As we saw in Chapter 12, one of the many uses of Monte Carlo simulation by

Transcription

1 Financial Modeling with Crystal Ball and Excel, Second Edition By John Charnes Copyright 2012 by John Charnes APPENDIX C Variance Reduction Techniques As we saw in Chapter 12, one of the many uses of Monte Carlo simulation by financial analysts is to place a value on financial derivatives. You now know that you can sharpen the point estimate of your derivative s value by using the brute force method of increasing the number of trials run during a simulation. However, there are also other relatively simple changes you can make to a model that provide dramatic increases in precision for a given number of simulation trials. Such changes made to a model are called variance-reduction techniques. This appendix, based on Charnes (2002), shows how the variance reduction techniques of antithetic variates (AV) and control variates (CV) can be used to sharpen the precision of your estimate of the value of an Asian call option. The best point estimate of the value of a derivative is usually the mean, or arithmetic average, of the derivative s discounted payoff taken over all trials of the simulation. One measure of the sharpness of the point estimate of the mean is Mean Standard Error, defined as Mean Standard Error Variance Trials (C.1) The precision of the mean as a point estimate is often defined as the half-width of a 95 percent confidence interval, which is calculated as Precision 1 96 Mean Standard Error (C.2) Lower values of Precision in Equation (C.2) correspond to sharper estimates. Increasing the number of trials is a brute-force method of obtaining sharper estimates. This reduces the Mean Standard Error by increasing the value of Trials in the denominator of Equation (C.1). However, highly precise estimates with the brute-force method can take a long time to achieve. So-called variance reduction techniques reduce Mean Standard Errorby decreasing Variance in the numerator of Equation (C.1) and can be used to speed up simulations by achieving a specified level of precision with a smaller number of Trials. 287

2 288 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL In this appendix, we consider two variance reduction techniques, AV and CV. The AV method is more widely applicable than CV, but when it can be used, the CV technique is much more efficient. The use of Monte Carlo simulation in pricing options was first published by Boyle (1977), but recently the literature in this area has grown rapidly. You can learn more about the use of variance reduction techniques from Fishman (2006), or Law and Kelton (2000). For a good discussion of variance reduction techniques applied to financial derivatives, see Boyle, Broadie, and Glasserman (1997), Glasserman (2004), or Glasserman and Li (2005). C.1 USING CRYSTAL BALL TO VALUE AN ASIAN OPTION T t 1 S t As described in Chapter 12, an Asian option is also called an average option because its price is linked to the average value of the underlying asset on specific dates. Suppose the prices of the underlying asset are denoted by S t,for t 0 1 T, where T is the expiration date of the option. The agreed amount for which the underlying is traded is called the strike price, denoted by K. An average-price Asian option has a payoff at time T based on the difference between the strike price and the arithmetic average price of the underlying asset. Specifically, the payoff is V max (A K 0), where A T. Financial analysts are often interested in determining the value of the option, denoted here as C A. In the Black-Scholes world view, a fair value for an option is the present value of the expected value of the option payoff at expiration under a riskneutral random walk for the underlying asset prices. Therefore, the formula used in a Crystal Ball model to generate daily asset prices on the ith trial of the simulation is S (i) t 1 S t exp (r 2 2)(1 252) 1 252Z (i) (C.3) for t 1 2 T, where r is the risk-free rate of interest, is the annual volatility of the asset prices, and Z (i) is a standard normal random variate. Equation (C.3) assumes that there are 252 trading days in a year. The general approach to using Crystal Ball to find the price of an average-price Asian option is straightforward (see the file AsianCallVarReduction.xls for specific details): 1. For each simulation trial, i 1 2 N, simulate sample paths of the underlying asset prices, S t,fort 0 1 T according to Equation (C.3); 2. Calculate the average price, A, of the underlying asset and payoff of the option at time T as V (i) max (A K 0); 3. Compute the present value of the cash flows of the option on each sample path, as C (i) A V (i) e rt 252 ; and

