Equivalent Measures

24 Equivalent Measures 24.1 CHANGE OF MEASURE IN DISCRETE TIME (i) In Section 19.2 the arbitrage theorem was applied to a simple portfolio over a single time step. The analysis led to the concept of pseudo-probabilities, which could be calculated from the starting value of the stock and a knowledge of the two possible values after a single time step. It was emphasized that these pseudo-probabilities are not to be confused with actual probabilities; they are merely computational devices, despite the fact that they display all the mathematical properties of probabilities. If the concept is extended from a single step model to a tree, there is a pseudo-probability assigned to each branch of the tree. In the simplest trees, there are only two pseudo-probabilities: up and down, which are constant throughout the tree. However, we will keep the analysis more general and consider variable branching probabilities. N 0 x 0 0 x N 1 x N N x n m x Figure 24.1 N -Step tree (sample space) Consider a large binomial tree as shown in Figure 24.1. The underlying stochastic variable x i can take the values at the nodes of the tree which are written x n m (n: time steps; m: space steps from the bottom of the tree). The entire set of values x 0 0 , .,x N N has pre- viously been referred to as the sample space ,orthe architecture of the tree. The set of probabilities of up or down jumps at each of the nodes is collectively known as the probability measure P. From a knowledge of all the branching probabilities (it is not assumed that they are constant throughout the tree), we can easily calcu- late the probabilities of achieving any particular node in the tree; these outcome probabilities can equally be referred to as the probability measure, since branching and outcome probabilities are mechan- ically linked to each other. We use the notation p n m to denote the time 0 probability under the probability measure P of x i achieving the value x n m at the nth time step. Clearly, we start with p 0 0 = 1. We have already seen that in solving any option theory problem, we have discretion in choosing sample space and probability measure. For example, the Cox–Ross–Rubinstein and the Jarrow–Rudd sample spaces use different probability measures, but yield substantially the same answers when used for computation (see Chapter 7). The purpose of this chapter is to explore the effect of changing from one probability measure to another. (ii) The Radon–Nikodym Derivative: Consider two alternative probability measures P and Q, i.e two sets of outcome probabilities p 0 0 , ., p N N and q 0 0 , .,q N N . Either of these could be applied to the tree (sample space ). Let us now define a quantity ξ n m = q n m / p n m for each node of the tree. A new N-step tree is shown in Figure 24.2, similar to that for the x n m but with nodal values 24 Equivalent Measures equal to the quantities ξ n m . This tree (), together with a probability measure, defines a new stochastic process ξ i . From these simple definitions, we may write E Q [x N | F 0 ] = N  j=0 x N j q N j = N  j=0 x N j ξ N j p N j = E P [x N ξ N | F 0 ] (24.1) And similarly for any function of x N : E Q [ f (x N ) | F 0 ] = E P [ f (x N )ξ N | F 0 ] (24.2) Simply by putting f (x N ) = 1 in this last equation gives E P [ξ N | F 0 ] = 1 (24.3) ξ i is a process known as the Radon–Nikodym–derivative of the measure Q with respect to the measure P. The somewhat misleading notation dQ/dP is normally used, but the word derivative (reinforced by the differential notation) must not be taken to denote a derivative as in analytical differential calculus; ξ i is a stochastic process. N 0 x 0 0 x N 1 x N N x n m x N 0 N N Figure 24.2 Sample space or Radon–Nikodym derivative Care needs to be taken with one point: each of the ξ n m is the quotient of two probabilities and if the denominator is zero at any node, then the calculations blow up. We must have either p n m and q n m both non-zero or both zero for each possible outcome. If the measures