0 t1 t2 T
Figure 10.2 Path-dependent option as multidimensional option
where P(St1,St2,ST) is the payoff andφ(St1,St2,ST) is the joint distribution function. Except for the most trivial examples, a multiple integral of this type would have to be evaluated numerically. The simplest way of doing this is by using a three-dimensional equivalent of the addition of strip-shaped areas: and here we see the beginnings of an intractable problem.
Suppose we divide the range of integration for each of our stochastic assets into just 10 slices. The number
of separate little calculations to perform the integration is 103; no matter, that’s what computers are for. But a three-step problem is too simple. A more realistic example might be a 1-year option with weekly monitoring, and we would then be looking at 1052separate little calculations. This is what Bellman called the curse of dimensionality.
The position is not improved if we decide to abandon numerical integration and just work with trees. Section 12.6 describes a binomial tree applied to two stochastic assets, and it is immediately apparent from its geometry that the number of calculations increases as N2. Extending todstochastic assets (dimensions) leads us to exactly the same dead-end that we hit with numerical integration.
The reader should now ask “if it’s so difficult, how come we have simple procedures for evaluating knock-out options, either analytically or using a simple (one-dimensional) tree?”.
The answer is that we have been clever enough to work out distributions for specific quantities such asSmaxorSmax or minorSgeometric av; but how would you value a 1-year option with weekly monitoring if the payoff is as follows:
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Knock-out if any sequence of five weekly stock prices has each price greater than the last.r
Otherwise, a call option.Here we seem to have no choice other than the multiple integral or tree approach; this is where Monte Carlo comes to the rescue.
(iv)Errors:Suppose we evaluate thed-dimensional integral using a total ofNdata points, allocating N1/d data points to each of thedvariables over which we integrate. Any calculus textbook covering numerical integration will explain how the error arising from the use of either the mid-point rule or the trapezium rule is inversely proportional to the square of the number of data points. The error in our multiple integration is therefore proportional toN−1/d.
10.2 BASIC MONTE CARLO METHOD
(i)Stock Price Simulation:In nearly the whole of this book, it is assumed that the risk-neutral stock price evolution follows equation (3.7):
St =S0e(r−q−12σ2)t+σ√t zi; zi ∼N(0,1)
Obviously, the way in which we use this formula will depend on the specific problem to be solved. In the following subsections we look at three simple examples: a European call, a knock-out call and an Asian option. For the European call, we need only the terminal values, so each simulation will just give us a random terminal valueST.
The other two options are path-dependent so each simulation will need to be an entire random path. This is approximated by a discrete path with successive values given by the same
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formula in a slightly modified form:St+δt=Ste(r−q−12σ2)δt+σ
√δt zi; zi ∼N(0,1)
We chose theδt to be the time between discrete monitoring or averaging points.
(ii)Estimates:The values zi in the last subsection are standard normal random numbers. Sec- tion 10.3 will explain how these are obtained. For each set of random numbers, a stock price path is calculated and the payoff of the option corresponding to this path is calculated; this is a single simulation. The process is repeated a large number of timesN, and an estimate for the expected value of the payoff is obtained by taking the average of the answers obtained for all the simulations. Finally we must take the present value of this expected payoff to get the option value.
This is in accordance with the following elementary statistical definitions and results:
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If fiis the option value calculated in a single simulation, then an unbiased estimate of the mean of all possible fi is equal to the sample averagef¯= 1 N
i
fi
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The unbiased estimate of the variance of all the fiis var[fi]=SD2= 1N−1
i
(fi− f¯)2
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The standard error of ¯f isSE=$
varf¯ = SD
√N
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The ¯f are normally distributed with mean f (the true option value) and variance SE2. (iii)Errors:From the last few lines it is obvious that the Monte Carlo method converges to theright answer with an error proportional toN−1/2; compare this with the error proportional to N−2/dfor numerical integration or trees. The key point is that the Monte Carlo error does not depend on the number of dimensions of the problem. We do not of course know the constants of proportionality for either error, but the variable term for multiple integration shoots up so quickly withdthat beyond a very few dimensions, Monte Carlo is more efficient.
