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