Pricing Formula Using Shifted Poisson and Linear Approximation
EPJ Web of Conferences 68, 0 00 06 (2014) DOI: 10.1051/ epjconf/ 20146800006 C Owned by the authors, published by EDP Sciences, 2014 Oscillatory Reduction in Option Pricing Formula Using Shifted Poisson
More informationOption Portfolio Modeling
Value of Option (Total=Intrinsic+Time Euro) Option Portfolio Modeling Harry van Breen www.besttheindex.com Email: h.j.vanbreen@besttheindex.com Introduction The goal of this white paper is to provide
More informationNumerical Methods for Option Pricing
Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly
More informationSimple formulas to option pricing and hedging in the Black Scholes model
Simple formulas to option pricing and hedging in the Black Scholes model Paolo Pianca Department of Applied Mathematics University Ca Foscari of Venice Dorsoduro 385/E, 3013 Venice, Italy pianca@unive.it
More informationVannaVolga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013
VannaVolga Method for Foreign Exchange Implied Volatility Smile Copyright Changwei Xiong 011 January 011 last update: Nov 7, 01 TABLE OF CONTENTS TABLE OF CONTENTS...1 1. Trading Strategies of Vanilla
More informationwhere N is the standard normal distribution function,
The BlackScholesMerton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at
More informationModelFree Boundaries of Option Time Value and Early Exercise Premium
ModelFree Boundaries of Option Time Value and Early Exercise Premium Tie Su* Department of Finance University of Miami P.O. Box 248094 Coral Gables, FL 331246552 Phone: 3052841885 Fax: 3052844800
More informationAmerican and European. Put Option
American and European Put Option Analytical Finance I Kinda Sumlaji 1 Table of Contents: 1. Introduction... 3 2. Option Style... 4 3. Put Option 4 3.1 Definition 4 3.2 Payoff at Maturity... 4 3.3 Example
More informationOption Pricing. 1 Introduction. Mrinal K. Ghosh
Option Pricing Mrinal K. Ghosh 1 Introduction We first introduce the basic terminology in option pricing. Option: An option is the right, but not the obligation to buy (or sell) an asset under specified
More information1 Geometric Brownian motion
Copyright c 006 by Karl Sigman Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM
More information9 Basics of options, including trading strategies
ECG590I Asset Pricing. Lecture 9: Basics of options, including trading strategies 1 9 Basics of options, including trading strategies Option: The option of buying (call) or selling (put) an asset. European
More information1 Capital Asset Pricing Model (CAPM)
Copyright c 2005 by Karl Sigman 1 Capital Asset Pricing Model (CAPM) We now assume an idealized framework for an open market place, where all the risky assets refer to (say) all the tradeable stocks available
More informationOPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options
OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis binomial model replicating portfolio single period artificial (riskneutral)
More informationLecture 6 BlackScholes PDE
Lecture 6 BlackScholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the riskneutral measure Q by If the contingent
More informationAmerican Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options
American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Early Exercise Since American style options give the holder the same rights as European style options plus
More informationValuation of American Options
Valuation of American Options Among the seminal contributions to the mathematics of finance is the paper F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political
More informationAn Introduction to Modeling Stock Price Returns With a View Towards Option Pricing
An Introduction to Modeling Stock Price Returns With a View Towards Option Pricing Kyle Chauvin August 21, 2006 This work is the product of a summer research project at the University of Kansas, conducted
More information