6 Martingales in continuous time Just as in discrete time, the notion of a martingale will play a key r^ole in our continuous time models. Recall that in discrete time, a sequence ; 1 ;::: ; n for which [j r j] < 1 for each r is a martingale if [ r jf r,1 ]= r,1 : The sequence ff r g n r= is called a ltration. In order to avoid abstract measure theory, we omit a detailed discussion of ltrations in continuous time. Instead we content ourselves with the following `working denition'. Denition 6.1 The symbol F t denotes the `information generated by the stochastic process on the interval [;t]'. If, based upon observations of the trajectory f(s); s tg, it is possible to decide whether a given event A has occured or not, then we write this as A F t : If the value of a stochastic variable can be completely determined given observations of the trajectory f(s); s tg then we also write Z F t : If Y is a stochastic process such that we have Y (t) F t say that Y is adapted to the ltration ff t g t. for all t, then we This denition is only intended to have intuitive content. Nevertheless, it is rather simple to use. xample 6. 1. Dene A by Then A F 18 but A = F 17. A = f(s) 3:14; 8s 18g:. For the event A = f(1) > 8g, A F s if and only if s 1. 3. The stochastic variable is in F s if and only if s 5. Z(s) = Z 5 (s)ds 4. If B t is Brownian motion and M t = max st B s, then M is adapted to the Brownian ltration. 5. If B t is Brownian motion and ~ Mt = max st+1 B s, then ~ M is not adapted to the Brownian ltration. Denition 6.3 Consider a probability space (; P) and a ltration ff t g t on this space. An adapted family fm t g t of random variables on this space with [jm t j] < 1 for all t is a martingale if, for any s t, P [M t jf s ]=M s :
Remarks: 1. Often we shall be sloppy about specifying the ltration. In all of our examples there will be a Brownian motion around and it will be implicit that the ltration is that generated by the Brownian motion.. Many stochastic calculus texts specify a probability space as (; ff t g t ; P), thereby specifying the ltration explicitly from the very beginning. 3. The natural ltration is the name given to the ltration for which F t consists of those sets that can be decided by observing trajectories of the specied process up to time t. There are other possibilities, but we always use the natural ltration corresponding to Brownian motion in our examples. Lemma 6.4 If fb t g t ff t g t, then is a standard Brownian motion generating the ltration 1. B t is an F t -martingale.. B t, t is an F t -martingale. 3. exp B t, t is an F t -martingale (called an exponential martingale). Proof: The proofs are all rather similar. For example, consider M t = B t, t. vidently [jm t j] < 1. Now Thus B t, B s Fs = (Bt, B s ) +B s (B t,b s ) Fs = (B t,b s ) Fs +Bs [(B t,b s )jf s ] = t,s: B t, t Fs = B t, B s + B s, (t, s), s Fs =(t,s)+b s,(t,s),s=b s,s: Theorem 6.5 (Optional Sampling Theorem) If fm t g t is a continuous martingale with respect to the ltration ff t g t and if 1 and are two stopping times such that 1 K where K is a nite real number, then M is integrable (that is has nite expectation) and Remarks: [ M jf 1 ]=M 1 ; P,a:s: 1. The term `a.s.' (almost surely) means with (P-) probability one.
