How to conduct Monte Carlo simulations to test validity of Black Scholes for a specific option?

Question

In reference to the original Black Scholes model, what approach is best to test the model in a rigorous way? Is there a standard approach that can accomplish this in a reasonable amount of time?

Details I require:

• number of trials,

• which software to use, formulas etc.

• any other information that I should be aware of

* This should be able to be done on a laptop with a Core i5 processor with a graphics card.

Answer 1

1. I recommend to use MATLAB / Excel for simplicity - depends which one do you already know.
2. Write down the SDE for geometic brownian motion (to simulate stock price over time) on paper, as quant_dev mentioned.
   
   1. Discretize it using i.e. forward Euler discretization (see Wikipedia), code up a MC simulation to simulate it for the time period you want to price your options.
   2. Don't forget to use the risk-free dynamics in the SDE, otherwise you wont converge to BS price.
3. Code the $f(S_T)$ payoff function for your option payoff.
4. Calculate the expected (average over simulations), discounted payoff.

With 10 000 simulations, or even 100 000, there should be a decent convergence of your simulation (error at $10^{-4}$) - your CPU should handle this in a few mins max.

Answer 2

Write out your model as an SDE, simulate it and compare the result with an analytical solution (if you've got one).

Answer 3

On the software end, if you want something quick/dirty I would personally go with Matlab/R/python however if you want something a bit more rigorous (e.g. payoff classes, "better" SDEs) something OO like C++ would really be the route to take.

The basic is fairly simple here's a quick sample of what it should look like:

double variance = vol*vol*expire; 
double rootVariance = sqrt(variance); 
double halfVar = -0.5*variance;

double SpotPlusOne = s*exp(r*expire+halfVar); 
double Spot; 
double runningSum=0;

for (unsigned long i=0; i < NumOfPaths; i++) 
{ 
double SN = SNByBoxMuller();
Spot = SpotPlusOne*exp(rootVariance*SN); 
double PayOff = Spot – strike;

PayOff = PayOff >0 ? PayOff : 0; 
runningSum += PayOff; 
}

double mean = runningSum / NumOfPaths; 
mean*=exp(-r*expire); 
return mean;

The SNByBoxMuller() is just the standard way of generating a random number from a standard normal distribution from Box Muller.

Answer 4

I think what you really need to do is test the results of the analytic/simulated solution against the actual AND future historical price. So you need to get historical data for the specific option. That, to me, is much more interesting since it will tell you if Black-Scholes is working vs. the reality. Of course you will need a large number of historical data points to tackle this.

Answer 5

It seems like everyone here is MC but you can use PDE methods as well.

Anyway there is two things that you can usually check, the Price and ... the Hedge (or replication price).

Let's look at the first case:

• If you have closed-form formulas (as is usually the case in the BS "fantas(ma)tic-wishfull thinking"-setting), then that's all you need. If not, then you are not comfortable with your math (or your model but this is another issue).

• If you don't have such an analytical solution at hand, then usually MC comes in naturally (as every one suggests here) but you could use PDE methods aswell (after all it was the original methods for derivation of the BS Call/Put options prices). And you have plenty of books and nice articles that will tell you how to proceed in both cases (especially in BS settings). An easy check that I recommand, is to compare the "closed-form formulas" vs "MC (and/or PDE)" in the vanilla cases. Moreover those methods provide a good introdution to the replication prices that you might be willing to check in the second case.

Note that by using finite difference (or element) methods for the PDE you get an error and that when using discretization sheme for SDE you get at the of the day "random variable" for your P&L. There it is really a matter of taste in my opinion both methods have pros and cons.

For the second case, that I called replication price then it is usually provided in a (at least in principle) straigthforward manner of the methods you used for the PDE and/or SDE discretization.

Still regarding the replication prices, the very recent article of Wilmott and Ahmad "Which Free Lunch Would You Like Today, Sir?: Delta Hedging, Volatility Arbitrage and Optimal Portfolios" is really illuminating in many ways and stays in the BS setting you want to stay within I think you should read it.