VALUING PATH-DEPENDENT, EUROPEAN-STYLE OPTIONS ON A SINGLE VARIABLE

In the previous section, I discussed three examples of path-independent exotic options. As was seen in these examples, it was relatively straightforward for one to develop analytical expressions to value such options. In

TABLE 5.3 Valuation of Nonlinear-Payoff Options

this section, I will extend my discussion to entertain the valuation of path- dependent exotic options. Since it is not often that one can easily arrive at accurate analytical expressions to value many of the path-dependent exotic options, I will use simplifying assumptions that are used in practice to evaluate such options whenever this is not possible. Additionally, in such instances, I will use simulations to redo the problem so as to show the reader the consequences of making such simplifying assumptions.

Averaging Options

^{[1]}

One path-dependent exotic option that gets traded in many asset classes quite extensively in the OTC market (and sometimes in the exchanges^{[2]}) is the arithmetic-averaging option. With this type of option, the option holder gets an in-the-money payoff that is a function of the average of the underlying prices. More precisely, the owner of an averaging call option will receive the following payoff on maturity date:

where the values of stock prices are averaged every Δt years and t + nΔt

=T^{[3]}

When arithmetic-averaging options first traded in the OTC market, practitioners quickly realized that because of the lognormal distribution assumption associated with future stock/index price movements, it was not possible to obtain the pdf of the arithmetic average of stock price movements due to the fact that the sum of lognormal distributions does not result in a lognormal distribution. Given the unavailability of cheap computing power when these options first started trading, practitioners were forced to think about ways of overcoming this deficit using approximations and assumptions. In the following subsections, I will review a couple of these methods and benchmark them against the use of a Monte Carlo method.

Geometric Averaging One simplifying assumption that Kemna and Vorst made in 1990 was that an arithmetic average can be well approximated by a geometric average (although in their paper they did assume a continuous time averaging as opposed to a discrete time averaging that I discuss here).^{[4]} Furthermore, since one can easily get the pdf for a geometric average when the underlying stock/index price movements are lognormal in nature, the authors felt that instead of valuing an arithmetic average call option, one could just as well value a geometric average call option as a good proxy. To do this, they first concluded that instead of valuing an option whose payoff is

(5.4a)

one should value an option whose payoff takes the form of

(5.4b)

Using this proxy payoff, they observed that the probability density function (pdf) of the natural logarithm of the geometric average is in fact the sum of the normally distributed pdfs – which in turn produces a normal pdf. More precisely, since

andhas a normal pdf with a mean of and variance, it readily follows that has a normal pdf with a mean of

(5.5a)

Similarly, it can be shown thathas a variance of

(5.5b)

With the aid of equations (5.5a) and (5.5b), one can conclude thathas a normal pdf with mean and variance .

Using these parameters, it can be easily seen that a call option premium for a geometric averaging option is given by the expression

(5.6a)

where and

It is easy to show that the formula to value the averaging put option takes the form

(5.6b)

The implementation of equations (5.6a) and (5.6b) are shown in Table 5.4.

Although the implementation in Table 5.4 shows the use of the geometric average as a proxy for the arithmetic average, by setting n (the number of averaging points) to 1, one can easily observe from equations (5.4a) and (5.4b) that the averaging option collapses to a vanilla option. As a consequence, one can expect to see the example outlined in Table 5.4 yielding the same answer as that in Table 3.4.

Moments Matching Since a geometric average can never be bigger than an arithmetic average, the consequence of using a geometric average to approximate an arithmetic average results in the geometric averaging call (put) option being lower (higher) than an arithmetic averaging call (put) option. To overcome this setback, Turnbull and Wakeman in 1991 proposed the use of moment matching to derive an approximate pdf for the arithmetic average. More precisely, they assumed thathas a normal pdf with meanand varianceand estimated both these parameters by matching the moments of the assumed lognormal pdf ofwith the actual values of the quantity.

