Black and Scholes Option Pricing Model and its Alternatives
CHAPER- BLACK AND SCHOLES OPION PRICING MODEL AND IS ALERNAIVES his chapter introduces and derives the Black and Scholes (BS) formula for option pricing. It also explains some of the other basic and popular alternative option pricing models. here are seven sections in this chapter. he first section covers the various factors that affect price of an option. he second section defines and explains some preliminary basic concepts that should be known before understanding the option valuation models. he third section discusses in detail the assumptions and derivation of the Black and Scholes model for pricing the options contracts. he fourth section deals with the option price sensitivities i.e. the response of option prices to the changes in the factors that determine the option prices. he fifth section explains some advantages and limitations of the Black and Scholes (BS) option pricing model. he sixth section explains in brief some of the other basic and popular models for valuing options. he seventh section summarizes the chapter..1. FACORS AFFECING OPION PRICES here are six factors that affect the price of an option, namely, the current stock price, the exercise price, the time to expiration, dividend or any income from the underlying asset, the volatility of the stock price and the risk free interest rate. At expiry, only two factors influence the option value i.e. the strike price and the asset price. If the asset price minus the exercise price is positive, a call will be worth the difference between the two but the put will be worthless. If the difference is negative a put will be worth the difference between the exercise price and the asset price and the call will be worthless. One can formally set out the limits to the value of a call or a put at expiry as follows:- For a call: C = max [0, S-X] For a put: P = max [0, X-S] where, S is the spot price and X is the exercise price at any point of time. he maximum function is necessitated by the choice of option exercise that rests with the purchaser, 4
who will not exercise the option unless it is economically advantageous to do so. hese limits apply to both American and European options at expiry. In determining the value of an option, it is usual to distinguish between intrinsic value and time value of options. An option s intrinsic value is based on the difference between its exercise price and the current (spot) price of the underlying asset. If the option is currently profitable to exercise, then it is said to have intrinsic value. he intrinsic value is given by: For a call: S-X For a put: X-S where, S is the spot price at any time t. Where the difference between the exercise price and the asset price is positive, the call option is said to have intrinsic value; for a put option the difference should be negative. For a European option, as exercise is possible only at maturity, the strike price is discounted to the relevant date in order to calculate the intrinsic value. he discounted strike price is then compared to the spot price of the asset. o a degree the concept of intrinsic value of a European option prior to maturity is redundant, as it is incapable of being exercised before maturity. he concept is relevant only as a means of identifying the sources of value for the option. In the case of American options where possibility of early exercise is permissible, the exercise price does not need to be discounted, as it will be paid in full at the date of exercise. If the asset price is below the exercise price the option would have no intrinsic value, instead it will have time value or extrinsic value, which is that part of the option premium, which is not intrinsic value. ime value is defined as: ime value = option premium - intrinsic value ime value reflects the price buyers are willing to pay for the possibility that at some time prior to maturity, the option may become profitable to exercise. For example, if an option is priced at Rs.10 with exercise price of Rs.100 and security price at Rs.105, the intrinsic value would be Rs.5 (105-100) and time value would be Rs.5 (10-5). 43
hus, strike price and the spot asset price are the two only relevant factors that affect the option s value at the date of maturity. However, if the option has time remaining to expiry, all factors mentioned above come into play. hese factors are briefly explained as follows: (1) Exercise Price (X) and Asset Price (S) Exercise [or strike] price is the price at which the underlying assets can be sold off by the holder of the call option at a certain fixed rate. If there is some time remaining to expiry, the pay off from a call option will be the amount by which the stock price exceeds the strike price. Clearly, the higher the stock price relative to the exercise price, the more valuable a call will be, and less valuable a put. () ime to Maturity (t) he longer the time left to maturity of an option the greater is the probability that at expiry the asset price will be substantially different from the exercise price and therefore, as this probability has some utility, the higher will be the value of the option. (3) Risk Free Interest Rate (r) For a call option, the exercise price represents a potential liability the owner of a call faces at expiration. Before expiration, the lower the present value of the liability associated with owning a call, the better for the call owner. herefore, call option prices increase with higher interest rates. he owner of a put option may exercise and receive the exercise price in exchange for surrendering the stock, so the exercise price represents a potential asset for a put owner. Lower the rate of interest, the higher the present value of a potential asset. herefore, the lower the interest rate, the higher the price of a put. (4) Volatility of the Underlying Security (σ) Roughly speaking, volatility of a stock price is a measure of how uncertain we are about future stock price movements. he more volatile the underlying asset price, the more valuable will be the option. his is because the greater the volatility, the greater the chance of the asset price rising substantially above (for a call) or below (for a put) the exercise price at expiry. It is true that the asset price may also fall (rise), but as the 44
