MA 573- Computational Methods of Mathematical Finance - Assignment 7 Solved

1.   Assume that stock prices follow under the risk-neutral measure the dynamics of Scott’s exponential Ornstein-Uhlenbeck process model,

 

with constant interest rate r. Describe the price of the European call option with payoff

 

at time t ∈ [0,T] as solution of a Cauchy problem of a PDE.

2.   Consider the Cauchy problem

                                                                         (1)

for t ∈ [0,T] and and x ∈R.

(a)    Write the probabilistic representation for f(x,t) as a conditional expectation of a function of a solution of a stochastic differential equation you should define.

(b)   Compute this conditional expectation.

(c)    Verify that f(x,t) you found in b) solves the PDE (1) and satisfies the terminal condition.

3.   Consider the option pricing problem

 

where the underlying stock price follows under the risk-neutral measure a CEV model

 

and the interest rate follows a CIR process

 

and the two Brownian motions are correlated, E[Wt1Wt2] = ρt. Describe the option price via the solution of a Cauchy problem of a PDE.

4.   While in general it is not possible to find a PDE formulation for option prices of path-dependent derivatives, this can be done sometimes by introducing auxiliary processes. If we can rephrase the option pricing problem in a way that the payoff depends only on the terminal value of the stock price and the auxiliary process. Specifically, the payoff of lookback options and barrier options can be rephrased in terms of the running maximum process

 

and Asian options in terms of the integral process

 

for instance

                      for all                 (up-and-out put),

 (lookback put),

(Asian call).

One can proof Itoˆ formulas for those processes, specifically we have for the stock dynamics

                                                        dSt = b(St)dt + σ(St)dWt,             S0 = s

that for sufficiently smooth (i.e., differentiable) f we have on the one hand

 

and on the other

 

Derive a PDE formulation for option pricing problems

                                              ,          

for some smooth payoff functions g, h by applying Itˆo’s formula to the discounted payoff. Note that for this purpose (as we focus on applications and not theory and rigorous proofs) you may assume that

(a)   The expectation of Itˆo integrals dWs is zero;

(b)   The expectation of a Riemann integrals ds is zero if and only if the integrand is zero;

(c)    The expectation of a Stieltjes integral dMs is zero if and only if the integrand is zero whenever M is increasing.

6 points per problems