As mentioned, the exchange-option example discussed earlier is a special case of a more generic class of options called spread options that trade on the exchanges, like the CME and the OTC markets. Although spread options trade in abundance in financial markets, what makes one spread different from another is the way the underlying spread is defined. One can, for example,^{[1]} find spreads in the CME whose differences are of the form

■ Wheat and corn futures expiring on the same month (sometimes called a cross-commodity spread).

■ March and May wheat futures (sometimes called a calendar spread).

For the purposes of my discussion, I will assume that a call (put) spread option purchaser gets a payoff that is of the form max[(ST – RT) – K, 0] (max[K – (ST – RT), 0]) on option maturity date, where ST and RT are the prices of the 2 independent underlyings at time T on which the spread underlying the option is written. Making the same distributional assumption as in the exchange options, to compute the expression for

observe that

As before, one can rewrite the above expression as follows:

Since the joint probability density function,, can be rewritten as a product of a conditional and marginal probability density function (i.e., , the above expression becomes

(5.20)

where,, andare all as defined in the section for exchange options. Since equation (5.20) cannot be succinctly written in an analytical form similar to that of an exchange option, this equation needs to be numerically evaluated. Due to the lack of cheap, high-powered computing technology, practitioners of the past had to resort to nonsimulation-based methods, which I will now discuss.

Spread as a Normal PDF One simplifying assumption that used to be made to get around this problem was to assume that the spread is normally distributed with a mean ofand a variance of (despite the fact that this distributional assumption is not true!)^{[2]} – where and are then computed using the method of moment matching. To see how this is done in practice, observe that

Thus, it can be easily summarized that

Similarly, since

It can be easily summarized that

Now that the values of and can be obtained from the above equations, one needs to obtain the expression used to value spread options. Given that is assumed to be normally distributed with a mean of and variance of , it readily follows that

(5.21)

The implementation of equation (5.21) is shown in Table 5.9.

TABLE 5.9 Valuation of Spread Options Assuming Normal PDF

Spread as a Difference of Two Lognormal PDFs Dsing Gaussian Quadrature While the above approximation is reasonable for at-the-money options, because of the nature of the assumptions one should not be surprised if this does not accurately value out-of-the-money and in-the-money options. To overcome this problem, Ravindran (1993a) provided a numerical approach to the original problem (i.e., not assuming any distributional form for SpreadT) by first reducing this two-dimensional problem using conditioning – an idea that I had used to derive the pricing formula for exchange options.

As the reader will realize, I started the section on spread options by deploying the same conditioning idea only to arrive at a point where I had to resort to numerical methods to progress further. Given this backdrop, I will henceforth focus my discussion on the use of the Gausssian quadrature method^{[3]} (see Abramowitz and Stegun (1965) and Burden and Faires (2010) for more details) to solve the integrals in equation (5.20).

Table 5.10 shows an example of the list of weights and roots for various values of n, where the larger the value of n the greater the accuracy of the approximation to the integral.

In order to now apply the Gaussian quadrature method to solve the one-dimensional integration problem I will, for the ease of reading, restate equation (5.20) which I am trying to numerically evaluate

(5.20)

To apply the Gaussian quadrature algorithm, one has to first transform the integral limits fromto ( – 1,1). Furthermore, as pointed out in footnote 28, the Gaussian quadrature method can only be applied to definite integrals. As such, one has to split the domaininto mutually

TABLE 5.10 Weights and Coefficients for Gaussian Quadrature

n

Roots

Weights

1

2.00000000

0.00000000

2

-0.57735027

1.00000000

0.57735027

1.00000000

3

-0.77459667

0.55555556

0.00000000

0.88888889

0.77459667

0.55555556

4

-0.86113631

0.34785485

-0.33998104

0.65214515

0.33998104

0.65214515

0.86113631

0.34785485

5

-0.90617985

0.23692689

-0.53846931

0.47862867

0.00000000

0.56888889

0.53846931

0.47862867

0.90617985

0.23692689

exclusive subdomains (0,M) and (M,), where M is chosen to be the median of the pdf of RT.^{[4]}

Doing this allows one to rewrite the first integral (and the term accompanying it) of equation (5.20) as

(5.22a)

While the first integral of equation (5.22a) is now a proper one (and hence can be easily mapped into the (-1,1) domain), the second integral is not. To transform the second integral into a proper one, one only needs to make the substitution. Doing this transforms the second

integral to

(5.22b)

Tables 5.11 to 5.12 show the implementation of the Gaussian quadrature algorithm to obtain the values of first integral (and the terms accompanying it) in equation (5.22a) and equation (5.18b) when n = 5.

