STOCHASTIC METHODS IN FINANCE 2021–22 STAT0013
Exercises 6 – Stochastic Calculus
1. Find the stochastic differential equation (SDE) satisfied by the square of a stock price that follows geometric Brownian motion. What is this process?
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Suppose that g(t) is a deterministic differentiable function for t > 0, with g(0) = 0. Show that a solution to the ordinary differential equation
dx(t) = ax(t)dt + bx(t)dg(t) with boundary condition x(0) = x0 ̸= 0 is
x(t) = x0eat+bg(t).
Hint: Write dg(t) as g′(t)dt and then use the variable separation
technique and take integrals on both sides of the equation.
(b) Show that the process Xt = x eat+bWt , where Wt is standard Brow- nian motion, satisfies the SDE
dXt =a+b2Xtdt+bXtdWt 2
with initial condition X0 = x.
3. Show that the Itoˆ process Xt = eWt e−t/2 (with Wt a standard Brownian
motion) satisfies the stochastic differential equation dXt = Xt dWt.
4. A zero-coupon government bond pays £100 at time T, and has price denoted by Bt. In the course so far we have assumed that the risk- free rate is constant and deterministic. In more advanced models, the risk-free rate can be modelled itself as a stochastic process. It has been
suggested that the short-term interest rate, rt, will not be constant over time but will in fact follow the stochastic process
drt =a(b−rt)dt+crtdzt
where a, b, c are positive constants and zt is a standard Brownian motion. Under this assumption, derive the SDE for the government bondpriceBt fort