To illustrate how a pricing problem is done when there are more than two underlying variables, I will discuss a variation of the path-dependent option on four variables that is popular in the investment sector. Known as lookback basket options, these options pay the owner on option maturity:

wherefor / = 1,2,...,n., ,, and. represent the prices of four correlated stocks at time with respective continuously compounded dividend rates and , respective annualized volatilities, and, and the correlation matrix of returns

As in the two-asset case, one needs to use the correlations to tie the generated standard normal variates. Suppose that one has already generated the four independent standard normal variates – ε, ω, λ, θ – for the simulation of the four respective asset prices. As pointed out by Ravindran (1997), the

TABLE 5.14 Using the Monte Carlo Method for Averaging Spread Options Valuation

standard normal variates associated for the asset prices (especially second, third, and fourth asset prices) need to be modified as follows:

In particular, when n = 2, Table 5.15 shows the implementation of the pricing of a lookback basket option.

The result given in cell B56 is obtained using one set of 4 correlated paths. To get a meaningful interpretation of this result, one needs to repeat this simulation for another 5,000 times and then take the average of all the results produced in cell B56 to arrive at a value of $33.42.

One last comment I would like to make before moving on to the next chapter relates to the ability to incorporate early-exercise features into all the options we discussed in this chapter. In practice, the good news is that many of the exotic options that trade in the marketplace tend to be European style in nature. As a consequence, the Monte Carlo technique presented in this chapter is typically sufficient for the reader to use to tackle the pricing

TABLE 5.15 Using the Monte Carlo Method for Lookback Basket Options Valuation

of exotic options. Having said that, it is tempting for the reader to question the necessity and value of discussing approximation methods. The reasons for this stem from the following:

■ One motivation for this book is to show the reader how quantitative methods are applied in practice and in this instance how such techniques are applied to the pricing of exotic options (regardless of the fact that the Monte Carlo method can easily do all of these things with very little effort).

■ Regardless of the fact that the use of the Monte Carlo method is the most common today, knowing such approximating methods (and improving on them) allows one to use them as great control variates to improve on the efficiency of the Monte Carlo runs – which as a consequence reduces run time.

■ In the event that one is able to find great approximation methods that produce very few relative errors compared to a Monte Carlo method, it is often much easier to use them (this a big reason why the Black-Scholes model is still coded as is in a program even though one can use Monte Carlo methods) – as it is much more efficient to use such methods vis-a-vis Monte Carlo methods, especially when running various stress tests.

Having said the above, if needed, one can still incorporate early-exercise features into the problems discussed here by using the methodologies identified in Chapter 3. Given that this not a frequently encountered problem, I will not discuss this here and refer the interested reader to Hull (2012) and Ravindran (1997).

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