option doesn t have to be exercised, most of the effect of the adverse movements is avoided. hus the greater the volatility, the greater the potential gain for the option holder, and therefore, the more valuable the option will be. (5) Income from the Underlying Asset (d) During the life of the option, some interest or dividend may have been paid on the underlying asset. he value of the underlying asset decreases as the interest or dividend is paid. hus, the value of the call option decreases and the value of the put option increases as interest or dividend payments on the underlying asset increases. o summarize, a call option premium, ceteris-paribus, will be higher, the higher the asset price relative to the exercise price, the higher the volatility, higher the rate of interest and the longer the time to maturity. Put option premium will be higher, the lower the asset prices relative to the exercise price, the higher the volatility, the lower the rate of interest and the longer the term to maturity... SOME BASIC CONCEPS o understand the option valuation models better, there must be some understanding of the basic concepts that are used therein. Present section deals with some of these basic concepts in brief. (1) Stochastic Processes Stochastic is derived from a Greek word τoχoς meaning aim, guess characterized by randomness 1. Any variable, whose value changes over time in an uncertain way is said to follow a stochastic process. Stochastic processes describe the probabilistic evolution of the value of a variable through time. Stochastic process is one whose behavior is non-deterministic, in that, a state doesn t fully determine its next 1 Randomness is a lack of order, purpose, cause or predictability. A random process is a repeating process, whose outcomes follow no describable deterministic pattern but follow a probability distribution. A random variable is any variable whose, value cannot be predicted exactly. A discrete random variable is one that has a specific set of possible values (example, total score when two dice are thrown). A continuous random variable is a variable that can take any one of a continuing range of values (example, temperature in a room is a random variable that is not discrete). he expected value E(X) (sometimes described as expectation) of a discrete random variable is the weighted average of all its possible values, taking the probability of each outcome as its weight. 45
state. Heuristically, the theory of stochastic processes describes the behavior of random variables overtime. he term stochastic or random process and time series are often used interchangeably. he probability characteristics of a stochastic process {X(t)} are completely specified if we determine the joint density function of a finite number of members of the family of random variables comprising the process. When t is fixed at a given point, X(t) has the conventional interpretation of a random variable, with associated (one-dimensional) probability density function. Specification of the stochastic process for X requires further specification of the joint density functions that relates Xs at different points of time. he joint densities provide a probabilistic specification of how X evolves overtime. his potentially complicated mapping can involve various combinations of discrete and continuous observations on X and t. he terms discrete stochastic process and continuous stochastic process are used to refer to the time intervals at which X(t) is observed. he theory of stochastic processes possibly started with the work of Abraham De Moivre in the 1730s, when he derived the normal distribution as the limit of skew binomial. A more traditional starting point would be in 187 with the work of Robert Brown, who observed that small particles, suspended in a liquid, exhibited ceaseless irregular motions. he important modern contributions in stochastic processes can be traced back to N. Weiner, in 1918. His role is recognized in the use of the term Weiner process to signify the fundamental building block of the theory of stochastic differential equations. In recognition of the early works of Brown, the term Brownian motion is often used synonymously. Stochastic processes can be classified as discrete time or continuous time. A discretetime stochastic process is one, where the value of the variable can change only at certain fixed points of time, whereas a continuous-time stochastic process is one where changes can take place at any time. Stochastic processes can also be classified as continuous variable or discrete variable. In a continuous-variable process, the underlying variable can take any value within a certain range, whereas in a discrete-variable process, only certain discrete values are possible. 46
() Stochastic Differential Equation It is a differential equation in which one or more terms is a stochastic process, thus resulting in a solution, which itself is a stochastic process. ypically, SDEs incorporate white noise 3, which can be thought of as the derivative of Brownian motion (or Weiner process). he earliest work on SDE was done to describe Brownian motion in Einstein s famous paper and Bachelier (1900). (3) Markov Stochastic Process Stochastic processes describe the probabilistic evolution of the value of a variable through time. A Markov process is a particular type of stochastic process, where only the present value of the variable is relevant for predicting the future 4. he past history of the variable and the way in which the present has emerged from the past is irrelevant. In effect, the Markov process is memory-less, as the past and future are statistically independent when the present is known. he Markov property implies that the probability distribution of the price at any particular future time is not dependent on the particular path followed by the price in the past. he Markov property of stock prices is consistent with the weak form of market efficiency. his states that the present price of a stock impounds all the information contained in a record of past prices. Consider a variable that follows a Markov stochastic process. Suppose that its current value is 10 and that the change in its value during one year is φ (0,1), where φ ( µ, σ) denotes a probability distribution that is normally distributed with mean µ and standard deviation σ. hen, the question arises that what is the probability distribution of the change in value of the variable during two years? he change in two years is the sum of the two normal distributions, each of which has a mean of zero and standard deviation of 1.0. Because the variable is Markov, the two probability distributions are 3 4 In calculus, a differential equation shows how a change in one or more variables is represented by a change in another. Differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and of its derivatives of various orders. See point number (5) under the heading Basic Concepts for the meaning of white noise. Statistical properties of the stock price history may be useful in determining the characteristics of the stochastic process followed by the stock price (e.g. its volatility). he point is that the particular path followed by the stock in the past is irrelevant. 47