. Notice in particular that if is a bounded stopping time then [M ]=[M ]. 3. The name Optional Sampling Theorem is used because stopping times are sometimes also called optional times. We illustrate the application of this by calculating the moment generating function for the hitting time T a of level a by Brownian motion. Proposition 6.6 Let B t be a Brownian motion and let T a = inffs :B s =ag (or innity if that set is empty). Then e,ta = e, p jaj : Proof: We assume that a. (The case a< follows by symmetry.) We apply the Optional Sampling Theorem to the martingale M t = exp B t, 1 t : We cannot apply it directly to T a as it may not be bounded. Instead we take 1 =and = T a ^ n. This gives us that Now [M Ta^n] =1: M Ta^n = exp B Ta^n, 1 (T a ^ n) exp (a) : On the other hand, if T a < 1, lim n!1 M Ta^n = M Ta, and if T a = 1, B t a for all t and so lim n!1 M Ta^n =. From the Dominated Convergence Theorem, Ta<1 exp, 1 T a + a = lim [M T n!1 a^n] =1: Taking =completes the proof. Warning: It was essential that there was a dominating random variable here. In this setting, the Dominated Convergence Theorem says that for a sequence of random variables Z n, with lim n!1 Z n = Z, if there is a random variable Y with jz n jy for all n and [Y ] < 1, then we may deduce that [Z] = lim n!1 [Z n ]. In this example, the dominating random variable is just the constant e a. 7 Stochastic integration and It^o's formula We already saw (in Lemma 6.4) a number of random variables that are martingales with respect to the same probability measure and adapted to the same ltration. All those martingales were expressed in terms of the underlying Brownian motion. In this section, we shall see that this situation is generic: if there is a measure Q under which the process M t is an F t -martingale, then any other Q-martingale adapted to the same ltration F t can be expressed in terms of M t (modulo some
technical assumptions). To understand this representation, we must rst understand the notion of stochastic integral. Processes that model stock prices are usually functions of one or more Brownian motions. Here, for simplicity, we restrict ourselves to functions of just one Brownian motion. The rst thing that we should like to do is to write down a dierential equation for the way in which the stock price evolves. The diculty is that Brownian motion is `too rough' for the familiar Newtonian calculus to be any help to us. Suppose that the stock price is of the form S t = f(b t ). Formally, using Taylor's Theorem (and assuming that f at least is nice), f(b t+t ), f(b t ) = (B t+t, B t ) f (B t ) (4) + 1! (B t+t, B t ) f (B t )+ : Now in our usual derivation of the chain rule, when B t is replaced by a Lipschitz function, the second term on the right hand side is order O(t ). However, for Brownian motion, we know that [(B t+t, B t ) ] is t. Consequently we cannot ignore the term involving the second derivative. Of course, now we have a problem, because we must interpret the term involving the rst derivative. If (B t+t, B t ) is O(t), then (B t+t, B t ) should be O( p t), which could lead to unbounded changes in y over a bounded time interval. However, things are not hopeless. The expected value of B t+t, B t is zero, and the uctuations around zero are on the order of p t. By comparison with the Central Limit Theorem, we see that it is possible that S t, S is a bounded random variable. Assuming that we can make this rigorous, the dierential equation governing S t = f(b t ) will take the form ds t = f (B t )db t + 1 f (B t )dt: It is convenient to write this in integrated form, S t = S + f (B s )db s + 1 f (B s )ds: (5) We have to make rigorous mathematical sense of the stochastic integral (that is, the rst integral) on the right hand side of this equation. The key is the following fact. Brownian motion has nite quadratic variation. Before proving this we dene total variation and quadratic variation. For a function f :[;T]!R, its variation is dened in terms of partitions. Denition 7.1 Let be a partition of [;T], N() the number of intervals that make up and () be the mesh of (that is the length of the largest interval in the partition). Write t i ;t i+1, for the endpoints of a generic interval of the partition. Then the variation of f is lim! 8 < : sup :()= N () 1 9 = jf(t j+1 ), f(t j )j ; :
