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Rethinking Implied Volatility

By Don Chance, Ph.D., CFA

This article originally appeared in the January/February 2003 issue of Financial Engineering News.

With the possible exception of Value at Risk, probably no topic has received more attention in risk  management research than implied volatility. This research can be classified into two major  categories: research on analytical methods for extracting the implied volatility and research on  methods for modifying existing pricing models to render the observed pattern of implied volatility  consistent with the chosen pricing model. The well-known volatility smile or skew is at the center of  much of this research. Unfortunately, there is little research that addresses the question of why the  existing research provides inconsistent and inexplicable results. The plethora of research on implied  volatility that makes no effort to address this question is astounding. I contend that the most fundamental question is how can the options market tell us that there is more than one volatility for an underlying asset?

The answer is really quite simple: It cannot, but more importantly, it does not.

The volatility smile/skew is inconsistent with an arbitrage-free world. Yet researchers devote countless hours to force-fitting existing models to produce prices that fit the smile/skew. I argue that we are neither asking the right questions nor approaching the problem in the right manner. We are afraid to go down a path we have been down before and thought we would never have to go down again. To understand my point, let us start with an explanation of the difference in two major classes of economic models.

Economic Models: Partial Equilibrium and General Equilibrium

Students of economics spend a great deal of time studying models of economic equilibrium. When markets are in equilibrium, the supply of assets equals the demand for assets, leading to the determination of a market-clearing price and a quantity held of each asset. These results are obtained by aggregation across all assets and market participants. Models of financial markets are a special class of economic models in which equilibrium is derived in the market for financial assets. Financial asset models produce financial asset prices, typically in the form of rates of return required by investors. These required rates of return are normally consistent with the notion that risk-averse investors would expect to earn the risk-free rate at a minimum and a higher return commensurate with the risk assumed, a factor known as the risk premium. When these models are developed in a framework in which all risky financial assets are priced, they are called general equilibrium models. Probably the best-known general equilibrium model of financial markets is the Capital Asset Pricing Model, or CAPM, of Sharpe (1964) and Lintner (1965).

Technically, the CAPM is not restricted to financial assets. It covers any risky asset that an investor  could own. Financial assets would obviously be included, but real estate and metals are good  examples of other non-financial assets that could come under the umbrella of the CAPM. Another  class of models is called partial equilibrium models. These models do not provide results by  aggregating across assets and market participants. They take the existence of a class of assets, the  prices of those assets, and the expectations and preferences of all market participants as given.  Partial equilibrium models then determine the price of one or more other assets relative to the  given class of assets. General equilibrium models show the big picture, while partial equilibrium models show a subset of the big picture. Securities in the form of stocks and bonds are the central assets in financial markets, and options are, of course, derivative assets. Think of securities as the main event, while derivatives are the side show, albeit an interesting and influential side show, that sometimes draws a bigger crowd. Option pricing models are typically partial equilibrium models. They nearly always take the assets in the market and their expected returns as given. Let us start by taking a brief look at the early models of option pricing.

The Archeology of Option Pricing

The first model for pricing options was the celebrated Bachelier (1900) dissertation, which assumed an arithmetic Brownian motion process for the stock. This assumption is generally deemed unacceptable for stocks, although arithmetic Brownian motion may be acceptable for options on spreads or other values that can be negative. Sprenkle (1964), under the assumption that the stock moves according to the more realistic geometric Brownian motion, derives the value of the option as the discounted expectation of its payoff at expiration. His formula requires the expected growth rate of the stock and a discount factor to reflect the investor’s risk aversion. Unfortunately, Sprenkle’s model does not use an interest rate, so the time value of money is ignored. Boness (1964) incorporates an interest rate and risk aversion, but discounts the option payoff at the stock’s discount rate. Each of these problems was solved by Samuelson (1965), but his model still requires discount rates for both the stock and option. All of these models assume that a general equilibrium model is available that would supply the missing link, which is the information on how risk-averse investors discount risky assets. None of these authors found the real insight provided by Black-Scholes (1973) and Merton (1973), which was that the option could be hedged by the stock, leading to the conclusion that the discount rate should be the risk-free rate.

The expected returns on the stock and option were not needed. It was indeed a joyous moment in the history of option pricing. But we know that the Black-Scholes model has problems. Fischer Black himself acknowledged this point and was quite amazed that the model was so successful. See Black (1989), which does not mention the volatility smile/skew.