To do this, they showed that

(5.7a)

This can be summarized more succinctly as

(5.7a)

TABLE 5.4 Using Geometric Averaging to Value Arithmetic Averaging Options

where

They went on to show that

(5.7b)

This can be summarized more succinctly as

(5.7b)

where

Solving equations (5.7a) and (5.7b) forand, they obtained

(5.8a)

(5.8b)

where and are as defined in equations (5.7a) and (5.7b).

Putting all these together, they showed that the call and put option pricing formulae are given by the expressions

(5.8c)

(5.8d)

where

and

Equations (5.8c) and (5.8d) have been implemented as shown in Table 5.5 where and have been defined in equations (5.8a) and (5.8b) respectively.

The formulae implemented in Table 5.5 is for n = 1. As can be seen, the result obtained matches that in Tables 5.4 and 5.3 (where the averaging option trivially collapses to the vanilla option). Furthermore, for higher values of n, one would need to extend the terms added to in cells BIO and B11.

Monte Carlo Method The geometric averaging and moment matching approaches are just two examples of what practitioners used to do to get around the issue of finding an analytical solution to price options when the pdf of the underlying (in this case the arithmetic average) cannot be exactly derived. While this was an issue before the availability of cheap computer hardware, this is no longer a concern in the current computing environment. Practitioners nowadays resort to the use of Monte Carlo simulations (also known as the brute-force approach) to solve such problems quickly and with a greater precision.

Table 5.6 shows the implementation of Monte Carlo simulations to value both averaging call and put options.

TABLE 5.5 Using Moments Matching to Value Arithmetic Averaging Options

TABLE 5.6 Using Monte Carlo to Value Arithmetic Averaging Options

FIGURES.1A Relative Errors Associated with Arithmetic Averaging Call Option Approximation

FIGURE 5.1B Relative Errors Associated with Arithmetic Averaging Put Option Approximation

Table 5.6 shows the implementation of the averaging option pricing for one simulated path when there are two averaging points.^{[5]} In practice, one has to redo this simulation 5,000 times and average the outcomes in cell B16 (cell B17) to obtain the premiums for the averaging call (put) option of $4.41 and $2.30 respectively. In the event that there are more than two averaging points, one has to extend the rows to entertain more random numbers (and more frequent jumps).

The relative errors associated with these methods are illustrated in Figures 5.1a and 5.1b for varying numbers of averaging points and the in- the-moneyness of the option.

One can easily make the following observations from Figures 5.1a and 5.1b:

■ The term ATM refers to an at-the-money option where the strike price is set equal to the spot price. On the other hand, the term OTM refers to an out-of-the-money option where the strike price is set to 25 percent out of the money. Thus for a call option when the spot price is $40, the OTM strike price is $50 (for the call option) and $30 (for the put option).

■ The relative errors that were computed to produce the graph were done so using the expression

■ The geometric average method is a better approximation to the Monte Carlo method than the two-moments method. Furthermore, both these approximations^{[6]} are bad when valuing out-of-the money options.

[1] These types of options are quite popular in the currency and commodity markets where hedgers are more concerned about the adverse movements in the average rates (prices) of the currency (commodity) markets during the month or quarter or year as opposed to daily movements. These are sometimes also called Asian options.

[2] Quite often, when averaging options traded on the exchange, they are written on underlyings that themselves trade as an average (e.g., average aluminum price, average gas price) thereby making such averaging options vanilla options. In such an instance, it suffices for one to use the formulae in Chapter 3 to price these options.

[3] In practice, it is not uncommon to find a payoff that is a function of the weighted average of the stock prices (where the different weights are given to different times) and the averaging is done over infrequent discrete time intervals.

[4] For a given set of four numbers 1,2, 3,4, the arithmetic average is given by the value , while the geometric average is given by the value 2.213. Additionally, it can be shown that the arithmetic average would never be lesser than the geometric average.

[5] Since the problem simplifies to the valuation of a vanilla call/put option when n = 1, the reader is referred to Table 4.9 for the valuation of vanilla options using the Monte Carlo method.

[6] The geometric averaging and the moment-matching approaches are just two of the many approaches that are used by practitioners to value averaging options. The reader would be interested to know that some of these other approaches that have not been discussed here are surprisingly accurate even for the OTM options.

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