### MF 796: Computational Methods in Mathematical Finance

Professors Christopher Kelliher and Eugene Sorets                                 Spring 2019

### Problem set # 3

Due: Wednesday, February 13, by 8am

1. Option Pricing via FFT Techniques The Heston Model  is  defined  by  the  following system of stochastic differential equations:

Assume the risk-free rate is 2%, the initial asset price is 250 and that the asset pays no dividends.

(a) Exploring FFT Technique Parameters Consider a European Call Option with strike 250 expiring in six months.

Additionally, assume you know that the parameters of the Heston Model are:

i. Calculate the price of the European Call option with many values for the damping factor, α. What values of α seem to lead to the most stable price?

ii. Using the results above, choose a reasonable value of α and calculate the price of the same European Call with various values of N and ∆k (or equivalently N and B). Comment on what values seem to lead to the most accurate prices, and the efficiency of each parameterization.

iii. Calculate the price of a European Call with strike 260 using various values of N and ∆k (or N and B). Do the same sets of values for N , B and ∆k produce the  best results? Comment on any differences that arise.

(b) Exploring Heston Parameters Assume the risk-free rate is 2.5%, the  initial  asset price is 150 and that the asset pays no dividends.