mathemactis

 

• Stochastic Calculus Financial Engineering

 

This is my requirements.

I have a exam online, I have total 3.5 hours to finish it once I start it.

So If you accept it you must finish it in 3 hours and send it to me.

I can start the exam anytime in the next 20 hours. I will keep online.

There is a sample of previous exam, so I want you to read it and tell me if you are able to do it.  I want to have a good grade. It’s ok if you can’t do it.

FE 6

1

0 Stochastic Calculus for
Finance Midterm

March 8,

2

018

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