Advanced Derivatives Course Chapter 4

Week 4: Random Walks – Application To Share Price Movements. Analogy To Brownian Motion (see also P.Wilmott, Chapter 3)

Lecture IV.1 Continues Random Variables The goal of the next two lectures is to introduce mathematical conceptions, like a Wiener process, and models of continuous random processes, which will be used for description of market share price movements. The reason why we need to model share price movements is that we want to implement a particular technique for pricing derivatives by using risk-less portfolio. This technique allows a derivative user to dynamically analyse not only the derivative’s value but the hedging strategy as well. Share prices are not trees. The discrete trees of the previous lectures are only an approximation to the way that prices actually move. In practice, a price can change at any instant, rather than just at some fixed tick-times when a portfolio can be appropriately rebalanced. The binary choice of a single jump ‘up’ or ‘down’ only becomes subtle as the ticks get closer and closer, giving the tree more and ever-shorter branches. But such trees grow too complex and we stop being able to see the wood. ‘The tree approach’ becomes even more difficult to use when we are dealing with perpetual derivatives for which there is no expiration date (like in the case of the Russian option which we will discuss in later lectures). But can we do better than simply growing trees? The rest of the course will provide us with a positive answer to this question. In order to understand how we model security price movements, it is necessary to understand random events that at any time moment can have an infinite number of possible outcomes. In probability theory, these are called continuous random variables. At the same time, we know ht at in reality shares do not take on an infinite number of values. Why, then, do we want to use continuous random variables to model them? By modelling random events with continuous random variables, we give ourselves access to a great deal of mathematical machinery that can simplify computations. The sacrifice, of course, is that we know share prices and share returns are not “continuous”. That is, there are only a finite number of possible share prices, and every price between two prices cannot be achieved. Despite this, the hope is, and experience tells us, these assumptions are “not too bad”. That is, while they are not perfect, the gain made in terms of mathematical efficiency and power often wins out over the losses incurred by making the continuity assumption. We now begin our study of random variables. The single binomial branching was the building block for our ‘realistic’ market. For the continuous world we need an analogous basis – something simple and yet a reasonable starting point for realism. The basic conception underlying finance in continuous time is normal distribution. Normal Distribution

Consider a trader who follows the price of an exchange-traded derivative asset V(t) in real time, using a service such as Reuters, Bridge, or Bloomberg. The price V(t) changes continuously over time, but the trader is assumed to have limited scope of attention and checks the market price every ät seconds (do not confuse a time interval ät with a product ä×t). We assume that ät is a small time interval. In other words, we approach modelling of a continuous process as a certain limit of a discrete process. More importantly, by analogy with binomial trees, we assume that at any time t there are two possibilities: 1. There is either an uptick and prices increase according to δ V (t ) = + a δ t ,

a > 0.

2. Or, there is a downtick and prices decrease by δ V (t ) = −a δt , where äV(t) represents the change in the observed price during the “small” time interval ät. This is where our experience with binomial trees becomes useful. All other outcomes that may very well occur in reality are assumed for the time being to have negligible probability. Then for fixed t, ät, the äV(t) becomes a binomial random variable . In particular, äV(t) can assume only two possible values with the probabilities P(δV (t ) = + a δt ) = p , P(δV (t ) = −a δt ) = 1 − p. The time index t starts from t0 and increases by multiples of ät:

t = t 0 , t0 + δ t ,..., t0 + nδ t ,... At each time point a new V(t) is observed. Each new increment äV(t) will equal either +a δ t or - a δ t . If the äV(t)’s are independent of each other, the sequence of increments äV(t), will be called a binomial stochastic process, or simply a binomial process. Now consider the following experiment with random variable V(t). We ask the computer to calculate many realizations of V(t). Then, beginning from the same initial point V(0) we plot these trajectories. Beginning from t0=0, in the immediate future, V(t) had only two possible values:

V (0) + a δt with probabilit y p , V ( 0 + δt ) =  V (0) − a δ t with probabilit y 1 − p. Hence, V(t) itself is binomial at t=0+ät. But, if we let some more time pass and then look at V(t), say, at t=2ät, V(t) will assume one of three possible values. More precisely, we have the following possibilities: V (0) + a δ t + a δt with probabilit y p 2 ,  V ( 2δt ) =  V (0 ) − a δ t + a δ t ......... 2 p (1 − p ),  V (0) − a δt − a δ t ......... (1 − p ) 2 .  That is to say, V(2ät) may equal V (0) + 2a δt ,  V ( 2δ t ) =  V (0), V (0 ) − 2 a δ t .  of these, the outcome V(0) is most likely if there is a 50-50 chance of an uptick. Indeed, for p=1/2 P(V (0) + 2a δt ) = 1 / 4, P(V (0)) = 1 / 2, P(V (0) − 2 a δ t ) = 1 / 4. Now consider possible values of V(t) once some more time elapses passed. Several more combinations of up-ticks and downticks become possible. For example, by the time t=5ät, one possible but “extreme” outcome may be V (5δt ) = V ( 0) + a δ t + a δt + a δ t + a δt + a δt = V ( 0) + 5 a δt . Another “extreme” may be to get five downticks in a row: V (5δt ) = V ( 0) − a δt − a δ t − a δt − a δ t − a δt = V ( 0) − 5a δt . More likely are combinations of up-ticks and downticks. For example, V (5δt ) = V ( 0) − a δt + a δ t − a δt + a δ t + a δt = V ( 0) + a δt . or V (5δt ) = V ( 0) − a δt + a δ t + a δt − a δ t + a δt = V ( 0) + a δt . are two different sequences of price changes, each resulting in the same price at time t=5ät. There are several other possibilities. In fact, we can consider the general case and try to find the total number of possible values the derivative V(nät) can take. Obviously, as n , V(nät) may take any of a possibly infinite number of values. A similar conclusion can be

reached if ät 0 and n while the product nät remains constant 1 (in physics this limit is called double scaling limit). In this case, we are considering a fixed time interval and subdividing it into finer and finer partitions. Probability Density Function

Assume that p=1/2 and that V(0)=0. Then, for fixed nät =T and “large” n, the distribution of V(nät) can be approximated by a distribution function whose probability density function is given by g (V (T ) = x ) =

1 2π a 2T

−

e

x2 2 a 2T

,

which is known as a normal probability distribution with mean 0 and variance 2 a2 T. In the option pricing the normal distribution is the most important of all distributions. It is of primary importance in share price modelling. The normal distribution will be used repeatedly throughout these lectures. As n goes to infinity, the random variable V(nät)=V(T) becomes a continuous random variable and will converge to what is called a Wiener process or a Brownian motion. Since we chose time T arbitrarily, the function V(T) is a normally distributed variable at all times. Above we studied random variables with a finite number of outcomes. In this situation we could easily define the probability of each outcome, even if there were millions of them. But as n goes to infinity, matters become more difficult because the number of outcomes is infinite: We can no longer sensibly talk about the probability of a single outcome. The point is that if we assign a positive probability to each outcome and try to sum them up, the net result is always infinite. Therefore, we cannot make sense of the probability of a single outcome for a continuous random variable.

1

To see that the derivative V(nät) can take an infinite number of values, notice that one possible outcome is and ät 0. Therefore, the derivative can take all values V (0) + na δt . The last term goes to infinity when n between V(0) and infinity. 2 The variance measures the dispersion of the outcomes about the mean of the random variable. Mathematically, Var(x)=E(x2)-E2(x), where E(.) is mean or expected value of x.

The way we handle this mathematically is to only consider ranges of events, such as all the returns between 10% and 11%. Therefore, when we think of continuous random variables, we can still think in terms of outcomes and probabilities, but rather than focusing on single outcome, we restrict our attention to ranges of outcomes. The probability that values of the function lie between x and x+äx is given as δ P = g ( x)δ x. The corresponding distribution function, which gives us the values of outcomes, does not have a closed-form formula. It can only be represented as an integral. We will denote this distribution function as N(0,a2T). Here the first argument in function is mean and the second argument is variance. Thus, the price in the example considered above can be presented as V(T)=N(0,a2T). Note that the function N(0,a 2T) is not an ordinary function. When we change T, the function changes randomly according to the normal distribution. That is, each time we start to change the function from the same moment t0=0 and the same value V(t0), it will follow a different path. Ordinary functions don’t do this. They always follow the same path.

