DERIVATIVES 'R US/V1N7/Option Pricing: Black-Scholes
VOLUME 1, NUMBER 7: April 3, 1995
This week's topic is the celebrated Black-Scholes model for pricing European options. Of course, this discussion and in fact all discussions here are fairly non-technical (I don t really want to put any equations in DRU), focusing on the basic principles, underlying intuition, and some trivia.
First off, let's call the model "B/S" (to be distinguished from "BS", which is so prevalent on the Internet) and when referring to the individuals, we ll call them Black and Scholes. B/S is a formula that relates the price of a European call (or put) to the variables that logically should affect its price. Intuition tells us that the stock price, exercise price, risk-free interest rate, volatility of the stock, time to expiration and dividends on the stock should affect the option's price. Interestingly, before B/S there was a widespread belief that the expected growth of the stock price ought to affect the option price. B/S demonstrates that this is not true, though it is not obvious on the surface why it's not true. I'll touch on that in a later issue. In short, B/S, like any good model, tells us what's important and what's not important. It doesn't promise to produce the exact prices that show up in the market, but it does a remarkable job of pricing options that meet all of the assumptions of the model. In fact, it's safe to say that virtually all option pricing models, even the extremely complex ones, have much in common with the B/S-based models.
Black and Scholes start by specifying a simple and well-known equation that models the way in which stock prices fluctuate. This equation, called Geometric Brownian Motion, implies that stock returns will have a lognormal distribution, meaning that the logarithm of the stock's return will follow the normal (bell shaped) distribution. Black and Scholes then propose that the option's price is determined by only two variables that are allowed to change: time and the underlying stock price (the other factors - the volatility, the exercise price, and the risk- free rate - do affect the option s price but they are not allowed to change). By forming a portfolio consisting of a long position in stock and a short position in calls, the risk of the stock is eliminated. This hedged portfolio is obtained by setting the number of shares of stock equal to the approximate change in the call price for a change in the stock price. This mix of stock and calls must be revised continuously, a process known as delta hedging, which I ll also cover in a later issue.
Black and Scholes then turn to a little-known result in a specialized field of probability theory known as stochastic calculus. This result defines how the option price changes in terms of the change in the stock price and time to expiration. Then they reason that this hedged combination of options and stock should grow in value at the risk-free interest rate. The result then is a partial differential equation or PDE, which is a fairly complicated expression containing derivatives (i. e., calculus derivatives, not financial derivatives; if you haven't had calculus, you won't follow this part). The solution is found by forcing a condition (called a boundary condition) on the model that requires the option price to converge to the exercise value at expiration. The end result is the B/S model.
Black and Scholes actually had some difficulty solving this equation. Though Black has a Ph.D. in applied mathematics from Harvard, he was not a specialist in differential equations and Scholes was only an economist. As many of you know, solving differential equations is often a matter of making educated guesses and taking advantage of prior knowledge of what the final solution might possibly look like. Black and Scholes benefitted from the fact that previous researchers had almost found the elusive formula. Their predecessors solutions looked remarkably similar to what we know as the correct formula. The final trick was found when their differential equation was recognized as of a form known in physics as the heat transfer equation. This equation already had a known solution, though there are quite a few complicated steps in getting to it.
Their difficulties did not end there. Black and Scholes had trouble getting the publishers of academic journals to care about their result. They were originally rejected by one distinguished economics journal, then another. Fortunately someone asked the editors of the original journal to take another look at it and the article was finally accepted. The rest was history. I would be surprised if there were many discoveries in economic and financial research that have had greater impact. It has taken many years but the offspring of the B/S model are now generating the prices of trillions of dollars of options. Though Black and Scholes have become wealthy, I think it is safe to say that had they been practitioners instead of academics primarily interested in pushing forward the frontiers of knowledge, they would likely have made many times more money.
In upcoming issues, I'll be exploring other topics in pricing derivatives.
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