Stochastic Calculus Made Easy

Transcription

1 Stochastic Calculus Made Easy Most of us know how standard Calculus works. We know how to differentiate, how to integrate etc. But stochastic calculus is a totally different beast to tackle; we are trying to play with the calculus of Random Variables. It s a field where Probability Theory and Calculus meet. Let s start the journey:- We will denote W(t) as the Standard Brownian Motion. Some properties are as follows:- 1. W(t) W(s) is normally distributed with mean 0 and variance t-s, for s<t 2. The process W has independent increments : for any set of times 0<, the random variables W ( W ( W ( W ( are independent 3. W(0) = 0 4. The sample paths are continuous function of t 5. It is not differentiable, the paths of a Brownian motion are so irregular, it s not possible to draw a tangent at any point in time 6. It s a Markovian Process : - the distribution of the future value W(t) given information up to time s<t depends only on W(s) and not on the past values. 7. Martingale Property: [ ( ] ( Some Stochastic Calculus What is d(? If this was Normal Calculus the answer would be WdW. But this not Standard Calculus, we are dealing with a Random Variable, so this is clearly not the correct answer, but it is more or less in line if not exactly similar. When we are dealing with Stochastic Calculus we always need to go to 2 nd order terms which were not necessary in Standard Calculus. This because of Quadratic Variation of Brownian Motion, which states:- (t, W) = ( ( = t For smooth differentiable functions this relationship will be = 0. Now I will try to prove that this relationship actually holds Let A = ( ( t (i) If I prove that eq(i) has Expected Value = 0 and Variance = 0, then I can say Almost Surely (a.s) that the Quadratic Variation relationship holds. So let s see how we can do that. A = ( ( t = [ ( ( ( ] = [ ( ( ( ] E[A] = E[ [ ( ( ( ] ] = E[ [ ( ( ] ] ( E[A] = [ ( ( ] - (..(ii) We know from the Properties of Brownian Motion that, W(t) W(s) is normally distributed with mean 0 and variance t-s, for s<t E[W(t) W(s)] = 0

2 Var(W(t) W(s)) = E[( ( ( ] - ( [ ( ( ] = E[( ( ( ] = t-s So eq(ii) becomes E[A] = ( - ( = 0.(iii) Var(A) = Var( [ ( ( ( ] ) = Var( ( ( ) Var(A) = ( ( ( ) = ( ( ) = ( ( - ( [( ( ] We can denote, ( ) = (Δ Z, where Z ~ N(0,1) Var(A) = ( ( (..(iv) Now how do we find (? We can do as the following:- ( = E[ ], as a ->0. Basically we need to do partial differentiate n times With the help of Moment Generating Functions, we know that - E[ ], where Z~N(0,1) = = a = + = (1 + ) = (2a) + (a + ) = ( 3a + ) = (3 + 3 ) + (3 + ) = ( ) As a->0, = 3. ( Eq(iv) becomes, Var(A) = ( ( = (. As Δt -> 0, this will also -> 0. Hence, Var(A) = 0 So we have proved the relationship of Quadratic Variation of Brownian Motion. Stochastic Differentiation Taylor Series is the key. A simple stochastic differential equation will be of the form:- ds = asdt + bsdw, where S is a function of W and t, S(W,t). The increments of S are dependent on a drift term which evolves with time and a random variable whose evolution is unknown to us. If there is another function V, such that V(S,t), how do we find dv? The answer is Taylor Series:- V(S+dS, t+dt) = V(S,t) (v) On this we are going to use ITO s rule which simply states:- dt.dt -> 0, dt.dw -> 0, dw.dw -> dt After applying the above rule eq(v) reduces to a simplified form and which is also called as Ito s Lemma dv = (asdt + bsdw) + + dv = ( + + )dt + bsdw (vi)

3 So with the help of above rules we can find the SDE of any process as long as we know the base asset price dynamics SDE. Stochastic Integrals Basics A general stochastic Integral is of the form I(t) = ( ( As this is not standard calculus we can t use the standard rules of calculus. But there is another way to look at this which was shown to us by Ito. We have to take a partition and break the above integral as a Riemann Sum, G(t) = ( ( ( ( ( Here ( 0, G(t) -> I(t) is a piecewise constant function within each interval. So as the partition goes down to How to solve Stochastic Integrations This is a very huge topic and it s often difficult to comprehend. I will try to present a few examples which may make the learning a little easier 1. Integrate Solution: - This is 1 of the most famous Stochastic Integrals and its essential for our learning. I will show you the easiest way to look at this. Let f(x) = Let f(w) =, then f (x) = 2x dx, this is standard calculus., the f (W) =?, here we have to use Ito s Lemma. Let s do a Taylor Expansion of f(w+dw) around dw df = + ( df = 2W dw + *2 dt, df = 2W dw + dt d = 2W dw + dt Let s integrate this from 0 to T = + T ( ( {We know that W(0) = 0} = (.(vii) Now the 2 nd question is, as this is Stochastic, this will have an Expectation and a Variance, what are those? E[ ( ] = (E[ ] T ) = (E[( ( ( ] T ) = (T T) = 0

4 So the Expectation of the above Integration = 0 Var( ( ) = 0.25Var( ) = 0.25(E[ ] - ( [ ] ) = 0.25( - ) = So the Variance of the above Integration = 2. Integrate Solution :- We will use the same steps that we had used before, using Taylor series:- d = 3 dw + Integrating both sides from 0 to T = ( - 3 = And now the question is, what is? How do we solve this? Let = Let there be a Riemann Sum, = ( = ( = ( - ( ( = ( = ( - ( ( = ( + (.{ as W starts frm 0 at t=0 and at any time before t=0 it will always be 0} ( + ( = ( = ( + ( - ( = ( + ( is nothing but T = + ( Now we need to simplify this equation = T + ( + ( = T - = (T - = ( )( = T + ( + ( +( = T + ( ( + ( +( + T -

5 = ( ( + ( ( +( ( -T) Hence can be re-written as ( (..(viii) We can say that -> as M ->, because that s when the Summation will converge to an integration. We can easily say that eq(viii) is just a summation of Normal R.V. s and hence Normally distributed. E [ ] = E [ ( ( ] = [( ]( = 0 Var[ ] = E [ ( ( ] = [( ] ( Var[ ] = [( ] ( = ( ( -> ( Hence, ~ N (0, ( ) So now we have seen 2 interesting Stochastic Integrals, and there are many more complex ones out there. This is just giving you a flavour of the unknown. When you work with Stochastic Integrals, you need to be careful regarding the Expectation and Variance of the Integral because the integral itself is Stochastic in nature. The below 2 properties are extremely useful:- 1. E[ ( ( ], 0 a < b T 2. Cov[ ( ( ( ( ] = E[ ( ( ], 0 a < b T