Black Scholes Option Pricing

Background

Major break-through in the valuation of derivatives came with two finance professors at MIT, Black and Scholes, came out with a formula that related the price of a call option to the price of the stock to which the option applies. Even though the model is not used by financial institutions today it still contains ideas that used in financial modelling.

The Black-Scholes formula is a partial differential equation that can be used to price the present value of an option under certain assumptions. The equation describes the Markov process of underlying asset price and it looks as given below:

There is an analytical solution of the equation, the walkthrough is given in this video:

Monte Carlo Simulation

Assuming the Brownian motion over this short period will be a normal (Gaussian) distribution with a mean of 0 and a variance of the time interval the iterative formula will be as follows:

Let's take the case of Option Call holder. The favourable cases of underlying asset movement (F) is given on the picture below:

Below you can find an implementation of Monte Carlo simulation of option pricing in python.

The program will generate a number of asset price paths. At option expiry time obviously the price will differ. Option price then will be the average of all gains received at expiry time.

Usefull Links

1. J.Hull Futures, Options and Other Derivatives 
2. http://pyvideo.org/video/1154/derivatives-analytics-with-python-numpy-0
3. https://www.enthought.com/store/ 
4. http://www.python.org/download/releases/2.7/ 
5. http://www.youtube.com/watch?v=i0sGAds8ztI&list=TLJV-9_NNgcRU 
6. http://www.personal.psu.edu/alm24/undergrad/bingqianMonteCarlo.pdf 
7. http://www.automatedtrader.net/glossary/Black-Scholes