fulfill this condition, we say that P and Q are equivalent probability measures. (iii) The Radon–Nikodym process has an additional interesting property which follows from equation (24.3). This equation holds true whatever the value of N, and since by definition ξ 0 = ξ 0 0 = 1, the process ξ i must be a martingale. (iv) It is a trivial generalization to extend equation (24.1) to the following which applies at step n: E Q [x N | F 0 ] = E P [x N ξ N | F 0 ] If we wish to express the expectation of x N at F 0 , subject to the condition that the process x i has previously hit a node with value x n m (or equivalently that ξ i has previously achieved ξ n m ), 276 24.2 CHANGE OF MEASURE IN CONTINUOUS TIME then we must introduce the probability of achieving x n m on each side of the last equation; but the probabilities must be expressed in the appropriate measure: q n m E Q  x N ; condition: x n = x n m   F 0  = p n m E P  x N ξ N N ; condition: x n = x n m   F 0  E Q  x N ; condition: x n = x n m   F 0  = 1 ξ n m E P  x N ξ N N ; condition: x n = x n m   F 0  (24.4) (v) From the foregoing paragraphs, we can distil the following general properties for a Radon– Nikodym derivative: r E Q [ f (x N ) | F 0 ] = E P [ξ N f (x N ) | F 0 ] r E P [ξ j | F i ] = ξ i i < j (martingale) r ξ 0 = 1; ξ i > 0 r ξ i E Q [ f (x N ) | F i ] = E P [ξ N f (x N ) | F i ] 24.2 CHANGE OF MEASURE IN CONTINUOUS TIME: GIRSANOV’S THEOREM (i) The probability measure in the tree of the last section was defined as the set of probabilities throughout the tree. In continuous time, the probability measure is the time-dependent fre- quency distribution for the process. For example, if W t is a standard Brownian motion, its frequency distribution is a normal distribution with mean 0 and variance t. The probability of its value lying in the interval W t to W t + dW t is dP t = 1 √ 2πt exp  − 1 2t W 2 t  dW t A change of probability measure which is analogous to changing the set of probabilities in discrete time simply means a change in the frequency distribution in continuous time. A Radon–Nikodym derivative can be defined which achieves this change in probability measure: ξ(x t ) = dQ t dP t = F Q (x t )dx Q t F P (x t )dx P t (24.5) where F P (x t ) and F Q (x t ) are the frequency distributions corresponding to the probability measures P and Q. As in the case of discrete distribution, the analysis breaks down if at any point we allow dQ t to remain finite while dP t is zero. Therefore only equivalent probability measures are considered, which ascribe zero probability to the same range of values of the variable x t . This is illustrated in Figure 24.3. Figure 24.3 Probability measures 277 24 Equivalent Measures The first frequency distribution is normal. The middle distribution is a little unusual, but is “equivalent” to the normal distribution. The third distribution is a very common one, but is not equivalent to the first, since it ascribes a value 0 outside the square range. (ii) We now investigate the process ξ T which is defined by ξ T = e y T ;dy t =− 1 2 φ 2 t dt − φ t dW t It is shown in Section 23.4(iv) that ξ T is a martingale and that ξ 0 = 1 and ξ t > 0. There- fore ξ T displays the basic properties of a Radon–Nikodym derivative which are set out in Section 25.1(v). The function ξ T will be applied as the Radon–Nikodym derivative in transforming a drifted Brownian motion dx P t = µ t dt + σ t dW P t (defined as having probability measure P) to some other stochastic process (defined as having probability measure Q). We are interested in dis- covering what that other stochastic process looks like, for the specific form of ξ T defined above. (iii) Before proceeding with the analysis, we recall a result described in Appendix A.2(ii). A normal distribution N (µT ,σ 2 T ) has a moment generating function given by M P () = E P  e x P T   F 0  = e µT + 1 2 σ 2 T  2 Furthermore, the moment generating function for the random variable x P T is unique, i.e. if x P T has the above moment generating