Texts: 2.
a. Consider the diffusion process for the state variable U, which evolves according to the Ornstein-Uhlenbeck process:
dU = -Udt + adWU,
(2.1)
where both a and o are constants. dWU is an increment in Brownian motion. Show that the solution of 2.1 can be written as:
Ut = ...
[10 Marks]
b. Consider a function V(t, S, r) where the two stochastic processes S(t) and r(t) evolve according to a two-factor model given by:
MPs + psr = sP drt = m - rdt + cdW2,
in turn, and where dW1dW2 = pdt. The parameters p, s, m, and c are constant. Let V(t, S, r) be a function on [0, T] with V(0, S, r) = v. Using Ito's lemma, compute the SDE for dV and deduce the integral form for V(T, S, r). [7 Marks]
c. Consider the RAND function in Excel, which generates uniformly distributed random numbers over 0,1. Show that:
√(N) ∑_(i=1)^N▒〖2 produces a standard Normal ~ N(0,1) and show that it is consistent with the Central Limit Theorem. [8 Marks]