independent. When we add two independent normal distributions, the result is a normal distribution in which the mean is the sum of the two means and the variance is the sum of the variances 5. he mean of the change during two years in the variable is therefore zero and the variance of this change is.0. he change in the variable over two years is therefore, φ (0, ). Similarly, the mean of the change during three years in the variable is zero, and the variance of this change is 3.0. he change in the variable over three years is thus, φ (0, 3). More generally, the change during any time period of length is φ ( 0, ). In particular, the change during a very short time period of length δ t is φ ( 0, δt). he square root signs in these results arise because, when Markov processes are considered, the variance of the changes in the successive time periods are additive. he standard deviations of the changes in successive time periods are not additive. he variance of the change in the variable in the example is 1.0 per year, so that the variance of the change in two years is.0 and the variance of the change in three years is 3.0. he standard deviation of the change in two and three years is and 3, respectively. Strictly speaking, one should not refer to the standard deviation of the variable as 1.0 per year. It should be 1.0 per square root of years. he result explains why uncertainty is often referred to as being proportional to the square root of time. 6 (4) Random Walk Random walk is a mathematical formulation of a trajectory that consists of taking successive random steps. Random walks are related to the diffusion models 7 and are a fundamental topic in discussions of Markov processes. Often, random walks are assumed to be Markov, but other more complicated walks also exist. he random walk theory says that stock price changes are independent of each other, so the past movements or trend of a stock price or market cannot be used to predict its future movement. In short, it says that a stock takes a random and unpredictable path. 5 6 7 he variance of a probability distribution is the square of its standard deviation. he variance of a one-year change in the value of the variable we are considering is therefore 1.0. John C. Hull, Options, Futures & Other Derivatives, fifth edition, Prentice Hall of India Private Limited. See point number (8) under the heading Basic Concepts for meaning of diffusion models. 48
(5) White Noise White noise is a sound with equal intensity at all frequencies within a broad band. Rock music, the roar of a jet engine, and the noise at a stock market are examples of white noise. he word white is used to describe this kind of noise because of its similarity to white light, which is made up of all different colors (frequencies) of light combined together. (Noise means random fluctuations/variations in price and volume that can confuse interpretations of market directions). Strict white noise or strong white noise is defined as a sequence of independent and identically distributed (IID) random variables with zero mean and finite variance. A weaker definition of white noise is a sequence of zero mean, finite variance of uncorrelated variables. A weak white noise is often called simply as a white noise. However, if noise is normally distributed then above distinction is useless. Infact, two variables are uncorrelated if and only if they are independent. (6) Brownian Motion he Scottish botanist Robert Brown gave his name to the random motion of small particles in a liquid. his idea of random walk has permeated many scientific fields and is commonly used as a mechanism behind a variety of unpredictable continuous time processes. he lognormal random walk based on Brownian motion is the classical paradigm for stock markets. Robert Brown made microscopic observations in 187 that small particles contained in the pollen of plants, when immersed in a liquid, exhibit highly irregular motions. Mathematically, a Brownian motion is a continuous stationary stochastic process B(t) having independent increments and for each t, B(t) is a Gaussian random variable with mean zero and variance t. It can be shown that B(t) is nowhere differentiable 8. his means that white noise, being thought of as derivative db(t)/dt of B(t), does not exist in the ordinary sense. White noise is a time derivative B(t) of Brownian motion. Brownian motion is among the simplest continuous-time stochastic processes, whose mathematical explanation was given by Weiner. Brownian motion, which was originally used as a model for stock price movements in 1900 by L. Bachelier, is a 8 his means that one can t define the rate of change (i.e. dy/dx). 49
