mathemactis

FE 610 Stochastic Calculus for Finance Midterm

March 8, 2018

Name:

• There are 4 problems, worth a total of 100 points.

• Simplify your answers as much as possible. If the result could be sim- plified further, you will potentially lose points.

• Showcase your work: providing just the answer will result in a minimum of points.

• Closed book (preferably). No internet enabled devices (except to take the test of course). Only one hand written sheet of notes should be used.

For instructor’s use only

Problem Points Score 1 25 2 25 3 25 4 25

Total 100

1

1. For a positive constant X(0) we have the process X(t) defined as:

X(t) = X(0) +

∫ t 0

W2(u)dW(u) +

∫ t 0

W(u)du

(a) Find [X,X](t)

(b) Find E[X,X](t)

2. Given a three-dimensional Brownian Motion, and for processes defined by:

X(t) =

∫ t 0

W1(u)W2(u)dW3(u)

Y (t) = Y (0) +

∫ t 0

δ(u)dW1(u) + W2(u)

Z(t) = X(t)Y (t)

with Y (0) being a positive constant and δ(t) an adapted process. Is Z(t) a martingale?

3. Let S(t) be the stock price governed by

S(t) = S(0)e ∫ t 0 (α(u)− 1

2 σ2(u))du+

∫ t 0 σ(u)dW(u)

with α(t) an adapted process and σ(t) a non-random process. Deter- mine the Quadratic Variation and Variance of S(t).

4. For a process X(t) governed by the dynamic:

dX(t) = X(t)dW(t)

with X(0) = 2. Determine:

E[e−tX2(t)]

2