function, then x P T must have the distribution N (µT ,σ 2 T ). The analogous results for a process x P t with variable µ t and σ t are obtained from the following modifications: µT = E P [x T | F 0 ] =  T 0 µ t dt σ 2 T = var[x T ] = E Q   T 0 σ 2 t dW t  2     F 0  =  T 0 σ 2 t dt The moment generating function result above can now be written more generally as M P () = E P  e x P T   F 0  = E P  exp    x 0 +  T 0 µ t dt +  T 0 σ t dW t      F 0  = exp    x 0 +  T 0 µ t dt  + 1 2  2  T 0 σ 2 t dt  (24.6) We will also use the following standard integral result which is a special case of the last equation with  = 1: E P  exp  x 0 +  T 0 µ t dt +  T 0 σ t dW t      F 0  = exp  x 0 +  T 0 µ t dt + 1 2  T 0 σ 2 t dt  (24.7) 278 24.2 CHANGE OF MEASURE IN CONTINUOUS TIME (iv) The effect of changing to probability measure Q by using ξ T as the Radon–Nikodym derivative is as follows: M Q [] = E Q  e x Q T   F 0  = E P  ξ T e x P T   F 0  = E P  exp  − 1 2  T 0 φ 2 t dt −  T 0 φ t dW t  exp    x 0 +  T 0 µ t dt +  T 0 σ t dW t      F 0  = E P  exp  x 0 +  T 0  µ t − 1 2 φ 2 t  dt +  T 0 (σ t − φ t )dW t      F 0  where we have used y 0 = 0 (since ξ 0 = 1). Use of equation (24.7) shows that the last equation may be written M Q () = exp  x 0 +  T 0  µ t − 1 2 φ 2 t + 1 2  2 σ 2 t − σ t φ t + 1 2 φ 2 t  dt  = exp    x 0 +  T 0 (µ t − λ t )dt  + 1 2  2  T 0 σ 2 t dt  (24.8) where we have arbitrarily defined λ t = σ t φ t . Comparing equations (24.6) and (24.8) leads us to the following conclusions which constitute Girsanov’s theorem. The effect on a drifted Brownian motion of changing probability measure by using the Radon–Nikodym derivative ξ t as defined above is as follows: r The measure P drifted Brownian motion is transformed into another drifted Brownian motion (with probability measure Q). r The variances of the two Brownian motions are the same. r The only effect of the change in measure is to change the drift by an instantaneous rate λ t which is defined above. Formally this may be written W Q T = W P T −  T 0 λ t d t or in shorthand dW Q T = dW P T − λ t dt The result is subject to the usual type of technical condition (the Novikov condition): E  exp  1 2  T 0 λ 2 t dt  < ∞ Girsanov’s theorem basically gives a prescription for changing the measure of a Brownian motion in such a way that it remains unchanged except for the addition of a drift. So what, you might ask? You can achieve the same effect by just adding a time-dependent term to the underlying variable W t ; what’s the big deal? We know already that the value of an option is the expected value of its payoff under some pseudo-probability measure. The value of the theorem is that it provides a recipe for applying this particular measure simply by adding a convenient drift term in the SDE governing the process in question. This procedure is explicitly laid out in the next section. 279 24 Equivalent Measures (v) Girsanov’s Theorem without Stochastic Calculus: This is a theorem of great power and usefulness in option theory, but it is worth a re-examination from the point of view of someone without a knowledge of stochastic calculus; it leads to an intuitive understanding which does much to demystify the theorem. Consider a stochastic variable x t (x 0 = 0) distributed as N(µt,σ 2 t). The probability distri- bution function of x t is n(x t ; µ, σ) = 1 √ 2πσ 2 t exp  − 1 2  x t − µt σ √ t  2  and the moment generating function has been shown to be M() =  +∞ −∞ e x t n(x t ; µ, σ)dx t = e µt + 1 2 σ 2 t  2 Remember that M() uniquely defines a distribution and all its moments can be derived from it. But by pure algebraic manipulation, the last equation could be written M() =  +∞ −∞ e x t n(x t ; µ, σ)dx t ≡  +∞ −∞ e x t  n(x t ; µ, σ) n(x t ; µ  ,σ)  n(x t ; µ  ,σ)dx t =  +∞ −∞ e x t  exp −1 2σ 2 t ((x t − µt) 2 − (x t − µ  t) 2 )  n(x t ; µ  ,σ)dx t =  +∞ −∞ e x t exp  − 1 2 φ 2 t − φ √ t  x t − µ  t σ √ t  n(x t ; µ  ,σ)dx t where φ = µ  − µ σ = e (µ  −φσ)t+ 1 2 σ 2 t 2 The conclusion to be drawn from this result is that any normal distribution, but always with the same variance σ 2 t, can be used to take the expectation of a function, but the function must be modified by multiplication by the factor exp[− 1 2 φ 2 t − φ √ t( x t −µ  t σ √ t )]. Alternatively expressed, an arbitrary choice of normal distribution (but always with the same variance) really only affects the drift term. This may be self-evident, if we remember that a normal distribution is entirely defined by the drift and variance, and it certainly takes some of the mystery out of Girsanov’s theorem. 