DERIVATIVES 'R US/V1N9/OPTION PRICING: NUMERICAL METHODS
VOLUME 1, NUMBER 9, April 17, 1995
This week's topic continues a series of essays on option pricing. We previously covered the Black-Scholes and Binomial models. This week we look at some of the other option pricing methods.
First let's consider why we might need another method. As described in Volume 1, Number 7, the Black-Scholes model is the solution to a partial differential equation (PDE). In some cases, the type of option is so complex that a solution to the PDE is not possible or is very difficult to find. When that is the case, it is nearly always possible to obtain the option price by a computationally intensive method that either solves the differential equation or simulates the possible outcomes and determines the average option price that would lead to the outcomes observed. These methods are called "numerical methods." We actually already covered one numerical method last week. Yes, the binomial model is a numerical method. It takes a set of possible final stock prices at the option's expiration and systematically determines the option price today that would lead to the various possible final option values at expiration. (You might want to review last week's essay.)
The binomial model solves a difference equation, which is a variation of a differential equation. An alternative method is called the finite difference approach. In this procedure, the differential equation can be solved by laying out a rectangle with rows representing a range of possible stock prices and columns representing the time to expiration. The rectangle, thus, looks like a grid representing various combinations of stock price and time to expiration. The column at the far right contains the option prices at the expiration for the range of possible stock prices. The differential equation can then be converted into a series of easily solvable algebraic equations. However, the solution can only be done for one column at a time. Thus, starting at the far right, one solves the equations, which then produces the option prices for the range of stock prices if the option has just a little time to go before expiration. Stepping back one column to the left, the user proceeds to solve the remaining equations until reaching the far left column, which gives the option price today for a range of stock prices. Reading off the option price for the actual current stock price gives us the actual current option price. This method, in effect, solves the differential equation for the option price at every point in time in the option's life for a range of reasonable stock prices. In order to get today's option price, however, one has to start at the end of the option's life and work successively back to the present.
There are several variations of the finite difference method. Admittedly in this non-technical essay, I can't do it full justice. But suffice it to say that the technique is not all that difficult if you're comfortable with algebra. However, its advantage over the binomial model is quite limited. In fact, I believe that most institutions use the binomial model or a variation thereof and get just as good results.
Another method used to price complicated options is the Monte Carlo simulation. In general Monte Carlo simulation is a method of analyzing the effects of business decisions under uncertainty. Random numbers are drawn from a probability distribution whose properties are comparable to those of the underlying variable of uncertainty, such as future sales, labor costs, etc. In option pricing, it is possible to simulate the stock prices that might occur at the option's expiration. For example, suppose the stock price today is 100 and the strike is 100. We draw a random number that produces a stock price at expiration of 105. Then the option would have a value at expiration of 5. Suppose the next random stock price drawn is 98. Then the option ends up worth 0. We proceed to draw many of these random stock prices and then average the corresponding set of option prices at expiration. Then we discount the average back to the present and that gives us the current option price.
How good is the answer? We can gauge its accuracy by comparing it against the Black-Scholes value for a standard European call. I ran a Monte Carlo simulation with only 1,000 random numbers of an option with a stock price of 72, an exercise price of 70, a risk-free rate of 6 %, a time to expiration of 42 days and a standard deviation of .32. I ran this 1,000 trial simulation three times and obtained prices of 4.46, 4.36, and 4.58 (remember that each price was obtained using 1,000 random stock prices). The correct answer using Black-Scholes is 4.46. Monte Carlo simulation, however, is most valuable for more complicated options such as Asian options, where the payoff is based not on the price of the stock on the expiration day but on the average price of the stock over the life of the option. Unfortunately, upwards of a million random numbers must usually be drawn.
Another method used in option pricing is the polynomial approximation. It is a well known rule in mathematics that virtually any well-behaved function can be approximated with a polynomial. Most option prices are represented by well-behaved functions, however shy those functions might be in revealing themselves to us. Thus, researchers have developed a set of models that are approximations to the actual formulas. As approximation, they vary in their accuracy, in some cases being extremely accurate over a given range of parameters and in others, showing severe biases. However, they do reduce the time and computational complexity and enable us in some cases to get accurate answers to the prices of complicated options.
In general numerical solutions are clearly inferior to models that gives us simple, easily computable, one-step formulas. Alas, real-world problems do not always lend themselves to such nice results, as the scientists and mathematicians among you can attest. Fortunately, computers have made numerical solutions more easily attainable.
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