Thus far, I have only discussed the generation of random variables from any desired pdf. In this section, I will discuss applications of simulations in practice. In doing so, I will address the second question that was posed in the beginning of this chapter. To apply simulations to solve problems in practice, one first needs to identify areas where assumptions are made about the nature of the pdfs being used. If this can be done, then it is a matter of simply applying what has been discussed in the earlier sections of this chapter. The tricky part, however, arises when one cannot find such pdfs. In such an event, trying to apply simulations to solve a problem becomes more an art than science. Whatever the problem, it is imperative for the reader to understand that all formulations require the simulations to be run many times before results can converge to any meaningful number – a reason why the results obtained over each simulated event are averaged over the total number of simulations. One way to obtain a more stable average is to increase the number of simulations, as the greater the number of simulations, the more stable the averaging process. However, doing that is not practical due to the time needed to produce the results. As a consequence, one is led to ask the third question posed in the beginning of this chapter – an issue that is discussed in the next section.

Valuing European-Style Options

In Chapter 3,1 discussed the valuation of European-style call and put options using the distributional assumptions for the underlying price/rate movements. To refresh the reader's memory, for a nondividend-paying stock, I assumed that lnST (natural logarithm of the stock price at a future time T) has a normal pdf with a mean ofand a variance of. By taking the expectation of max[Sт – K, 0] with respect to ST and discounting it byone was able to arrive at the expression in equation (3.4a) to value a European-style call option on nondividend-paying stocks.

In this example, I revalue the option implemented in Table 3.2 using simulations and show that I can arrive at similar (but not identical) results. To do that, one should first observe that the distributional assumption for ST can be rewritten as, where r is a normal standard variate.^{[1]} Table 4.10 shows the implementation of valuing a European-style call and put option on a nondividend paying stock for one path.

Since basing the option value on one simulated path is not enough, running this over 5,000 paths and then averaging over these 5,000 results yields a call option value of 6.56 and a put option value of 2.41. This compares favorably with the results in Table 3.2 (where the value of the call and put options were shown to be 6.88 and 2.36 respectively).

TABLE 4.10 Valuing European-Style Options on Nondividend-Paying Stocks

[1] To see this, observe from the distributional assumption of ln5T that since has a normal pdf with a mean ofand a variance of , it readily follows thathas a standard normal pdf. Letting r represent a standard normal variate, one can rearrange the equation to arrive at the required result.

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