stochastic process, say B(t), characterized by the following three properties: (i) Normal increments: B(t)-B(s) has a normal distribution with mean 0 and variance t-s. Moreover, if s=0 that B(t)-B(0) has normal distribution with mean 0 and variance t, (ii) Independence of increments: B(t)-B(s) is independent of the past, (iii) Continuity of paths: B(t), t>0 are continuous functions of t. hese three properties alone define Brownian motion, but they also show why Brownian motion is used to model stock prices. Property (ii) shows that stock price changes will be independent of past price movements. his is an important assumption made in stock price model used in option pricing. (7) Generalized Weiner Process Norbert Weiner developed a rigorous theory for Brownian motion, the mathematics of which was to become a necessary modeling device for quantitative finance. he starting point for almost all financial models, the first equation written down in most technical papers, includes the Weiner process as the representative for randomness in asset prices. Weiner process is a particular type of Markov stochastic process with a mean change of zero and a variance rate of 1.0 per time period t. It has been used in physics to describe the motion of a particle that is subject to a large number of small molecular shocks and is sometimes referred to as Brownian motion 9. o adequately motivate Stochastic Differential Equations (SDEs), it is necessary to develop the basic concept of a Weiner process as the limit of a discrete normal random walk, defined by a SDE. he discrete random walk, without drift, has the form X ( t + 1) = X( t) + Z( t + 1), where X ( 0) = 0, and t {1,,3,...} where Z(1), Z(), Z(3),. form a stochastic process of independent random variables with the standard normal probability distribution: Z(t) ~N[0,1]. In other words, over the unit time interval ( t =1) Z(t) is a normal random variable with mean 0 and variance of one. 9 A variable z follows a Weiner process if it has the following two properties: (1) the change δ z during a small period of time δt is δ z = δt where is a random number drawn from a standardized normal distribution φ (0,1) and () the values of δ z for any two different short intervals of time δt are independent. 50
Consider what happens to a normal random walk as the time interval tshrinks. In this case, Z( t + t) is no longer N[0,1], but rather N[ 0, t], where the random walk now has the form X( t + t) = X( t) + Z( t + t) = X( t) + Z( t) t Re-expressing in difference form and using: lim t o t = dt gives the stochastic differential equation (SDE) for the standard Weiner process: dx ( t) = Z( t) dt dx ( t) = dw( t) W(t) is usually referred to as a standard Weiner process. In other words, the basic Weiner process (or Brownian motion) is where stochastic processes of a variable that is subjected to numerous small shocks are described. he asset price could be such a variable. Considering X to be a random variable and t to be time, over a small interval of time, t, the random variable X, can change by X. If X follows Wiener process (i.e. Brownian motion), the change in X over a small interval of time, i.e. X, will be related to t by the equation: X = t At the limit, this becomes: X = t where is a random sampling from a standardized normal distribution, with a mean of zero and a standard deviation of one 10. With assumed to be a standard normal 10 If we take w instead of X and Z instead of, as is used by some authors, then the above equations can be written as: w = Z t And at the limit, this becomes: = w Z t 51
variable, it follows that X must be normally distributed with a mean of zero, a variance of t and a standard deviation of t. hus, in effect, we have a variable X, that is changing randomly by X, because it is dependent upon another random variable t (may be the effect of randomly arriving new information), that has an expected value of zero, a variance of t and a standard deviation of t. herefore, a Wiener process, w (or X), is a process describing the evolution of a normally distributed variable. he drift rate of the processes is zero and the variance rate is 1 per unit time. his means that if the value of the variable is x at time zero, at time it is normally distributed with mean x and standard deviation. he ensemble of sample paths for X(t) conform to the evolution of a random variable that is standard normal on the unit time interval. he Weiner process w(t) can be immediately generalized to allow for non-zero drift and variance that differs from t. When a trend or drift term µ and standard deviation σ ( 1) are admitted then the arithmetic Gaussian stochastic process (with constant coefficients µ andσ ) is defined: dx( t) = µ dt + σdw( t) his constant coefficient process is also referred to as an arithmetic or absolute Brownian motion. he ensemble of time paths for this process is also normal but differs from the standard Weiner process by allowing for a different amount of variation around a constant trend (i.e. has a drift term). he basic Weiner process can be used to construct a wide range of stochastic differential equations, each of which is associated with a different specification of the joint density of the stochastic process. he construction of the Weiner process requires these densities to be functionally connected to the normal. A Generalized Weiner process is the basic Weiner process with the addition of a drift parameter. When a variable is a financial asset, say S, it is appropriate to assume a positive drift because risky assets should offer a positive return to compensate investors for the risk that they bear. hus, drift is analogous to expected return. he drift parameter µ represents the change in S per small unit of time, t. If we are considering 5
only the drift, the S resulting from an expected absolute return of µ per small unit of time, would be µ t. However, when combined with the Weiner process, we have stochastic processes of a random variable, which has both a drift rate and a basic Weiner process. hat random variable has an expected change for two reasons: first, our expectation of return is over a small interval of time µ t; and second, the random change σ t is described by the basic Weiner process. hus, a small change in the asset price, following a Generalized Weiner process, over a small interval of time can be defined in terms of w by the following stochastic differential equation:- S = µ t + σ t As S is normally distributed, the mean (expected value) is µ t, the standard deviation is σ t and the variance of S is σ t. hus, σ becomes the variance per unit of time; σ is known as volatility. he Generalized Weiner process incorporates both the basic Wiener process and a drift element. he drift element is deterministic (i.e. non-random) and the basic Weiner process is the stochastic element. In general, the drift and the standard deviation can be functions of both the state variable and time, such that α [ x, t] is the drift and β [ x, t] is the volatility. Consider a simple form of state dependence, where β [ x, t] = σx, with µ and σ being constants: dx( t) = µ Xdt + σxdw( t) dx( t)/ X = µ dt + σdw( t) α [ x, t] = µ X and In this case, the instantaneous rate of change (dx/x) follows a Gaussian process. For this reason, the terms geometric process or geometric Brownian motion are used to identify this process. It can be shown that for the geometric Gaussian process the paths of X(t) correspond to a process that is log-normally distributed at each point of time. BS model uses the geometric Gaussian process to describe the behaviour of the stock price. he process is often encountered in a variety of option pricing situations because geometric Brownian motion usually leads to a relatively simple closed-form solution. In addition Nelson (1990) has shown that allowing for µ = µ[x] and σ = σ[x] produces 53
