Consider a Brownian motion W(t) with t 0 and consider two stock prices de- scribed by S 1(t) and S 2(t) which fulfill the following stochastic differential equations (SDEs) dS 1(t) =1S 1(t)dt +1S 1(t)dW(t) dS 2(t) =2S 2(t)dt +2S 2(t)dW(t), with 1, 2 Rand 2 > 1 > 0. a) For f (x) =log x, derive the SDE satisfied by the process f (S 1(t)). b) Without further calculation, what is the process followed by f (S 2(t))? c) Find the SDE satisfied by Y(t) =g(S 1(t),S 2(t)) =ln(S 1(t)/S 2(t)) when = 1 =2. What type of stochastic process is Y(t) undergoing? Describe the parameters of this process.

5. Consider a Brownian motion W(t) with t > 0 and consider two stock prices de- scribed by S (t) and S2(t) which fulfill the following stochastic differential equations (SDE) = ds (t) = MS (t)dt +0S(t)dW(1) dS 2(t) = u2S 2(t)dt + 02S2(t)dW(t), - with uj, M2 ER and o2 > 0, > 0. a) For f(x) = log x, derive the SDE satisfied by the process f(S1(t)). b) Without further calculation, what is the process followed by f(S2(1))? c) Find the SDE satisfied by Y(t) = g(Si(t), S2(t)) = In(S (t)/S 2(t)) when u = Mi = M2. What type of stochastic process is Y(t) undergoing? Describe the parameters of this process. [10]