As indicated above, the main advantage of the Monte Carlo method is its ability to capture the nonlinearities in the portfolio value created by the presence of options in the portfolio. This can be important for portfolios
–
–
liquidated value = bS0 (1+r1)(1+r)n–1+(1+r1)(1+r2)(1+r)n–2 1+r1
( )(1+r2)(1+r3)(1+r)n–3 … (1+ri) . i=1
∏
n+ + +
>
with significant options positions, and especially for portfolios that include exotic options. Additional advantages are that it can be used with a range of different distributions for the market factors and can be used to measure the risk of dynamic trading strategies, such as LaVaR.
For short holding periods such as one day, an offsetting factor is that the nonlinearities are much less important. This is true because: (i) for short holding periods, the typical changes in the values of the market factors are much smaller; and (ii) linear (and quadratic) approximations work better for small changes in the variables. Thus, the delta-normal method and the delta-gamma-theta-normal method described in Chapter 14 often work well over short holding periods.
An advantage of the Monte Carlo method over the historical simula- tion method is that it is not constrained to rely on relatively small samples (small N). Thus, the problem of relying on small samples to estimate the tail probabilities that are inherent to historical simulation is not shared by Monte Carlo simulation. However, the number of Monte Carlo trials required for estimating the ␣ quantile accurately can be surprisingly large, especially when ␣ is small.
If the Monte Carlo simulation method has such advantages, then why is its use not universal? The answer is that it does suffer from one signifi- cant drawback. Earlier, it was carefully stated that there is no conceptual difficulty in extending the Monte Carlo method to handle realistic port- folios with many different instruments. However, the computational bur- den of the procedure can present a practical problem. Letting N denote the number of hypothetical future scenarios used (i.e., the number of samples or “draws” in the simulation), K the number of market factors, and M the number of instruments in the portfolio, a naïve application of the Monte Carlo simulation method will involve generating N K pseudo-random variables and performing N M valuations of financial instruments. Such straightforward application of the Monte Carlo method is termed full Monte Carlo. In actual applications, N might exceed 10,000, K can range from 2 or 3 to 400 or more in systems that employ detailed modeling of yield curves in different currencies, and M can range from only a few to thousands or even tens of thousands. Using reasonable values of N 10,000, K 100, and M 1000, a straight- forward application of the Monte Carlo simulation method would involve generating one million pseudo-random variables and performing 10 million valuations of financial instruments. For the interest-rate swap
“book” of a large derivatives dealer, M could be 30,000 or more, so a naïve use of the Monte Carlo method would involve hundreds of millions of swap valuations.
× ×
= = =
Options portfolios typically do not involve so many positions, and for them K is often small. However, the value of options with American-style exercise features typically requires computations on “trees” or lattices, and the prices of some exotic options may be computed using various numerical methods. For this reason, using full Monte Carlo with even a moderately sized options portfolio can be time consuming. In contrast, the delta-normal method requires only that the instrument’s deltas be evaluated once, for the current values of the market factors. The difference between doing this and performing NM instrument valuations can be the difference between sec- onds and days of computation.
NOTES
As in Chapter 3, the calculations of the futures prices of the S&P 500 and FT-SE 100 index futures contracts use the cost-of-carry formula F Sexp[(r d)(T t)], where S is the current index value, r is the interest rate, d is the dividend yield on the portfolio underlying the index, and T t is the time until the final settlement of the futures contract. For the S&P con- tract, the parameter values are r 0.05, d 0.014, and T t 0.3836, while for the FT-SE 100 contract they are r 0.05, d 0.016, and T t 0.3836.
This calculation of the futures contract settlement payments in equa- tion (6.1) and the similar calculation for the FT-SE 100 index futures con- tract ignore the fact that positive cash flows would likely be deposited at interest and negative cash flows would either incur financing costs or require a reduction in the investment in the portfolio. We translate the FT- SE 100 futures contract settlement payments to U.S. dollars using (6.3) instead of (6.2) primarily for convenience, because if we used (6.2) we would be forced to simulate the entire path of index values and exchange rates during the month. However, (6.3) is not necessarily a worse assump- tion than (6.2) and may be better. Equation (6.2) ignores any pound bal- ance in the margin account and assumes that the pound flows in and out of the account are converted to, or from, dollars each day. Equation (6.3) assumes that the pound flows in and out of the account are allowed to accumulate until the end of the month and then converted to, or from, dol- lars at the month-end exchange rate. It is almost certain that neither assumption is exactly correct. As a practical matter, the exchange-rate risk of the futures contract is sufficiently small that the difference between these two assumptions does not have a significant impact on the calculated value-at-risk, and we use (6.3) because it is more convenient.
=
– –
–
= = – =
= =
– =
The need for large numbers of Monte Carlo trials can be reduced through use of variance reduction techniques. While these are beyond the scope of this book, Glasserman, Heidelberger, and Shahabuddin (2000) provide an introduction to them in the context of value-at-risk estimation.
Most Monte Carlo approaches are based on the multivariate normal and lognormal distributions, though there has been some limited work out- side this framework. Recently, Hull and White (1998) and Hosking, Bonti, and Siegel (2000) have suggested approaches that involve simulating multi- variate normal random vectors and then transforming them to have mar- ginal distributions that better fit the known properties of changes in financial market factors. The heavy reliance on the normal and lognormal distributions seems to be driven by two considerations. First, these distribu- tions are convenient and tractable. Second, while distributions with fat tails relative to the normal are available, the data provide little guidance about which fat-tailed distribution should be selected. The whole issue is how much probability should be in the tails, but almost by definition there have been very few realizations in the tails. Thus, it is difficult to resolve this issue by looking at the data, and the designer has little basis to select and/or parameterize a particular fat-tailed distribution.
The use of Monte Carlo simulation to evaluate the risk of time- and path-dependent portfolio strategies, such as that underlying LaVaR is pushed by Dembo, Aziz, Rosen, and Zerbs (2000) under the rubric Mark- to-Future. An alternative approach to adjusting for liquidity is described by Berkowitz (2000).