the result that the geometric Gaussian process is the continuous limit of the popular generalized autoregressive heteroscedasticity (GARCH) discrete stochastic process. (8) Diffusion Process here can be two types of continuous-time stochastic processes: diffusion processes and jump processes. Continuous-time diffusion process is where past stock prices if continuously observed, can be graphed without lifting the pencil from the paper. Such a process is said to have a continuous-sample path. he second type of process is the jump process, studied by Cox and Ross (1976), which follows a constant drift except at random times when it undergoes a jump (of perhaps random magnitude). Diffusion processes are (strong) Markov processes, continuous in both X and t, that is, it is a Markov process in which both time and the state variable X are continuous. hey are constructed from Weiner processes that are continuous time representation (or version) of a discrete-time random walk (both types of processes, that is, diffusion and Weiner obey the martingale property 11 ). When expressed in the form of a SDE, diffusions can be used to concisely specify the joint density functions for the stochastic process. In probability theory, diffusion process is a solution to a SDE. It is a continuous time Markov process with continuous sample paths. A sample path of a diffusion process mimics the trajectory of a molecule, which is embedded in a flowing fluid and at the same time subjected to random displacements due to collisions with other molecules i.e. Brownian motion. he position of this molecule is then random 1. (9) Lognormal Distribution A variable has a lognormal distribution if the natural logarithm of the variable is normally distributed. hus, if S t /S t-1 is log normally distributed then ln (S t /S t-1 ) is approximately normally distributed. A variable that has lognormal distribution can take any value between zero and infinity. Figure.1 below shows the shape of a lognormal distribution. Unlike normal distribution, it is skewed to the right so that the mean, median and mode are all different. It has no negative values and the probability of 11 1 A martingale is a stochastic process, which has no drift. A key property, which a martingale has is that its expected value at a future time is the same as its value today. Its probability density function is governed by an advection-diffusion equation. 54
extremely high values is asymptotically zero, as one would expect a random variable describing, among other things, security price relatives. While the lognormal distribution only approximates stock returns, it has two great virtues. First, it is mathematically tractable, so we can obtain solutions for the value of call options if stock returns are log normally distributed. Second, the resulting call option prices that we compute are very good approximations of actual market prices. probability density asset prices Figure.1: A Lognormal Distribution (10) Stationarity Stationary process is a stochastic process, whose joint probability distribution doesn t change when shifted in time or space. As a result, parameters such as the mean and variance, if they exist, also don t change overtime or position. In other words, a stochastic process is said to be stationary if its mean and variance are constant over time and value of the covariance between the two time periods depends only on the distance or gap or lag between the two time periods and not the actual time at which the covariance is computed. In the time series literature, such a stochastic process is called weakly stationary, or covariance stationary, or second-order stationary, or in wide sense, stochastic process. Stationarity is used as a tool in time series analysis, where the raw data are often transformed to become stationary. One needs to assume stationarity of the data to be able to make inference from the sample set. Strictly, stationarity would mean, that in addition to first and second moment, even the third moments are stationary/constant/non-changing. 55
(11) ime Series Analysis A financial time series, in discrete time, is a sequence of financial variables such as asset prices or returns observed at discrete points of time, for example, the end of a trading day or last trading day of a month. Most models assume that the spacing between points is fixed, for example, models of daily returns assume that returns are observed between consecutive trading days. In order to recover fixed spacing between time points due to weekends, holidays or periods when trading is suspended; a sequence of trading days different from sequence of calendar days is typically introduced. Not all financial variables can be represented with the fixed periodicity described here. For instance, in most markets intraday trades are randomly spaced as trading occurs when the correct match between buy and sell orders are found. When considering high frequency data (i.e. data related to individual trades), the assumption of periodic and fixed discrete time points must be abandoned. hus, time series is a sequence of data points, measured typically at successive times, spaced at (often uniform) time intervals. ime series analysis comprises methods that attempt to understand such time series, often either to understand the underlying context of the data points (where did they come from? What generated them?) or to make forecasts/predictions. ime series forecasting is the use of a model to forecast future events: to forecast future data points before they are measured. (1) Ito Process Ito shows the relationship between an SDE for some independent variable and SDE for a function of that variable. One of the starting points for classical derivatives theory is the lognormal SDE for the evolution of an asset. An Ito process is a stochastic process, a generalized Weiner process with normally distributed jumps. Generalized Weiner process is a continuous-time random walk with a drift and random jumps at every point of time. It is defined to be an adapted (or nonanticipatory process i.e. one that cannot see into the future ) stochastic process, which can be expressed as the sum of an integral with respect to Brownian motion and an integral with respect to time. 56
