Stochastic Calculus. Kevin Sinclair. August 2, 2016

Transcription

1 Stochastic Calculus Kevin Sinclair August, 16 1 Background Suppose we have a Brownian motion W. This is a process, and the value of W at a particular time T (which we write W T ) is a normally distributed random variable. In a standard Brownian motion W = and W T has a variance equal to T. We can integrate with respect to a Brownian motion. From an intuitive point of view, this means adding up all the increments in the process that occur between the time limits. Hence for some function of time C t an integral such as θ T = C t dw t is the continuous version of the Riemann sums T θ T = C i (W i+1 W i ). i= Brownian motions are an example of stochastic processes. This is a large complicated field of mathematics, but fortunately only a small subset of the universe of stochastic processes are relevant to finance. These are called Ito processes, and they are of the general form dψ = µdt + σdw, where µ and σ might be functions of time or randomness. 1

2 The axioms of cross-variation Brownian motions have many peculiar properties, and they have their own branch of calculus. Cross-variation can be thought of as a quantification of the difference between ordinary calculus and stochastic calculus. Below are a set of rules that detail how to handle cross-variations. Suppose we have four stochastic processes A, B, C and D. The cross-variation between A and B is a process, and it is written as A, B. 1. The cross-variation is commutative: A, B = B, A. It s linear: A + C, B = A, B + C, B 3. For a constant α, we have αa, B = α A, B and α = 4. A, A is written as A 5. A, B, C = 6. If W 1 and W are standard Brownian motions with an instantaneous correlation between them of ρ t, we have W 1, W t = This is written in differential form as ρ s ds. d W 1, W t = ρ t dt. 7. If we create two new processes θ and φ defined by then and in differential form, θ t = C s da s and φ t = θ, φ t = C s D s d A, B, D s db s, d θ, φ t = C s D s d A, B t

3 3 Definition of Ito s lemma Suppose we have a function f(x), where x is a scalar space variable, and X is a stochastic process in that space. Taylor s theorem states that a discrete change in f is given by f = f x x + 1 f x x + lower order terms In continuous calculus, x dx and second and lower order terms, giving df = f x dx Suppose we form a stochastic process φ, given by φ t = f(x t ). This means we take the value of the process X at any time, plug that value into the function f, and this gives us the value for the process φ at the same time. How does φ change as X changes? Ito s Lemma in integral form states that at time T, φ T = f(x T ) = f(x ) + f x dx t + 1 f x d X t Note that here the second order term does not disappear, as in standard calculus. This is not surprising, since cross variation rule 6 tells us that for a single Brownian motion W, d W, W t = dt. This suggests that a general second-order stochastic term will actually be linear in t, and so will not tend to zero as t goes to dt. In differential form Ito s Lemma is dφ = f x dx + f x d X We can extend this to a two variable case; suppose we have g(x, y) t, and we form the process ψ t = g(x t, Y t ) from the stochastic processes X t and Y t. Ito s lemma for ψ becomes 3

4 ψ T = g(x T, Y T ) = g(x, Y ) g T x d X t + T x dx t + y dy t g x y d X, Y t + 1 g y d Y t 4 Using Ito 4.1 Example 1: Two Brownian motions Suppose we have a function g(x, y) = xy, and a process φ = g(w, Z) where W and Z are two standard Brownian motions with a correlation ρ t between them. We want to know how φ evolves. The first order differentials we require are x = y and y = x, and hence the second order differentials are g x = g y = and g x y = 1, If we substitute these into the Ito s lemma for dφ, we get dφ = ZdW + W dz + d W, Z But from rule 6, d W, Z = ρdt, giving dφ = ZdW + W dz + ρdt. 4. Example : A lognormal process Now let us suppose that g(x, y) = e λx+σy, where λ and σ are constants. We want to know how the process ψ = g(t, W ) evolves. Process of this form are called log-normal 4

5 processes, since if ψ = e λt+σw, we have ln ψ = λt + σw. From this we can see that the random variable ln ψ t is normally distributed, giving rise to the name. The first order differentials of the function g are: x = λeλx+σy and y = σeλx+σy, and hence the second order differentials are g x = λ e λx+σy, g y = σ e λx+σy, g x y = λσeλx+σy. The differential form of Ito s Lemma for the process ψ is then dψ = λe λt+σw dt + σe λt+σw dw + λ eλt+σw d t + σ eλt+σw d W + λσe λt+σw d t, W We can eliminate some of these terms using the cross-variation rules. Rule 4 means that d t = d t, t, and since rule 6 states that d W, W = t, this can then be written as d t = d t, W, W. 5

6 According to rule 5, any cross-variation of this form must be equal to zero. Hence t and by the same argument t, W are equal to zero. Ito s lemma for dψ can now be written as dψ = λe λt+σw dt + σe λx+σy dw + σ eλx+σy dt Substituting ψ back in, dψ = λψdt + σψdw + σ ψdt This can be rearranged to give the evolution of ψ as ( ) dψ ψ = λ + σ dt + σdw as required. In the specific case where λ = σ, then the expression for dψ will not contain a term in dt, and ψ is said to have no drift. Such processes are called martingales, and they have the property that their expected value at some time in the future is equal to the current value of the process. This will be of crucial importance in the fundamental theorem of derivative pricing. 6