3 Appendix C: Variance Reduction Techniques Use the average of the present values over the sample paths, C A N i 1 C (i) A N as the point estimate of the option s value, and use the variance of the distribution of the values C (i) A to obtain the precision of C A with Equation (C.2). Crystal Ball takes care of the housekeeping details in the steps shown previously, so that in practice all we need to do after running the simulation model for a given number of trials is look at the Crystal Ball Forecast Window statistics to obtain the forecast cell mean, which is C A, and the Mean Standard Error, which is a measure of precision. C.2 ANTITHETIC VARIATES The method of antithetic variates for variance reduction is based on the fact that if Z (i) has a standard normal distribution, then so does Z (i). Therefore, if we replace Z (i) in Equation (C.3) with Z (i), we also get a valid sample from the distribution of stock prices at time T. In using antithetic variates with the procedure given above, we construct two intermediate estimates in Step 3, C A (Z (i) ) and C A ( Z (i) ), then a final estimate, CA AV (C A C A ) 2 as the point estimate in Step 4 here. Because of the way we use Z (i) and Z (i), the estimates C A and C A both have the same expected value; however, because the two estimates are negatively correlated, the distribution of CA AV has a lower variance than the variance of either estimate by itself. Thus, antithetic variates gives an estimate that has the expected value we are trying to find, but with a smaller Mean Standard Error than the estimate obtained without using a variance reduction technique. C.3 CONTROL VARIATES The method of control variates replaces the evaluation of an unknown expected value with the evaluation of the difference between the unknown quantity and a related quantity whose expected value is known. Here, the unknown quantity of interest is the value, C A, of an average-price Asian call option whose payoff at expiration is max (A K 0), where A is the arithmetic average of the underlying asset prices during the holding period. The related quantity with known expectation is the value, C G, of an Asian option whose payoff is max (G K 0), T where G ( t 1 S t) 1 T is the geometric average. Because of the lognormality of the stock price model, an analytic expression is available for C G, but not for C A (see Kemna and Vorst 1990 for details).

4 290 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL The values of interest here are denoted as C A E[C (i) A ], and C G E[C (i) G ], where C (i) A and C(i) G are the discounted option payoffs for a single simulated path of the underlying for options that pay off on the arithmetic and geometric means, respectively. Because an unbiased estimator of C A is given by C A C G E C A C G C CV A C A (C G C G ) which becomes the point estimate to be used in Step 4 in Section C.1. Using C G as a control variate reduces variance because it steers the estimate toward the correct value. C.4 COMPARISON The spreadsheet file AsianCallVarReduction.xls shown in Figure C.1 contains a Crystal Ball model for estimating the value of an average-price Asian call option, along with reports on the performance of the model for options having different strike prices and volatilities of the underlying asset. Figure C.2 shows the distribution of FIGURE C.1 call option. Crystal Ball model for estimating the value of an average-price Asian

5 Appendix C: Variance Reduction Techniques 291 FIGURE C.2 Distribution of the discounted cash flows for simulations with no variance reduction, antithetic variates (AV), and control variates (CV) for an at-the-money Asian call option having initial price $55, strike price $55, underlying volatility 30 percent, and time to expiration 126 days.

6 292 FINANCIAL MODELING WITH CRYSTAL BALL AND EXCEL TABLE C.1 Means, standard errors, and increases in precision from the simulation model in AsianCallVarReduction.xls using no variance reduction, antithetic variates (AV), and control variates (CV) to obtain estimated values of an average price Asian call option having initial price $55, strike price $55, underlying volatility 30 percent, and time to expiration 126 days. Standard Relative Estimation Method Mean Error Precision No Variance Reduction Antithetic Variates (AV) Control Variates (CV) C.5 CONCLUSION the discounted cash flows for simulations with no variance reduction, antithetic variates (AV), and control variates (CV) for an at-the-money Asian call option having initial price $55, strike price $55, underlying volatility 30 percent, and time to expiration 126 days. All of these distributions have means that are estimates of the option s value, but only the distribution of values from the simulation with no variance reduction resembles the distribution of cash flows that might be generated by the option. The other two simulations yield distributions that have the desired mean, but the shapes of the distributions are not indicative of the potential cash flows of the option. However, the mean standard errors computed from each of the three distributions are comparable. Table C.1 shows the mean, standard error, and increase in precision for each estimation method when the simulation model was run for N 10,000 trials. The antithetic variates method slightly more than doubled the precision (halved the standard error). The control variates method gave an estimate that is roughly 20 times more precise than (has a standard error that is 4.8 percent of) that achieved with no variance reduction. Variance reduction techniques offer potentially large increases in the precision of estimated derivative values. The method of AV is generally less effective than CV, but AV can be easily applied to more types of derivatives than CV because CV requires that a control variate is available, such as the value of the geometricaverage option that was used here. Interest in use of Monte Carlo methods for derivatives pricing is increasing because of the flexibility of the method in handling complex financial instruments. Monte Carlo simulation will continue to gain appeal as financial instruments become more complex, workstations become faster, and simulation software is adopted by more users. The use of variance-reduction techniques along with the greater power of today s workstations can help to reduce the execution time required for achieving acceptable precision to the point that simulation can be used by financial traders to value derivatives in real time.