Monte Carlo Methods
Go to the companion website for more details.
The Chapter 16 spreadsheet contains an example using the Black-Scholes-Merton model for a European call option on a nondividend-paying stock with spot price S = 100, strike K = 105, risk-free rate 3 percent, with volatility 14.4 percent, and one year to exercise. This option has value $4.90. Let's try to replicate the BSM result using Monte Carlo. We have the following representation from Example 17.2:
We can exponentiate this for easier simulation as follows:
In Excel, we substitute for ε = norminv(rand()),
Taking T = one year, Column A under the Monte Carlo tab contains 1,000 simulated prices for this stock S. Pressing F9 refreshes this spreadsheet calculation. As you can see, these solutions are distributed closely to the BSM solution of $4.90. Changing the parameter values on the spreadsheet will generate various problems that you can use to check against the analytic solution provided by the BSM formula.
Let's now take a look at correlated Brownian motion. Suppose we have two stocks, each following a lognormal process ...
Get Investment Theory and Risk Management, + Website now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