24.3 BLACK SCHOLES ANALYSIS (i) The stochastic differential equation governing a stock price movement is assumed to be dS t = µ t S t dt + σ t S t dW RW t (24.9) where µ t is the drift observed in the real world and the superscript RW indicates that the Brownian motion is observed in the same real world. Our objective now is to find the measure under which the discounted stock price S ∗ t is a martingale; S ∗ t = S t B −1 t where B t is the zero coupon bond price. The reason we want to find the measure is that the value of an option can be found by taking the expectation of its payoff under this measure. With variable (but non-stochastic) interest rates, we can define the value of the zero coupon bond in terms of continuous, time-dependent interest rates r t as B −1 t = exp(−  1 0 r τ dτ ), so 280 24.3 BLACK SCHOLES ANALYSIS that d(B −1 t ) =−B −1 t r t dt. The process for S ∗ t can then be written dS ∗ t = d  S t B −1 t  = B −1 t dS t + S t d  B −1 t  = S t B −1 t  (µ t − r t )dt + σ t dW RW t  (ii) Girsanov’s theorem tells us that we can change the probability measure by changing the drift of the Brownian motion. Writing dW RW t = dW Q t − λ t dt, the last equation becomes dS ∗ t = S t B −1 t  (µ t − r t − λ t σ t )dt + σ t dW Q t  where Q is a new measure. This is a Q-martingale if the coefficient of dt is zero, i.e. if λ t = µ t − r t σ t (24.10) The term on the right-hand side of this last equation will be familiar to students of finance theory as the Sharpe ratio. It is normally referred to in option theory as the market price of risk. An important and much used property of λ t is that it is the same for all derivatives of the same underlying stock. Consider a stock whose process is given by dS t = µ t dt + σ t dW RW t where µ and σ are functions of S t . Now consider two derivatives; Ito’s Lemma means that we can write the processes for these as f (1) t = µ (1) t dt + σ (1) t dW RW t ; f (2) t = µ (2) t dt + σ (2) t dW RW t Let us construct a portfolio consisting of f (2) t σ (2) t units of the derivative f (1) , and − f (1) t σ (1) t units of f (2) . The portfolio value is π t = f (1) t f (2) t σ (2) t − f (1) t f (2) t σ (1) t = f (1) t f (2) t  σ (2) t − σ (1) t  A change in the value of this portfolio over an infinitesimal time step dt is dπ t = f (2) t σ (2) t d f (1) t − f (1) t σ (1) t d f (2) t = f (1) t f (2) t  σ (2) t µ (1) t − σ (1) t µ (2) t  dt since the dW RW t terms cancel. But if the return is not stochastic (i.e. is risk-free), then the return must equal the interest rate: σ (2) t µ (1) t − σ (1) t µ (2) t σ (2) t σ (1) t = r t or µ (1) t − r t σ (1) t = µ (2) t − r t σ (2) t (= λ t ) (iii) Let us now return to equation (24.9) and rewrite this in terms of the measure Q, using the above value for λ t . Simple substitution gives us dS t = r t S t dt + σ t S t dW Q t (24.11) In a nutshell, we have changed the real-world SDE by changing to the alternative measure Q which turns the discounted stock price into a martingale; the effect of this switch is merely to replace the real-world stock drift by the risk-free interest rate. The measure is therefore usually referred to as the risk-neutral measure. This analysis is simply a sophisticated re-statement of the principle of risk neutrality on which we based the first three parts of this book. (iv) Continuous Dividends: The effect of a continuous dividend rate q is easy to include in the above framework. We use constant rates for simplicity. The effect