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Option Pricing Theory

Options have existed—at least in concept—since antiquity. It wasn’t until publication of the Black-Scholes (1973) option pricing formula that a theoretically consistent framework for pricing options became available. That framework was a direct result of work by Robert Merton as well as Black and Scholes. In 1997, Scholes and Merton won the Nobel Prize in economics for this work. Black had died in 1995, but otherwise would have shared the prize.
Option pricing theory—also called Black-Scholes theory or derivatives pricing theory—traces its roots to Bachelier (1900) who invented Brownian motion to model options on French government bonds. This work anticipated by five years Einstein’s independent use of Brownian motion in physics.

Research picked up in the 1960’s. Typical of efforts during this period is Samuelson (1965). He considered long-term equity options, and used geometric Brownian motion to model the random behavior of the underlying stock. Based upon this, he modeled the random value of the option at exercise. The model required two assumptions. The first was the expected rate of return α for the stock price. The second was the rate β at which the option’s value at exercise should be discounted back to the pricing date. These two factors depended upon the unique risk characteristics of, respectively, the underlying stock and the option. Neither factor was observable in the market place. Depending upon their degree of risk aversion, different observers might propose different values for the factors. Accordingly, Samuelson’s formula was largely arbitrary. It offered no means for a buyer and seller with different risk aversions to agree on a price for an option. Black and Scholes got around the problem with a completely new approach.

Consider an options trader who is about to sell an option. She intends to dynamically hedge the exposure until the option expires. What price should she charge for the option? Black and Scholes propose that she charge the cost of dynamically hedging the short option. Significantly, given certain simplifying assumptions, they proposed that this cost could be known in advance.

In the real world, dynamic hedging is an uncertain undertaking. Depending upon the underlier and your hedging strategy, you might adjust your hedge a few times a week, or a few times a day. Between those adjustments, a lot can happen. If the underlier moves, you might rehedge immediately, or you might wait. If you wait, the underlier might move back, saving you the need to rehedge, or the underlier might keep moving in the same direction, causing a large loss. Real world dynamic hedging entails risk.

As a practical matter, there is a limit to how frequently a trader can rehedge, however, as the frequency of rehedging increases, dynamic hedging becomes more predictable. Using stochastic calculus and certain simplifying assumptions, Black and Scholes took the limiting case as the frequency of rehedging approaches infinity. In that limiting case, the cost of dynamic hedging is independent of the actual path taken by the underlier’s price. It depends only upon the price’s volatility. If that volatility is constant and known in advance, the cost of dynamic hedging a short option is certain. Being certain, it entails no risk, so it can be discounted at a risk free rate to obtain the price of the option.

Based upon this approach, Black and Scholes derived a partial differential equation for valuing claims contingent on a traded underlier. The equation is general. By applying different boundary conditions, it can be solved to price any such contingent claim. Black and Scholes applied the boundary conditions for a European call option on a non-dividend-paying stock and obtained their famous (1973) option pricing formula.
John C. Cox and Stephen A. Ross made an important contribution with their method of risk neutral valuation. Consider again Samuelson’s (1965) approach to pricing options, where he modeled an underlying stock price with some expected return α and discounted option values at exercise back to the pricing date with some rate β. There was nothing theoretically wrong with such an approach. Indeed, if it were based upon the same assumptions as the Black-Scholes approach, it would produce consistent option prices. This lead Cox and Rubinstein to an interesting conclusion. The two approaches were equivalent, yet one required α and β as inputs whereas the other did not. They concluded that the effects of α and β must somehow cancel. As long as α and β reflect the same degree of risk aversion, they do not affect the option price—and this must be true no matter what degree of risk aversion they reflect. If they can reflect any degree of risk aversion and still yield correct option prices, then they can be based upon an assumption of no risk aversion whatsoever. If an investor were risk neutral, he would require no excess return for taking risk. He would discount all cash flows—irrespective of their risk—at the risk free rate. The factors α and β would both equal the risk free rate. This brilliant insight was the Cox and Rubinstein risk neutral approach to option pricing.

 

The risk neutral approach opened the door to a host of option valuation techniques that used binomial trees or the Monte Carlo method to model future asset values. Rather than attempt to ascribe “realistic” expected returns and “realistic” discount rates in the analyses, users could treat all financial assets as having expected returns equal to the risk free rate. They could discount all cash flows at the risk free rate. The risk neutral assumption is not reflective of the real world. Real investors are not risk neutral, but this doesn’t matter. Correctly implemented, the risk neutral assumption produces correct option prices.

Cox and Ross didn’t immediately perceive how profound risk neutral valuation would be. They buried it in the middle of a (1976) paper on pricing options with jump processes. But three years later, they teamed up with Mark Rubinstein to publish a (1979) paper that used risk neutral valuation to develop the method of binomial trees. The mathematics of risk neutral valuation was formalized in continuous time by other authors to become the method of equivalent martingale measures. Today, this is the predominant methodology for derivatives pricing in complete markets.

Work on financial engineering spawned the  field of financial engineering. Practitioners, called financial engineers, design and implement derivatives pricing models. Top financial engineers are highly paid professionals who typically hold advanced degrees in mathematics or physics. Financial engineers are informally called quants or rocket scientists.

The Black Scholes approach and generalizations employ partial differential equations, so they are sometimes called the differential equations approach. Those differential equations often have closed-form solutions, leading to simple pricing formulas such as the original Black-Scholes (1973) formula. Other times, the differential equations need to be solved numerically using techniques such as the Monte Carlo method.

The risk neutral approach tends to entail extensive use of stochastic calculus with changes of measure between a “real world” and a “risk neutral” world. For this reason, it (and analogous approaches) tend to be called the stochastic calculus approach. It can lead to closed form solutions, but numerical solutions tend to be the norm. It is more flexible than the Black-Scholes approach. Sometimes, it can be used to price derivatives that the Black-Scholes approach cannot.

Techniques of financial engineering have been extended to fixed income derivatives, which generally require the modeling of entire term structures. They have also been extended to commodities markets, where risk neutral valuation becomes problematic.

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