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MF 796: Computational Methods in Mathematical Finance
创建日期：2024-07-17 17:41:51
所属分类：数学mathematics
标签： MF 796
Computational Methods in Mathematical Finance

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MF 796: Computational Methods in Mathematical Finance

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

Professors Christopher Kelliher and Eugene Sorets

Problem set

Due: Wednesday, February 13, by 8am

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

The characteristic function for the Heston Model is known to be: 

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 month. 

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

σ = 0.2
ν0 =  0.08
κ = 0.7
ρ = −0.4
θ = 0.1

i.Calculatethe 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?

Mathematical Finance

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.数学金融作业代写

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 priceis 150 and that the asset pays no dividends.

σ = 0.4
ν0 = 0.09
κ = 0.5 
ρ = 0.25
θ = 0.12

i.Using these parameters, calculate Heston Model prices for three-month options at a range of strikes and extract the implied volatilities for each strike. Plot the impliedvolatility σ(K) as a function of strike.

ii.Usethe FFT pricing technique to obtain prices of 150 strike calls at many  Extract the implied volatility for each and plot the term structure of volatility by plotting time to expiry on the x-axis and implied volatility on the y-axis.

iii.Holdingall other parameters constant, vary each of the model parameters and plot the updated volatility skews and term structures. Comment on the impact that each parameter has on the skew and term structure.

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