com.imsl.test.example.math

Class FeynmanKacEx2

java.lang.Object

com.imsl.test.example.math.FeynmanKacEx2

public class FeynmanKacEx2
extends Object

Applies a diffusion model for options pricing.

In Beckers (1980) there is a model for a Stochastic Differential Equation of option pricing. The idea is a "constant elasticity of variance diffusion (or CEV) class" $$ dS = \mu S dt + \sigma S^{\alpha/2}dW,\; 0 \le \alpha \lt 2. $$ The Black-Scholes model is the limiting case \(\alpha \to 2\). A numerical solution of this diffusion model yields the price of a call option. Various values of the strike price K, time values, \(\sigma\) and power coefficient \(\alpha\) are used to evaluate the option price at values of the underlying price. The sets of parameters in the computation are:

1. power \(\alpha = {2.0, 1.0, 0.0}\)
2. strike price \( K = {15.0, 20.0, 25.0} \)
3. volatility \(\sigma = {0.2, 0.3, 0.4}\)
4. times until expiration = {1/12, 4 /12, 7 /12}
5. underlying prices = {19.0, 20.0, 21.0}
6. interest rate \(r = 0.05\)
7. \( x_{\min} = 0, x_{\max} = 60 \)
8. \( nx = 121, \, n = 3 \times nx = 363\)

With this model the Feynman-Kac differential equation $$ f_t +\mu(x,t) f_x +\frac{\sigma^2(x,t)}{2}f_{xx}-\kappa(x,t)f=\phi(f,x,t), $$ is defined by identifying:

• \(x:\; S\)
• \(\sigma(x,t):\; \sigma x^{\alpha/2};\; \frac{\partial \sigma}{\partial x}= \frac{\alpha \sigma}{2}x^{\alpha/2-1} \)
• \(\mu(x,t):\;rx\)
• \(\kappa(x,t):\; r\)
• \(\phi(f,x,t) \equiv 0\)

The payoff function is the "vanilla option", \(p(x) = \max(x-K, 0)\).

See Also:

Code, Output

Constructor Summary

Constructors

Constructor and Description
FeynmanKacEx2()

Method Summary

Modifier and Type	Method and Description
static void	main(String[] args)

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

FeynmanKacEx2

public FeynmanKacEx2()

Method Detail

main

public static void main(String[] args)
                 throws Exception

Throws:

Exception