If a function is `nice', for example dierentiable, then it has bounded variation. Brownian motion has unbounded variation. Denition 7. The quadratic variation of a function f is dened as q:v:(f) = lim! 8 < : sup :()= N () 1 jf(t j+1 ), f(t j ))j 9 = ; : Notice that quadratic variation will be nite for functions that are much rougher than those for which the variation is bounded. Roughly speaking, nite quadratic variation will follow if the uctuation of the function over an interval of length is order p. We can now be more precise about the quadratic variation of Brownian motion. Theorem 7.3 Let B t dene denote Brownian motion and for a partition of [;T] S() = N() Btj, B tj,1 : Let n be a sequence of partitions with ( n )!. Then js( n ), T j! as n!1: Proof. We expand the expression inside the expectation and make use of our knowledge of the normal distribution. So, rst observe that js( n ), T j = Write n;j for Btn;j N ( n) n o Btn;j, B tn;j,1, (t n;j, t n;j,1 ) :, B tn;j,1, (t n;j, t n;j,1 ). Then js ( n ), T j = N ( n) n;j + j<k n;j n;k : Note that since Brownian motion has independent increments, and [ n;j n;k ]=[ n;j ] [ n;k ]= if j 6= k; h n;j = Btn;j, B tn;j,1 4, Btn;j, B tn;j,1 i (t n;j, t n;j,1 )+(t n;j, t n;j,1 ) Now for a normally distributed random variable with mean zero and variance, it is easy to check that [jj 4 ]=3,so that n;j = 3(tn;j, t n;j,1 ), (t n;j, t n;j,1 ) +(t n;j, t n;j,1 ) = (t n;j, t n;j,1 ) ( n )(t n;j, t n;j,1 ) : :
Summing over j js( n ), T j N (n) ( n )(t n;j, t n;j,1 ) = ( n )T! as n!1: This result is not enough to dene the integral R f(b s )db s in the classical way, but it is enough to allow us to essentially mimic the construction of the (Lebesgue) integral, at least for functions for which [f (B )] L 1 [;T]. However, although the construction of the integral may look familiar, its behaviour is far from familiar. We rst illustrate this by dening R T B sdb s. From classical integration theory we are used to the idea that f(x s )dx s = N (),1 lim ()! j= Let us dene the stochastic integral in the same way, that is B s db s = N (),1 lim ()! j= Consider again the quantity S() of Theorem 7.3. S() = = N () N (), Btj, B tj,1 f(x tj ), x tj+1, x tj : (6) B tj, Btj+1, B tj : (7) n, o B t j, B t j,1, B tj,1 Btj, B t;j,1 N (),1 = B T, B, j= B tj, Btj+1, B tj : The left hand side is T (by Theorem 7.3) and so letting ()! and rearranging we obtain B s db s = (B T, B, T ) Remark. Notice that this is not what one would have predicted from classical integration theory. The extra term in the stochastic integral corresponds to S(). In equation (6), we use f(x tj ) to approximate the value of f on the interval (t j ;t j+1 ), but in the classical theory we could equally have taken any other point in the interval in place of t j and, in the limit, the result would have been the same. In the stochastic theory this is no longer the case. On the problem sheet you are asked to calculate two further limits
1. The limit as ()! of N(),1 j= B tj+1, Btj+1, B tj :. lim ()! N (),1 j= Btj + B tj+1,btj+1, B tj : By choosing dierent points within each subinterval of the partition with which to approximate f over the subinterval we obtain dierent integrals. The It^o integral is dened as f(b s )db s = N () lim ()!, f(b tj ) B tj+1, B tj : The Stratonovich integral is dened as N () f(btj )+f(b tj+1 ) f(b s ) db s = lim ()!,Btj+1, B tj : The Stratonovich integral has the advantage from the calculational point of view that the rules of Newtonian calculus hold good. From a modelling point ofview, at least for our purposes, it is the wrong