It is instructive to look at three simple examples which we use later in this chapter. In each of the three cases (a simple European call, a knock-out call and an Asian call), a 20,000 shot simulation run is already yielding reasonable option prices with errors in the region±1% of the option prices and running times on a laptop of less than a minute. Now compare the alternative calculation methods for each example:
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The call option is a one-dimensional problem, having no path dependence. The numerical integration alternative would divide the range of integration into 20,000 slices; or we could build a tree with 20,000 steps. Either way, we would have produced an incredibly accurate answer, compared with the±1% we have produced here. Alternatively, we could have solved the integration analytically: this is called the Black Scholes model.r
The knock-out option has 52 monitoring points. We can derive a continuous distribution for the maximum value of the stock price and hence evaluate a continuously monitored knock-out10.2 BASIC MONTE CARLO METHOD
option analytically. Alternatively, we can build a simple tree to evaluate the option [not a 52-step tree as this would be too small for accurate answers (see Section 16.5), but some multiple of 52]. We saw in Section 10.1 that this simple knock-out is a “special case”.
Suppose instead that we tried to solve this problem by multiple integration or trees, which is what we would have to do if the knock-out feature were more complex. We devote two points to each dimension, i.e. divide the domain of integration into two slices in each dimension or equivalently, construct a multidimensional tree with only two steps for each asset. This would be hopelessly inadequate for an accurate answer, but would still take 252≈5×1015 calculations. Our Monte Carlo calculations consisted of 20,000 shots where each shot is a path of 52 steps, i.e. about a million calculations.
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The arithmetic average (Asian) option also has 52 monitoring points. Unlike the simple knock-out example, we cannot derive a distribution for a state variableSarithmetic avand must therefore go straight to the long-winded computation (this is not true for the geometric average option but these are not really traded in the market). Here again the Monte Carlo method takes about 106 calculations to yield an error of around ±1%, while numerical integration would take 5×1015calculations for an inadequate answer.So how does Monte Carlo manage to be so much more powerful than numerical integration, which we normally think of as a fairly efficient procedure? The answer is a combination of two parts: first, by its very nature the majority of Monte Carlo paths used in a simulation are the most probable paths. The calculation procedure wastes little time in exploring regions where paths are unlikely to fall. By contrast, tree methods spend equal calculation time on a remote node at the edges as they do on a highly probable node at the center. Similarly, numerical integration spends as much effort on a strip at the edge of a distribution as on one in the center.
The second part of the answer lies in the arithmetic effects of exponentiation. Suppose that in a one-dimensional tree, only 70% of the calculations really contribute appreciably to a pricing. If this is repeated in each of 52 dimensions, then only 0.752=0.000001% of the multidimensional tree contributes to the pricing; the rest are wasted calculations.
(iv)Strengths and Weaknesses of Monte Carlo:From what has been written in the last couple of pages it is clear that Monte Carlo is the only feasible approach to solving the general multidimensional problem (except for a few special cases). We have described this in terms of general path-dependent options, but there are other multidimensional pricings where Monte Carlo is the method of choice: most particularly spread options involving several assets, for which analytical methods are inadequate.
Quite apart from these theoretical considerations, Monte Carlo has immensely wide appeal.
Just about any final payoff can be accommodated and the method can be manipulated by computer programmers with little knowledge of either mathematics or finance. It is the ultimate fall-back method, which works when you cannot think of anything else, or do not have the time or inclination to be analytical. We have all at some time got fed up with working on a problem and instead just switched on the simulator overnight, to find a highly accurate answer on our screens in the morning.
Despite all these advantages, Monte Carlo does have an Achilles heal: American options.
Recall the binomial method of calculating an American option price. Starting at the maturity date, we roll back through the tree by discounting the expected option values to previous nodes [see Section 7.3(iv)]. At each node we compare the discounted-back value with the exercise value at that point; if the exercise value is larger, we assume that the American option is
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terminated at this node, and we substitute the payoff for the option value. The key point is that the method inherently allows us to compare the option value with the payoff value at each point in the tree. With the Monte Carlo method, we can calculate the payoff at each point on our simulated path, but there is no way of comparing this with the option value along the path.
Inevitably, methods have been found around this inherent problem and encouraging results have been claimed; but one cannot help feeling that it is better to go with an approach that is intrinsically better suited to American option pricing. Water can be pushed uphill, but why bother if you don’t have to?