An Ito process is a generalized Wiener process in which the parameters α the expected return (or the drift) and σ, variance are a function of the underlying variances. In a generalized form the Ito process is [ x, t] t + σ[ xt] t x = α, where, x is a random variable and t is time. If the underlying variables are asset price (S) and time (t) the equation becomes [ S, t] t + σ[ S t] t S = α, hus, as the underlying variables change, so the absolute rate of drift and the variability of that drift change. For example, as S gets bigger, so will α andσ ; and as t gets larger, so will α andσ. o convert the general Wiener process into an Ito process, we must (1) construct the drift so that it must be a function of asset price and time; () construct the stochastic element so that it is also a function of asset price and time. For the first step we will specifyµ as the expected rate of return expressed in decimal form, so µ S is the absolute return (µ S =α ). hus, for a small period of time t, the expected absolute return will be S = µ S t, which at the limit is S = µ S t. Although the investor s degree of uncertainty will be independent of the asset price, the absolute return will be larger, the higher the asset price.hus, the actual dispersion of the expected asset price will depend upon the magnitude of the current asset price. Consequently, the standard deviation of the absolute change will be larger, the higher the asset price. hus, the variance of the actual change in the asset price is given as: σ S t and the resulting instantaneous variance becomes: σ S hese results arise because, if we specify that the standard deviation is proportional to the asset price, the variance has to be proportional to the square of the asset price. We 57
now have the deterministic expected element of return µ S t expressed as a function of the asset price and time and expressing variability in terms of standard deviation, we also have a stochastic element of the variability of the asset price σ S t, as a function of the asset price and time. Such a model of the movement in the price of an asset can be given as: S = µ S t + σ S t which is an Ito process because the expected returns and the variability of those returns are a function of the underlying variable, the asset price (S) and of time (t). As this Ito process gives us the absolute variability, it can be applied to asset prices. (13) Ito s Lemma Derived by Japanese mathematician Kiyoshi Ito in 1951, Ito s lemma 13 forms an integral foundation to stochastic calculus. It is used to find the differential of a stochastic function and when compared to normal calculus, Ito s lemma is considered as the chain rule of stochastic calculus. Ito s lemma tells us the SDE for the value of an option on a particular asset. It is a rule for calculating differentials of quantities dependent on stochastic processes. Conventionally, calculus is applied to functions that are deterministic. In effect, the familiar rules associated with dy/dx, such asy = x dy/ dx = x, only apply when x is known. When x is a random variable the usual rules of calculus no longer apply. Ito s lemma provides a method for evaluating the total derivative of a function of a stochastic variable that follows a Markov diffusion process. By applying Ito s lemma, the call option pricing problem is transformed from a Stochastic Differential Equation (SDE) problem that is not solvable in closed form with standard techniques into a deterministic Partial Differential Equation (PDE) problem that can be solved. It is a theorem of stochastic calculus that shows that second order differential terms of a Weiner process can be considered to be deterministic when integrated over a non-zero time period. Ito s lemma applies only to continuous (path) processes. It formalizes the fact that the random (Gaussian) part in the log of the stock price has a variance that is 13 A dictionary definition shows that a lemma is a proposition used in the demonstration of another proposition. It is also considered to be a proof or a theorem. 58
proportional to the (infinitesimal) time interval, the standard deviation is therefore, proportional to the square root of this time interval. Also known as Ito-Doeblin theorem, an Ito s lemma is a theorem in stochastic calculus that explains that if one has a random walk, in say y, and a function of that randomly walking variable, call it f(y,t), then one can easily write an expression for the random walk in f. A function of a random variable is itself random in general. (14) Risk Neutral Valuation It is a general result, which is commonly used in pricing of derivatives. Risk neutral valuation states that any security dependent on other traded securities can be valued on the assumption that investors are risk neutral. It means that investor s risk preferences have no effect on the value of a stock option when it is expressed as a function of the price of the underlying security. It explains why the BS equation doesn t involveµ (the expected rate of return). In a risk neutral valuation two simple results hold: (1) the expected return from all securities is the risk free interest rate and () the risk free interest rate is the appropriate discount rate to apply to any expected future cash flow. he prices we get are correct not only in a risk neutral world, but in other worlds as well. When we move from a riskneutral world to a risk-averse world, two things happen. First, the expected growth rate in the stock price changes and secondly, the discount rate that must be used for any payoffs from the derivative also changes. It so happens that these two changes always offset each other exactly. A risk neutral world is a particularly easy world to work with because the expected return on all stocks in a risk neutral world is the risk free interest rate. In such a world, the investor requires no compensation for risk; they are all indifferent to risk. Both the no-arbitrage argument and the risk-neutral valuation lead to the same BS equation. (15) Perfect Hedging Perfect hedge means a hedge, which completely eliminates the risk of another investment. Perfect hedge is a hedge that completely offsets any gain or loss from an 59
