MONTE CARLO APPROXIMATIONS: AS EXOTIC AS THE NAME GOES???

I get easily carried away by all things European; the architecture, the art, the culture, the accents, the visage (of women strictly). No surprise that out of assortments of quantification and approximation techniques, the Monte Carlo method catches my attention more. It's soooo Italian.

Monte Carlo in general is the name of the administrative area under the Principality of Monaco. Of course I know that because of Wikipedia and because I saw Selena Gomez strutting around it in a sunshine dress and certainly because Monaco rings of one of the finest racing circuits and records some of the most celebrated Formula 1 races ever.

However, the Monte Carlo Simulation seemed like too much unnecessary work for estimating Option prices. Most of you would remember it from your Financial Economics Exam (MFE) of SOA and I suppose its applications in Finance are just as many as the only two pages dedicated for it in the ASM Manual.

The Monte Carlo process on a general look for option pricing would be this:

Assuming a log-normal equity price process, the simulation technique is conveniently straightforward:

1. Generate a random number U(0,1);
2. Determine the inverse by assuming a normally distributed white noise process of N(0,1);
3. Multiply the inverse by volatility (sigma) and add the mean (mu).
4. The exponential of the sum then multiplied by the initial stock price would be the projected Stock price.
5. This price is would then be used to calculate the gain/loss of the option at a given strike and then discounted at the risk free rate to provide a simulated risk free estimate of the option value.

For a call option with spot and strike price of $100 both, mean/risk free rate (mu) of 8% and volatility (sigma) of 20% with a time horizon of 1 year, the black scholes estimate would be $13.27. The monte carlo approach attempts to match this unbiased estimate with increased precision as number of simulations increase and averaged to equal the mean value of the option.

The scatter below shows 10,000 simulated values of the call option with minimum value of 0 and maximum as high as 130.

For such a simplified assumption for the equity process (lognormal with constant volatility and risk free rate of return) the Monte Carlo estimation seems ridiculously excessive, time consuming and a unnecessary burden on processing power. But imagine a more complex function or industry model that cant be explained in a single line mathematical function and therefore could not produce a continuous plot. The Monte Carlo approach would then produce a reasonable illustration of the variability and possible shape of the distribution.

And so the technique is being used in medical sciences and engineering sciences a lot simulating drug behaviors, fluid mechanics or anywhere randomness is both complicated and crucial to understand. While the above example would be more appropriate for a "Monte Carlo for Dummies" book, the applications would be endless in much more wider roles like Economic Capital, ALM, Risk Management etc.