choice. To see why, think of what is happening over an innitesimal time interval. We might be modelling, for example, the value of a portfolio. We readjust our portfolio at the beginning of the time interval and its change in value over the innitesimal tick of the clock is beyond our control. A Stratonovich model would allow us to change our model now on the basis of the average of the value corresponding to current stock prices and the value corresponding to prices after the next tick. We don't have that information when we make our investment decisions. Consider then the It^o integral. We have evaluated it in just one special case. We increase our repertoire in the same way as in the classical setting by rst considering the value on simple functions. Denition 7.4 A simple function is one of the form f(b s )= n i=1 a i (B s ) Ii (s); where I i = [s i ;s i+1 ), [ n i=1 I i = [;T), I i \I j = f;g if i 6= j and the functions a i satisfy [a i (B s ) ] < 1. By our denition, f(b s )db s = n, a i (B s ) B si+1, B s : i=1
Now, just as for regular integration, we approximate more general functions by simple functions and pass to a limit. We have to be sure, however, that the integrals converge when we pass to such a limit. This will not be true for all functions that can be approximated by simple functions. The next Lemma helps identify the space of functions for which we can reasonably expect a nice limit. Lemma 7.5 Suppose that f is a simple function, then 1. R t f s(b s )db s is a continuous F t -martingale.. 3. " " # f(b s )db s = sup tt f(b s ) ds: # f(b s )db s 4 f(b s ) ds: Remark: The second assertion is the famous It^o isometry. It suggests that we should be able to extend our denition of the integral to functions such that R t [f s(b s ) ]ds < 1. Moreover, for such functions, all three assertions should remain true. In fact one can extend the denition a little further, but the integral may then fail to be a martingale and this property will be important to us. Proof. The third assertion follows from the second by an application of a famous result of Doob: Theorem 7.6 (Doob's inequality) If fm t g tt then sup tt M t 4 MT : is a continuous martingale, We omit the proof of this remarkable Theorem. We conne ourselves to proving the second assertion of Lemma 7.5. denition we have and so " # f(b s )db s f(b s )db s = = = n, a i (B s ) B si+1, B s ; i=1 By our n 4,! 3 a i (B si ) B si+1, B si 5 i=1 " n, # a i (B si ) B si+1, B si i=1 ",, # a i (B si )a j (B sj ) B si+1, B si Bsj+1, B sj : i<j +
Suppose that j > i,then a i (B si )a j (B sj ), B si+1, B si, Bsj+1, B sj = a i (B si )a j (B sj ), B si+1, B si, Bsj+1, B sj Fsj = Moreover, ha i (B si ), B si+1, B si i = h h, a i (B si ) Bsi+1, B si ii Fsi = a i (B si ) (s i+1, s i ) : Substituting we obtain Notation: We write " # f(b s )db s = = n a i (B si ) (s i+1, s i ) i=1 f(b s ) ds: H = ff : R + R! R : f s (B s ) ds < 1g: Theorem 7.7 Let F t denote the natural ltration generated bybrownian motion. There exists a unique linear mapping, J, from H to the space of continuous F t - martingales dened on [;T] such that 1. If f is simple,. If t T, 3. J(f) t = J(f) t sup tt J(f) t f s (B s )db s ; = f s (B s ) ds; 4 f s (B s ) ds: Sketch of proof: The last part follows from Doob's inequality once we know that J(f) is a martingale. The second assertion follows almost as our proof of existence of Brownian motion. We rst take a sequence of simple functions such that jf s, f (n) s j ds! as n!1:
One then checks that (with probability one) the uniform limit of J(f (n) ) exists on [;T]. (This is an easy consequence of Lemma 7.5.) We write J(f) t = f s (B s )db s : Having made some sense of the stochastic integral, we are now in a position to try to make sense of the chain rule for stochastic calculus. Theorem 7.8 (It^o's formula) For f such that @x H, f(t; B t ), f(;b )= @x (s; B s)db s + @s (s; B s)ds + 1 Notation: Often one writes this in dierential notation as df t = f t db t + _ f t dt + 1 f t dt: @ f @x (s; B s)ds: Outline of proof: To simplify notation, suppose that f _ t. The formula then becomes f(b t ), f(b )= @x (B s)db s + 1 Let be a partition of [;t] with mesh. Then f(b t ), f(b )= N(),1 @ f @x (B s)ds:, f(btj+1), f(b tj ) : We apply Taylor's Theorem on each interval of the partition. f(b t ), f(b )= for some points j N(),1, N (),1 f 1 (B tj ) B tj+1, B tj + N (),1 + 1 3! f (B tj ), B tj+1, B tj f ( j ), B tj+1, B tj 3 [t j ;t j+1 ]. If f is uniformly bounded, then the third term tends to zero in probability (xercise). The rst converges to the It^o integral and the second, by Theorem 7.3, converges to 1= R t f (B s )ds. xample 7.9 Use It^o's formula to compute [B 4 t ]. Let us dene = B 4 t. Then by It^o's formula d =4B 3 t db t +6B t dt; and, of course, Z =. In integrated form,, Z = 4B 3 s db s + 6B s ds:
Taking expectations, the expectation of the stochastic integral vanishes (by the martingale property) and so [ ]= 6[B s]ds = 6sds =3t : The most common model of stock price movements is given by geometric Brownian motion, dened by S t = exp (t + B t ) : Applying It^o's formula, dst = S t db t +, + 1 S t dt S =1: This expression is called the stochastic dierential equation for S t. It is common to write such symbolic equations even though it is the integral equation that makes sense. Writing = + =, geometric Brownian motion is a martingale if and only if = and [S t ] = exp(t). It is convenient tohaveaversion of It^o's formula that allows us to work directly with S t (that is to write down a stochastic dierential equation for f(s t ) for example). We now know how to make our original heuristic calculations rigorous, so with a clear conscience we proceed as follows: f(s t+t ), f(s t ) f (S t )(S t+t, S t )+ 1 f (S t )(S t+t, S t ) f (S t )ds t + 1 f (S t ) S t db t + S t dt +S t db tdt = f (S t )ds t + 1 f (S t ) S t dt: (We have used that t(b t+t, B t )=o(t) which is usually written symbolically as dtdb t =.) As before, allowing f to also depend on t introduces an extra term f(s _ t )dt. Writing this version of It^o's formula in integrated form gives then: f(t; S t ),f(;s )= = @x (u; S u)ds u + @x (u; S u)s u db u + + @u (u; S u)du+ 1 @x (u; S u)s u du @u (u; S u)du + 1 @ f @x (u; S u) S u du @ f @x (u; S u) S u du Warning: Be aware that the stochastic integral with respect to S will not be a martingale with respect to the probability under which B t is a martingale except in the special case when =. Toactually calculate it is often wise to separate the martingale part by expanding the `stochastic' integral as in the last line. It is left to the reader to justify the following more general version of It^o's formula.
Theorem 7.1 If Y t satises dy t = a(y t )db t + b(y t )dt; and then = f(t; Y t ); d = f (t; Y t )dy t + _ f(t; Y t )dt + 1 f (t; Y t )a(y t ) dt: Remark: Notice that M t = Y t, Y, b(y s )ds is a martingale with mean zero. From the It^o isometry, we know that the variance is [M t ]= a(y s ) ds R t The expression a(y s) ds is the quadratic variation of M t, often denoted hmi t or [M] t. Suppose now that we have two stochastic dierential equations, dy t = a(y t )db t + b(y t )dt; : Write and Then the covariance is given by M Y t M 1 Z t = d =~a( )db t + ~ b( )dt: = M Y t = M = h, 4 M Y t a(y s )db s ~a(z s )db s : + M a(y s )~a(z s )ds,, M Y t, M i R t The quantity a(y s)~a(z s )ds, often denoted hm Y ;M Z i t or [M Y M Z ] t, is called the covariation of M Y and M Z. Notice that,hmi t is an F t -martingale and so is In this notation we have M t M Y t M,hM Y ;M Z i t :
