Topaz: Monte Carlo Method for Option Pricing Under Black-Scholes model and Merton Jump Diffusion Model

Purpose

• The package "Topaz" computes the price of a European put option under the Black-Scholes model and the Merton jump diffusion model. The Monte-Carlo method was used.

Specifications

• Name: Topaz.
• Author: Feng Chen.
• Finishing date: 10/13/2012.
• Languages: MATLAB.

Simple Example

• Parameters: $$\begin{equation} T = 0.25, \quad \sigma = 0.15, \quad r = 0.05, \quad K = 100, \end{equation}$$
where $T$ is the maturity, $\sigma$ is the volatility, $r$ is the interest rate, and $K$ is the strike price of the option.
• Initial condition: $$\begin{equation} P_0(S) = \max\{K-S,0\}. \end{equation}$$

Quick Start

• Compiling and running:

  matlab Topaz_BS
  

• Graphics:
  
• CPU: Intel(R) Core(TM) i5-4570S CPU @ 2.90GHz
• OS: Ubuntu 14.04.1 LTS.
• Release: MATLAB R2014b.

References

• Black-Scholes model.
• Monte Carlo method.
• Feng Chen, Jie Shen, and Haijun Yu. A New Spectral Element Method for Pricing European Options Under the Black–Scholes and Merton Jump Diffusion Models, Journal of Scientific Computing, Volume 52, Number 3, 499-518, (2012).

Code Highlights

randn('state',4);
B = sqrt(T) * randn(1,n);
S = S0' * exp(r*T-sigma*sigma*T/2d0+sigma*B);
P0 = exp(-r*T)*sum(subplus(K-S),2)/n;