Lecture IV.2 Brownian Motion This will be the only lecture with some direct “physical” input. The name Brownian motion has its origins in a physical description of the motion of a heavy particle suspended in a medium of light particles. The light particles move around rapidly, and as a matter of course, occasionally randomly crash into the heavy particle. Each collision slightly displaces the heavy particle; the direction and the magnitude of this displacement is random and independent from all the other collisions, but the nature of this randomness does not change from collision to collision (in the language of probability theory, each collision is an independent, identically distributed random event). Brownian motion is a zigzag trajectory of the microscopic particle under the continuous buffeting of a gas observed by botanist Robert Brown. Mathematically, the process W(t) (in the previous lecture we considered a random process V(t) which is a particular example of the process W(t)) is a Brownian motion if and only if 1) W(t) is continuous and W(0)=0; 2) The value of W(t) is distributed as a normal random variable N(0,t); 3) The increment W(s+t)-W(s) is distributed as a normal N(0,t) and is independent of the history of what the process did up to time s. Just for your information, a Brownian motion is a particular type of Markov process. In its turn, a Markov process is a particular type of random process where only the present value of a variable is relevant for predicting the future. It does not matter what scale you examine Brownian motion on – it looks just the same. That is, the randomness does not smooth out as we zoom in. The Brownian motion model for particles fits real-world observations of “Brownian motion particles” and thus can be regarded as a mathematical explanation for this behaviour. Now, surprisingly, share prices have many characteristics in common with Brownian motion particles. To see the relation, imagine prices as heavy particles that are jarred around by lighter particles, trades. Indeed, each trade moves the price slightly. Brownian Motion As Share Model Taken as it is, BM has mean zero, whereas the share of a company normally grows at some rate – and historically we expect prices to rise if only because of inflation. But we can add in a drift by hand. In a small time interval ät, the change in the value of a share S can be presented as follows δ S = µδt + σ ⋅ ε δ t , where ì and ó are constant and å is a random drawing from a standardized normal distribution (i.e. a normal distribution with mean zero and variance equal to 1). Thus äS has a normal distribution with: • Mean E (δ S ) = µδt ;

•

Standard deviation σ (δ S ) ≡ Var (δS ) = σ δ t ;

•

Variance Var(δS ) = σ 2δ t.

The value of S during a relatively long period of time T can be regarded as the sum of the increases in S in n small time intervals of length ät, where n=

T . δt

This procedure is known as binning. Thus, n

S (T ) = S (0) + ∑δ S i , i =1

where

δSi = µδt + σεi δt . We get E (S (T )) = S (0) + µT , Var( S (T )) = σ 2 T . Compare with V(T) from the previous lecture. Let us introduce the following notation δW = ε δt for the change over period ät of a Brownian-Wiener process W. It obeys the following relations: E (δ W ) = 0, Var(δW ) = δ t. Then δ S = µ ⋅ δ t + σ ⋅ δW .

The last formula describes arithmetic Brownian motion shown on Fig.

Arithmetic Brownian Motion 100.04

100.03

100.01

Share

100.02

100

99.99

99.98

Time

Geometric Brownian Motion It is tempting to suggest that a share price follows a generalized (or arithmetic) BrownianWiener process; that is, that it has a constant drift rate and a constant variance rate. One obvious problem with such a model is that arithmetic Brownian motion describes variables, which may become negative. This might be suitable for describing net cash flows, however, it is absolutely inappropriate for shares, since the last cannot become negative. Moreover, this model fails to capture a key aspect of share prices. This is that the expected percentage return (dS/S) required by investors from a share is independent of the share’s price. If investors require a 14% per annum expected return when share price is €10, then they will also require a 14% per annum expected return when it is €50. Clearly, the constant expected drift-rate assumption is inappropriate and needs to be replaced by the assumption that the expected drift, expressed as a proportion of the share price, is constant. The later implies that if S is the share price, the expected drift in S is ìS for some constant parameter, ì. Thus, in a short interval of time, ät, the expected increase in S is ìSät. The parameter ì is the expected rate of return on the share, expressed in decimal form. Also, a reasonable assumption is that the variance of the percentage return in a short period of time, ät, is the same regardless of the share price. In other words, an investor is just as uncertain as to his or her percentage return when the share price is €50 as when it is €10. Define ó2 as the variance rate of the proportional change in the share price. This means that ó 2ät is the variance of the proportional change in the share price in time ät and that ó2 S 2ät is the variance of the actual change in the share price, S, during ät. The instantaneous variance rate of S is therefore ó 2 S2 .

Geometric Brownian Motion Time 230 210 190

150 130

Share

170

110 90 70 50

These arguments suggest that S can be represented as geometric Brownian motion (shown on picture): δ S = µS δt + σSδ W This equation is the most widely used model of share price behaviour. GBM is related to ABM according to the following formula δS = µδt + σδ W , S which shows that while the share price, S, cannot get negative, the return, äS/S, can. The variable ó is usually referred to as the instantaneous share price volatility. The variable ì is its instantaneous expected rate of return. We will return to the discussion of geometric Brownian motion in Lecture VII.2. Try To Answer The Following Questions 1) Can you explain the conception of continuous random processes (random walks) in the context of share price movements? What are the modelling assumptions we have to make to use these processes for describing share price behaviour? 2) Which probability distribution underlies continuous random processes? Can you characterize this distribution? 3) What is Brownian motion? Why (arithmetic) Brownian motion, taken as it is, is not suitable for share markets? 4) What is the most widely used model of share price behaviour? Can you discuss its features as compared with ordinary (or arithmetic) Brownian motion?