Theorem 7.11 (Integration by parts) Let t = Y t with Y; Z as above, then d t = Y t d + dy t + dhm Y ;M Z i t : Proof: We apply the It^o formula to (Y t + ) and Y t second two from the rst to obtain and Z t, and subtract the Y t, Y Z = Y s dz s + Z s dy s + a(y s )~a(z s )ds; which is the integrated form of the result. We nowhaveavery large number of continuous time martingales in our hands. For any reasonable function f, M t = Z f(s; B s )db s is a martingale with respect to the Brownian probability and adapted to the Brownian ltration. It is natural to ask if there are any others. The answer is provided by the martingale representation theorem which says, essentially, no. Theorem 7.1 (Brownian martingale representation theorem) Let ff t g tt denote the natural ltration of Brownian motion. Let fm t g tt be a squareintegrable F t -martingale. Then there exists an F t -adapted process s such that with probability one, M t = M + s db s : The process s is essentially unique which leads, with a little work, to Theorem 7.13 (Levy's characterisation of Brownian motion) If M t is a continous (local) martingale with quadratic variation hmi t = t (with probability one), then M t is a standard Brownian motion. We'll use this to `prove' one more important result. Recall that in the discrete setting we were able to reduce pricing options to calculating expectations once we had found a probability measure under which the discounted stock price was a martingale. The same will be true in the continuous world, but it will no longer be possible to nd the martingale measure by linear algebra. The key now will be Girsanov's Theorem. Theorem 7.14 (Girsanov's Theorem) Suppose that B t is a P-Brownian motion with the natural ltration F t. Suppose that t is an F t -adapted process such that Dene exp L t = exp, 1 t dt < 1: s db s, 1 s ds
and let P (L) be the probability measure dened by P (L) (A) = Z A L t (!)P(d!): Then under the probability measure P (L), the process fw t g tt, dened by is a standard Brownian motion. Notation: We write W t = B t + dp (L) dp Ft = L t : s ds; (L t is the Radon-Nikodym derivative of P (L) with respect to P.) Remarks: 1. The condition 1 exp t dt < 1 is enough to guarantee that L t is a martingale. It is clearly positive and has expectation one so that P (L) really does dene a probability measure.. Just as in the discrete world, two probability measures are equivalent if they have the same sets of probability zero. vidently P and P (L) are equivalent. 3. If we wish to calculate an expectation with respect to P (L) we have (L) [ t ]=[ t L t ]: This will be fundamental in option pricing. Outline of proof: We have already said that L t is a martingale. We don't prove this in full, but we nd supporting evidence by nding the stochastic dierential equation satised by L t. We do this in two stages. First, dene Then =, s db s, 1 d =, t db t, 1 t dt: s ds: Now we use Theorem 7.9 applied to L t = exp( ). dl t = exp( )d + 1 exp() t dt =, t exp( )db t =, t L t db t :
Now we integrate by parts (using Theorem 7.1) to nd the stochastic dierential equation for W t L t. Since dw t = db t + t dt; d(w t L t ) = W t dl t + L t dw t + dhm W ;M L i t = W t dl t + L t db t + L t t dt, t L t dt = (L t, t L t W t ) db t : Granted enough boundedness (which is guaranteed by our assumptions), W t L t is then a martingale and has expectation zero. Thus, under the measure P (L), W t is a martingale. Now the quadratic variation of W t is the same as that of B t, and we proved in Theorem 7.3 that with P-probability one, the quadratic variation of B t is just t. Now P and P (L) are equivalent and so have the same sets of probability one. Therefore W t also has quadratic variation t with P (L) -probability one. Finally, by Levy's characterisation of Brownian motion (Theorem 7.1) we have that W t is a P (L) -Brownian motion as required. We now try this in practice xample 7.15 Let t be the drifting Brownian motion process t = B t + t; where B t is a P-Brownian motion and and are constants. Then taking = =, under P (L) of Theorem 7.13 we have that W t = B t + t= is a Brownian motion, and t = W t is then a scaled Brownian motion. Notice that, for example, whereas P t = P B t +tb t + t = t + t ; P (L) t = P (L) W t = t: We are nally in a position to describe the Black-Scholes model for option pricing.