DERIVATIVES 'R US/V1N8/OPTION PRICING: THE BINOMIAL MODEL
VOLUME 1, NUMBER 8, April 10, 1995
Last week I presented a brief overview of the Black-Scholes (B/S) model for pricing European options. This week I'll discuss the binomial model.
First off, let's ask the question of why we need another model when we've got B/S. Technically, we don't for standard European options. But the binomial model serves two useful purposes. One is that the mathematics involved in deriving the B/S model are quite difficult. This is unfortunate since the intuition is not all that difficult. One simply holds the underlying asset and sells a call in the appropriate proportions such that the call price change and the asset price change offset, leaving a riskless position that should grow in value at the risk-free rate. The position must be revised continuously since the correct number of units of the asset per option changes constantly. The binomial model lets us see how this happens without having to place ourselves in a world of continuous portfolio revision. Let me explain how that works and then I'll conclude with the other reason why we need the binomial model.
Suppose a stock is priced at $100. We propose that it can go up to $112 or down to $92. Consider a call on the stock with a strike of $100 that expires when the stock gets to either $112 or $92. What is a fair price for the call today? The answer is simple though it takes a few steps.
Suppose you purchase 60 shares of stock and sell 100 calls (that's actually one call contract covering 100 shares). How much will that cost? Sixty shares of stock will cost 60($100) = $6,000 but the 100 calls will pay for a portion of that cost depending on what you can get for them. When the option expires, the stock will be at either $112 or $92, meaning that the call will expire worth either $12 or $0. So the value of your position will be either (1) 60 shares worth $112 each minus 100 calls worth $12 each for a total of $5,520 or (2) 60 shares worth $92 each for a total of $5,520 (note that your calls expire worthless). Thus, you're hedged and that $5,520 value should represent a return equal to the risk-free interest rate. Suppose the risk-free rate is 5 %. That means your initial outlay should be $5,520/1.05 = $5,257. Since the 60 shares of stock cost $6,000, the 100 calls should cost $743 or $7.43 each. In other words, 60 shares of stock purchased at $100 each, less 100 calls sold at $7.43, would cost $5,520 and would return $5,520 one period later with no risk, a rate of 5 %, the current rate on alternative risk-free investments.
If the call isn't priced at $7.43, you could earn an arbitrage profit by either selling it if it's overpriced and buying the shares, or buying it if it's underpriced and shorting the shares. Thus, $7.43 is the only call price at which everyone would agree that nothing is to be gained by engaging in riskless arbitrage.
If we added a period to the problem and let the stock go from 112 or 92 to some other values, we could maintain the hedge by adjusting the number of calls per share. We would ultimately earn the same risk-free return over each of the two periods, if and only if the call is selling at its correct, arbitrage-free price. Each time we allow the stock to make one more move, we are adding what is called a time step.
In fact, we can continue to add time steps, letting an actual option's real life be represented by a large number of these time steps. Assuming we let the number of time steps become large, we would find, by appropriately setting the percentages by which the stock can go up and down, that the binomial price will bounce around and converge to the B/S price. My experience has found that it usually takes a minimum of 50 time steps for standard European options. Obviously a computer is necessary for any calculations involving more than a few time steps but programming the binomial model is simple and requires surprisingly few lines of code.
Therein lies the other advantage of the binomial model. We hardly need it to verify the B/S price but often we are interested in options that are not standard European. American options, for example, are exercised early and, therefore, should sell for higher prices than their European counterparts. The binomial model enables us to check the various points in an option's life for the possibility of early exercise. If early exercise makes sense at any point in time, we assume the option holder would do so and the computed values are then replaced by the exercise values. Only one additional line of code is required in most programs. Other more complex options of the type often seen in over-the-counter markets today can also be priced by using the binomial model or a variation thereof but the code does become complex and the speed slows down considerably.
The binomial model provides us with a computational procedure that transparently solves a difference equation, which is similar to the differential equation that I mentioned last week as being the key toward obtaining the B/S formula. In a future issue, I'll look at some other methods of pricing options that are used when the binomial model isn't fast enough.
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