existing investment position. An example is the short sale of an owned security in order to lock in an existing profit and transfer it to a subsequent tax year. (16) Probability Distribution Probability distributions are mathematical models of the likelihood (probability) of uncertain events happening. hey describe the way probability associated with a random variable are distributed across all the possible ranges of that variable. he probability P is defined by historical frequencies: the probability of an event is the frequency at which it happened in the past. hey are important in option pricing because they enable us to evaluate the amount on uncertainty surrounding the future price of the asset underlying the option. (17) Mean Reversion he idea behind mean reversion is that a variable will eventually revert back to its mean or average value. For example, if a stock price has steadily been increasing over a period of time, it will eventually decrease back to its mean value. (18) Monte Carlo Simulation he Monte Carlo method encompasses any technique of statistical sampling employed to approximate solutions to quantitative problems. Stanislaw Ulam, John von Neuman and Nicholas Metropolis are the three individuals, who are credited with inventing this concept in 1946, with Ulam and Metropolis later publishing the first paper on Montecarlo simulation several years later in 1949. he method was named after the casino of the same name. o consider Montecarlo simulation, it is useful to think of it as a general technique of numerical integration, which is what makes it a very reliable method to get approximations of option prices, which follow a partial differential equation. Similar to standard Montecarlo simulation methods are the Quasi-Montecarlo simulation, with the exception of one aspect. Montecarlo methods utilize pseudorandom numbers to generate the output, whereas, Quasi-Montecarlo utilizes quasirandom number sequences, also referred to as low discrepancy sequences in order to generate the output. 60
(19) Multivariate In statistics, multivariate as opposed to bivariate is a generalization, which merely shows that something is dependent on several variables as opposed to univariate (one) and bivariate (two). (0) Partial Differential Equations (PDEs) hese are equations to be solved in which the unknown element is a function, but whereas a standard differential equation has only one variable, PDEs function is of one or several variables. his makes PDEs somewhat trickier to solve than normal differential equations as increased dimensions make increased computations. In finance, the concept of PDEs is equally as important as applications within engineering and physics as many of the derivatives we encounter, require a solution to a PDE. From the simple Black and Scholes to the pricing of exotics, one must solve the PDEs using the applicable boundary conditions in order to derive the prices for these options. For plain vanilla European options, one can solve the Black-Scholes-Merton PDE using the boundary condition of a call or a put: Call: c=max(s-x,0) Put: p=max(x-s,0) where S is the stock price and X is the exercise price at which the option will get exercised..3. HE BLACK AND SCHOLES OPION PRICING MODEL he theory of option pricing has a long history, dating back to Bachelier (1900), Samuelson (1965) and several others, before the seminal contribution of Black and Scholes (1973) and Merton (1973). Many theoretical contributions were made by Bachelier: he developed the continuous time random walk process; he solved for the passage problem for such a process; he derived the probability law for diffusion in the presence of an absorbing barrier; and he priced various types of options. Sixty five years after Bachelier s remarkable study, Samuelson (1965) revisited the problem of 61
pricing a call. He replaced Bachelier s assumption that the price of the underlying asset follows a continuous random walk (leading to negative asset prices) with the assumption that the price of the underlying asset follows a geometric continuous random walk. He discounted the option payoff at the expected rate of return of the underlying asset and comes close to obtaining the Black-Scholes-Merton option pricing formula. However, the concept of risk-neutral pricing had to wait for the breakthrough by Black and Scholes (1973) and Merton (1973)..3.1. Assumptions of the Black and Scholes Option Pricing Model Black-Scholes model is perfectly accurate, for the inputs provided, if the assumptions are met. If the assumptions are met, it deduces the best possible present value (PV) estimate of a stock option, given the inputs. he original Black and Scholes model is based on the following assumptions: (1) European Options Black and Scholes assume that options are European. A European option is one that can be exercised only on a specified future date. In contrast to it, there are American options that can be exercised at any time up to the date on which the option expires. he original BS model assumes options to be European options. It can be extended to American options with a few adjustments. () No Dividends he model assumes that the underlying stock pays no dividends or other distributions. If this assumption is not made then it can affect the value of a call since when the stock goes ex-dividend the stock s price goes down by an amount reflecting the dividend per share, which reduces the value of calls and increases the value of puts. Later, the analysis is extended by assuming that the underlying asset pays dividends and accordingly the BS formula is adjusted for dividends, as is done by Merton. (3) Stock Price Behavior Most people agree that stock prices move randomly because of the efficient market hypothesis, but all say the same two things. First, history of the stock is fully reflected 6
