Monte Carlo Methods

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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 but which have correlation ρ. I will provide an application further on, but for now, let's concentrate on the task of simulating two lognormal stock prices that are also correlated. The sheet labeled Corr BM provides the details. Here is the intuition: the first step is to generate the Wiener processes. Columns A and B do this, generating 1,000 observations on , where T is one year in this case. If we wanted, say, monthly frequency, then we would use the command in Excel: Normsinv(rand())/12. We then construct a vector containing the drift components , where μ is the risk-neutral rate (the risk-free rate in Black-Scholes).

The second step constructs the covariance matrix. In this example, I assume that the first stock has annual volatility equal to 14.4 percent while the second stock has annual volatility equal to 17 percent and that the correlation between their returns is 50 percent. The covariance matrix V is therefore the product:

Here, S is a diagonal matrix of volatilities and C is the correlation matrix. Making substitutions:

Thus, we have

We need the Cholesky decomposition of V:

Recall that . Therefore, when we construct the correlated Brownian motion in columns C and D, we multiply the 1,000 × 2 matrix formed by columns A and B by the 2 × 2 Cholesky matrix, adding as we go along. Columns C and D give this result.

In the last step, given in columns E and F, we exponentiate and multiply by the starting values for the stocks, in this case, $100. We can check our work if desired by estimating the covariance and variances of the simulated stock prices and compare these to V. Columns G and H are call options on each stock. The average of these calls should be close to the values calculated using BSM.

So far, this has been primarily an exercise in constructing correlated Brownian motions. Let's now explore the implications of the correlation property. Suppose we want to hold a call option on each stock. Does the value of the portfolio of call options depend on the correlation between the underlying returns?

On an intuitive level, the prices of the individual options should not depend in any way on the correlation between the returns on the underlying stocks. However, the value of the portfolio of calls could be sensitive to the correlation. To understand why, take the case in which the correlation ρ = –1 so that the returns are perfectly negatively correlated. This would suggest that the two stock prices move in opposite directions. Thus, when one option has high value, the other is likely to have low value. In this case, the portfolio of options will tend to have a mix of high and low value options at any given time, indicating that lower correlations lead to more diversification. On the other hand, when the correlation is close to 1, then the two options will also tend to track very closely in value at any given time period. It would make sense, therefore, to observe very low volatility across option value in the portfolio of options when correlations are high (and vice versa).

I constructed a Monte Carlo experiment that generates correlated Brownian motions, ranging the correlation from −0.95 ≤ ρ ≤ 0.95. For each value of ρ, I simulated 1,000 values for the two stocks reconstructing V to reflect the new value for ρ. At each of these 1,000 points, the difference between the two call values is computed and the standard deviation of this difference across the 1,000 calls is saved. The experiment therefore generates a volatility estimate for the call differences across the values of ρ. The plot is given in Figure 17.7.

Figure 17.7 Call Portfolio Volatility

As we can see, low values for ρ correspond to bigger differences in the two call option valuations and that the variability in their difference declines as ρ approaches 1. While this result was intended to illustrate how to generate correlated Brownian motion, it also illustrates that a portfolio of options will take on properties related to those on the underlying portfolio of stocks.