The reader should note a few things from Tables 5.11 and 5.12. In order to cut down on repetition, I only identify changes in Table 5.12 vis-a-vis the formulae in Table 5.11. Furthermore, the weights and roots of the tables given in cells A21:B26 are for n = 5 (see also Figure 5.2).

One can apply a similar approach to solve the second integral (and the term accompanying it) of equation (5.20), namely,

so as to make it Gaussian-quadrature ready. Putting all the pieces together yields value of 7.52. Clearly for more accuracy, the value of n needs to be increased.

Monte Carlo Simulation The idea of using Gausssian quadrature (or any other nonsimulation-based numerical method) to solve a one-dimensional integral worked very well when one had to solve a spread-option problem that was not analytically tractable – during the times when cheap computers were not available. Given the presence of inexpensive, powerful hardware nowadays, it is reasonable to want to take advantage of such cheapness and resort to simulations to value spread options. Other additional benefits for using simulations, despite the existence of efficient numerical methods are

■ The use of a generalized methodology to value the entire book of options in the process making it easier to maintain the source-code instead of having many one-off special cases for each type of option.

■ The ability to easily add new products with more complex payoffs.

Given this backdrop, I now discuss the use of Monte Carlo simulations to value the same options. As the reader will recall, all the option valuations using Monte Carlo simulations were done on a single variable. Clearly one

TABLE 6.11 Valuation of the First Integral in Equation (5.22a)

TABLE 5.12 Valuation of Equation (5.22b)

FIGURE 5.2 Relative Error from Normal Approximation

has to adapt this to entertain the fact that there are now two variables that may be correlated to each other. To do this, one has to first observe that when using Monte Carlo simulations to value options on a single variable, one had to first simulate lognormal prices (this is easily done once one is able to generate standard normal variates obtained from the generated uniform (0,1) variates). Similarly, to generate two lognormal correlated variates, one needs to be able to generate two correlated standard normal variates. To do this, one needs to first generate two independent standard normal variates, ε and ω, where in the context of the spread-option problem the first (ε) is used for generation of the price RT. To use a correlated standard normal variate ω* for ST, one would have to use relation ^{[5]}

Besides this minor change, the rest of the implementation is no different from what was done for pricing an option on a single variable. Table 5.13 shows the implementation of a Monte Carlo simulation for pricing a spread option.

As in the previous Monte Carlo examples, a call option value of $1.844 is obtained using a pair of simulated paths for the two underlying stocks. If this simulation is repeated for another 5,000 times, the values can be seen to converge to 7.48 (a value that is close to 7.52). Since the Monte Carlo method is supposed to mimic the results obtained using the Gaussian

TABLE 5.13 Using the Monte Carlo Method for Spread Options Valuation

quadrature method (as the reasons for the introduction of the Monte Carlo method has nothing to do with increasing the efficiency over the Gaussian quadrature method), I will compare the effectiveness of the approximation associated with assuming that the spreads are normally distributed vis-a-vis not making any distributional assumption for the behavior of spreads. Figure 5.2 shows how the relative error looks for varying levels of correlation where the relative error was computed using the expression

[1] In theory, this spread can also be a difference of a function of one variable (e.g., average price of the first variable) and a function of the second variable (e.g., the highest price of the second variable) – where both these functions do not have to be identical.

[2] Ifand lnTT each has a normal pdf, thenwill also have a normal pdf, as the difference between two normal pdfs will also be a normal pdf. However, this is not the same ashaving a normal pdf.

[3] The Gaussian quadrature method allows one to solve the integral numerically using the weights, roots(examples of which are given in Table 5.10 for n = 1,2,3,4,5) and the approximation. To perform an integral over an arbitrary interval (a, b) (i.e.,), one needs to first map the integral into a ( – 1,1) interval using the expressionwhich can now be evaluated using the weights, roots, and the approximation In addition to Table 5.10, weights and roots for more values of n are given in the online content.

[4] The choice of M does influence the accuracy of the approximations. In practice I have found that a good choice for M when dealing with a lognormal pdf is the median of pdf. More precisely, if InY has a normal pdf with a mean ofand a variance of , then it can be easily shown that the mean, median, and mode of Y are given by the expressions,, andrespectively.

[5] ω* is also a standard normal variate (i.e., a normal pdf with a mean of 0 and variance of 1). In addition to this, it can be easily seen that when p is -1, 0, and 1, ω* is – ω, ε, and ω respectively.

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