in the present price and second, markets respond immediately to new information about the stock. With the previous two assumptions, changes in a stock price follow a Markov process in which only the present value is relevant for predicting the future. So, the stock price model states, that the predictions for the future price of the stock should be unaffected by the price one week, one month or one year ago. In the real world, stock prices are restricted to discrete values and changes in the stock price can only be realized during specified trading hours. Nevertheless, the continuousvariable, continuous-time model proves to be more useful than a discrete model. Another important observation to note is that the absolute change in the price of a stock is by itself, not a useful quality. For example, an increase of one Rupee in a stock is much more significant on a stock worth Rs. 10 than a stock worth Rs. 100. he relative change of the price of a stock is information that is more valuable. he relative change will be defined as the change in the price divided by the original price. A continuous stock price is assumed which is needed so that hedgers can adjust their hedge ratios, whenever stock prices change in order to keep a risk-free hedge. Large jumps in the stock prices make this needed revision in the hedge ratio impossible to achieve 14. he random process assumed by the model is called Geometric Brownian Motion (GBM). GBM assumes that the stock had drift plus shock. If a discrete price path is considered, it means, at each step the stock drifts up by some expected return plus random shock (where greater volatility implies greater shock). In other words, at each step, the stock will follow the linear drift but will end up shocked above or below the trend line. he model assumes that the asset prices are log normally distributed (driven by a single Brownian motion with a constantσ andµ ) and therefore, the continuously compounded 14 Some researchers believe that the existence of jumps in stock prices is one reason for some mispricings by the BS model, since jumps are equivalent to saying that the true distribution of returns has fatter tails than assumed by the lognormal distribution. A significant amount of empirical evidence on stock returns suggest that there are more large price changes than warranted by a lognormal distribution. In essence, this means that the true variability of the stock is not measured accurately by the variance of the returns, creating a bias in the model. his bias is more severe as the stock price moves away from the strike price. 63
asset returns are normally distributed 15. his assumption needs further explanation, which will help in understanding the model. Black and Scholes assume a Markov stochastic process for the behavior of a stock price over time. It is known as a Generalized Wiener process and describes the evolution of a normally distributed variable. he drift 16 of the process is a per unit time and the variance rate is b per unit time where a and b are constants. his means that if the value of the variable is x at time zero, at time it is normally distributed with mean x + a and standard deviation of b. It describes the stochastic process of a variable, like asset price, that is subjected to numerous small shocks. It keeps on changing randomly through time because of the effects of numerous independent random shocks caused by the receipt of new pieces of information. o satisfy the needs 17 of a model for asset prices, it is necessary for the Generalized Wiener process to be replaced by a more general type of stochastic process, known as Ito process. Doing so, we are provided with a deterministic expected element of return µ S t expressed as a function of asset price and time, and the variability in terms of 15 16 17 It may be noted that natural logarithms of the price relatives are the continuously compounded rates of returns. Drift is a statistical term referring to the bias, which exists within pricing of various derivative products. With regard to stochastic processes, the term drift is used to represent the positive or negative trend in the time series of the stochastic variable. he drift variable is strongly time dependent and is often used to adjust derivative prices to try and reflect random behavior. A drift rate of zero implies that the expected value of a variable at any time in the future, is equal to its current value. Common drift comes in the form of risk free rate minus the dividend yield (r-d). Needs of the model: (a) Different securities have different degrees of volatility, thus the impact of will be different between securities as reflected by the volatility of the security. his problem is easily addressed by scaling byσ, the annualized standard deviation of S. hus, σ still has an expected value of zero, but now has a standard deviation ofσ 1 = σ. hus, equation (.3) at the limit becomes: S = σ t Now S has an expected value of zero, a standard deviation of σ t and a variance ofσ t. (b) he second problem arises because the expected value of the random variable is zero and thus, the expected change in S, S is also zero. Yet the expected return from holding risky assets must be positive, on average, in order to reward investors for bearing risk. We must therefore, adapt the basic Wiener process to a generalized Wiener process to account for positive expected return. (c) Here,α is the absolute return per unit of time and is constant, but it is not independent of the asset price. However, investors require a percentage rate of return dependent upon the risks involved, which is therefore independent of the level of the asset price. hus, if investors need an 8% returns on a given asset, they require 8% whether the asset price is Rs 10 or Rs. 100. hus, the desired stochastic process of asset prices should incorporate an absolute return which is a function of the asset price, but a rate of return, which is independent of the asset price. o satisfy our needs for a model of asset price, it is necessary for the generalized Wiener process developed so far to be replaced by a more general type of stochastic process, known as an Ito process. 64