of a dividend is that the holder of the shares receives a cash throw-off. It was shown in Chapter 1 that this can be 281 24 Equivalent Measures incorporated into the calculations by writing the stock price as S t e +qt . The discounted share value is therefore S ∗ t = S t e qt e −rt so that dS ∗ t = S ∗ t  (µ − (r − q)) dt + σ dW RW t  or from the previous analysis dS t = (r − q)S t dt + σ S t dW Q t (24.12) (v) Forward Price: This can be written F t = S t e (r−q)(T −t) . We saw in the last subsection that S t e −(r−q)t is a Q-martingale, i.e. dS ∗ t = S ∗ t σ dW Q t . Multiply both sides by e (r−q)T to give dF t = σ F t dW Q t (24.13) Using the results of Section 23.4(iv) gives F t = e − 1 2 σ 2 t+σ dW Q t (24.14) 282 25 Axiomatic Option Theory 25.1 CLASSICAL VS. AXIOMATIC OPTION THEORY (i) In the first three parts of this book, option theory was developed from a very few key concepts: (A) Perfect Hedge: An option may be perfectly hedged. Previously, it was just assumed that this is possible, and it was shown in Section 4.4 that if such a hedge exists, then it must be a self-financing portfolio. Now, using arbitrage arguments we have shown that a discounted derivatives price is a martingale. It was also shown that the discounted value of a self-financing portfolio consisting of stock plus cash is also a martingale; the martingale representation theorem therefore proves that an option can be perfectly hedged. The reader who has the inclination to play with these two set of arguments as explicitly laid out in Sections 4.3 and 21.5 will quickly realize how closely the two analyses are related. Stochastic theory has just added a lot of fancy words. (B) Risk Neutrality: For discrete models, the derivation of risk neutrality is very closely related in classical and axiomatic option theory. In both cases, we start by examining a single step: in the classical case, we get the result in Section 4.1 by saying that a portfolio consisting of an option plus a hedge must be risk-free and therefore have a return equal to the interest rate; in the axiomatic case, we use the arbitrage theorem to prove the same result. For continuous models, the arguments appear to diverge rather more. We inferred risk neutrality in the classical case from the fact that the real-world drift does not appear in the Black Scholes equation. In axiomatic theory, risk neutrality falls out of the application of Girsanov’s theorem and a consideration of the properties of martingales. (C) The Black Scholes Equation: This was derived in Section 4.2 by constructing a continuous time portfolio of derivative plus hedge and requiring its rate of return to equal the interest rate. In Section 23.5 it appears as a consequence of the fact that the discounted option price is a P-martingale, which in turn is a consequence of the arbitrage theorem. Both derivations are critically dependent on Ito’s lemma, which was introduced with a lot of hand waving in Section 3.4 and which is of course a central pillar of stochastic calculus. It is not possible to derive an options theory without some recourse to stochastic calculus, albeit the very rough and ready description of Ito’s lemma given in Chapter 3. (D) Risk-neutral Expectations: Use of these to price options was introduced with a minimum of fuss (or rigor) in Section 4.1. Using the axiomatic approach, it was shown in Section 22.3(iv) to result from the fact that the discounted option price is a P-martingale (and hence from the arbitrage theorem). (ii) At this point the reader faces the awkward question “was it all worth it?” Despite our rather robust approach, stochastic calculus has been seen to be a tool of great subtlety; but we don’t seem to have any additional specific results to what we had before. Without beating about the bush, our view is that if someone is interested in equity-type options (including FX and commodities), he is likely to find most results he needs through the classical statistical approach to option theory. His main problem will be reading