Package 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:

  • Constructor Details

    • FeynmanKacEx2

      public FeynmanKacEx2()

  • Method Details