the technical 25 Axiomatic Option Theory literature, which tends to use stochastic calculus whether or not it is needed to explain option theory. But for anyone interested in interest rate options, a knowledge of stochastic calculus is indispensable. Having said this, the next two sections deal with two topics which demand fairly advanced stochastic techniques. The first is the question of whether an American option is indeed a hedgeable instrument, which is by no means self-evident from the last few chapters. The second is the so-called stop–go paradox, which lies at the very heart of option theory; its resolution is quite subtle and it puzzled theorists for some while. The remainder of the chapter uses stochastic calculus to re-derive a number of previous results more elegantly and sometimes more convincingly. 25.2 AMERICAN OPTIONS The axiomatic option theory developed in this part of the book has so far only examined the case of European options. American options are analytically more difficult to handle; but they are also commercially more common, so we need to be sure that the mathematical rules do not break down when the possibility of early exercise is allowed. For example, if the possibility of early exercise were to destroy the martingale property of the discounted option price, the option might no longer be hedgeable! This subject will need a little bit of a mathematical detour, as we first need to introduce some new concepts. (i) Consider an American option in a discrete time framework. f n is the no-arbitrage value of the option at step n and K n is the exercise value at the same point in time; f ∗ n and K ∗ n are the discounted values B −1 n f n and B −1 n K n . Suppose our model has N steps and the option has got as far as step N − 1 without being exercised. There are two possibilities at this point: r It is not worth exercising the option, in which case its value is the same as for the corre- sponding European option: f ∗ N −1 = E Q [ f ∗ N | F N −1 ] where Q is the risk-neutral probability measure. r It is better to exercise immediately and receive the payoff K N −1 . Taking these two possibilities together gives the price of an American option at N − 1as f ∗ N −1 = max[K ∗ N −1 , E Q [ f ∗ N | F N −1 ]] Generalizing this to any point in the process gives f ∗ n = max  K ∗ n , E Q [ f ∗ n+1 | F n ]  (25.1) (ii) Snell’s Envelope: This expression for f ∗ n clearly means that the nice martingale property of a discounted option price has been destroyed; this is a little worrying, given the extent to which the martingale property was exploited in the European case. From its definition in equation (25.1), f ∗ n is a Q-supermartingale, i.e. f ∗ n ≥ E Q [ f ∗ n+1 | F n ]. Consider this result written in a slightly different format: f ∗ n =  E Q [ f ∗ n+1 | F n ]ifE Q [ f ∗ n+1 | F n ]>K ∗ n K ∗ n if E Q [ f ∗ n+1 | F n ]<K ∗ n 284 [...]... Kolmogorov backward equation with appropriate boundary conditions, using the method of images This gave the appropriate distribution function to work out the risk-neutral expectations of the payoffs An equivalent but much cooler derivation of the same result is now given using stochastic calculus (i) Reflection Principle for Standard Brownian Motion: This very slick theorem relies on the symmetry properties... become dxt = (u t − xt )(r − q) dt + (u t − xt ) σ dWtQ dvt = vt q dt + vt (ϕt − σ ) dWtQ 297 (25.26) 25 Axiomatic Option Theory Let vt = vt e−qt Using Ito’s lemma, we can show that vt is a martingale; or equivalently expressed: + vt = e−q(T −t) E Q [vT | Ft ] = e−q(T −t) E Q [X T | Ft ] (25.27) (iv) The problem can now be presented as a stochastic control problem as follows: r If dxt = (u t − xt )(r − . Probability measures 277 24 Equivalent Measures The first frequency distribution is normal. The middle distribution is a little unusual, but is equivalent . zero for each possible outcome. If the measures fulfill this condition, we say that P and Q are equivalent probability measures. (iii) The Radon–Nikodym process