The Biggest Hole in Black-Scholes

The Black-Scholes model solved the option pricing problem without resorting to a general  equilibrium theory. By invoking the principle that no arbitrage profits should be available, the  model provided the framework for pricing a financial instrument through the process of  replicating its payoffs using instruments with known prices. [Interestingly, the Black-Scholes  model was not the first model for pricing derivatives that was based on the principle that no  arbitrage profits could be earned. Well before the days of Black and Scholes, agricultural economists knew that the price of a futures contract should be the price of the underlying spot asset increased by the costs of holding it and reduced by any implicit yield on the asset. This argument follows from the fact that the asset can be purchased and hedged using futures to produce a risk-free position that should yield the risk-free rate over and above any costs of storage less any yield. It is difficult to pinpoint who first identified this relationship, but see Blau (1944-45) for an early discussion.] The Black-Scholes model is, thus, a partial equilibrium model. The price of an option is determined relative to the price of the underlying, taking into account interest rates and other factors exogenous to the model. It is probably safe to say that the derivatives industry would be stuck in the psychedelic 60s, and many talented mathematicians would still be teaching freshman algebra for $20,000 a year had Black, Scholes and Merton not made their contribution. But the Black-Scholes model has been both a blessing and a curse. It may well be a Pandora’s Box that has caused us to think that we neither can nor should ever look back.

As we said earlier, the Black-Scholes model produces implied volatilities of traded options that can vary by exercise price for a given underlying asset. How should we respond to such a finding? First, we could suggest that the Black-Scholes model is not correct. Since there cannot be more than one volatility of the underlying asset, the model must be incorrect. Case closed. But industry has not responded in that manner. Indeed it has responded in quite the opposite, embracing the Black-Scholes model and seeking to improve on it. The volatility smile notwithstanding, select 20 articles at random on implied volatility and I would be surprised if one of them argued that the Black-Scholes model is fundamentally incorrect. The closest any would come is to argue that perhaps the stochastic process of the underlying is improperly specified. Jumps and fat tails are commonly thought to be the explanation of the smile/skew. But such a conclusion tells us only that other models, perhaps with different stochastic processes, can accommodate multiplied implied volatilities. That should be just as unsettling. The implied volatility is a catch-all that reflects anything important omitted from the model.

Consider a world in which the stock market reveals volatilities but not prices. Then the stock price would catch any omitted inputs in the option pricing process. Options on stocks with different expirations would have multiple implied stock prices. We would then be looking at an implied stock price smile or skew. And that would be absurd. So what is missing? The Black-Scholes model is indeed incorrect. It may well be the case that the stochastic process is improperly specified. But there is something else. The Black-Scholes model is incorrect because of the very reason why the Black-Scholes model is so highly regarded. The Black-Scholes model tells us that the price of the underlying stock and investors’ feelings about risk do not matter in the pricing of options. These things should not matter because options can be perfectly replicated with the underlying stock and risk-free bonds. The model cannot identify the demand for options or distinguish it from the demand for stocks and risk-free bonds, because stocks and risk-free bonds, held in the right proportions, are equivalent to options. This ability to replicate is the glue that holds the model together. It says that some instruments are perfect substitutes for other instruments. A subtle point in this premise is that any one option is a perfect substitute for any other option. What is not so obvious in the Black-Scholes model is the fact that the model says nothing about the demand for options or why anyone would want to buy or sell an option, other than to earn an arbitrage profit. It does not tell us how many options an investor would hold in his or her portfolio. It does not do these things because it is a partial equilibrium model and not a general equilibrium model. The Black-Scholes model gives us only the price of an option, given that demand, preferences, asset prices and interest rates have been determined exogenously.

I emphasize that the reason the Black-Scholes model tells us nothing about the demand for options is that any option on a given underlying is a perfect substitute for any other option in the Black-Scholes world. In fact, in a general equilibrium framework, you can even substitute seemingly different securities. For example, in the CAPM world two stocks with equivalent beta coefficients are effectively the same stock, because they make the same contribution to an investor’s portfolio. In a general equilibrium framework, it could be possible to substitute an option on one stock for an option on another stock. If the assumptions under which the Black-Scholes model is derived do not hold sufficiently in practice, then any option on a given underlying is not a perfect substitute for any other option on the same underlying. If that is the case, then an investor may have needs that can be met only by a particular option. Then that option will have a greater value than some other option on the same stock with a different exercise price. If the Black-Scholes model is then used to obtain the implied volatilities, the former will show a higher implied volatility. To capture the demand for options, we need a general equilibrium model, but that would require that we impose restrictions on investors’ preferences. The prices of options would, thus, no longer be preference-free.

We would be back to a world that probably blends the Samuelson model with the Capital Asset Pricing Model. This does not, however, mean that option prices would violate the no-arbitrage rule. General equilibrium models are consistent with a world of no arbitrage opportunities. Put-call parity and various static trading strategies that lead to boundaries on option prices would indeed hold. But it is unlikely that continuous-time dynamic trading strategies that lead to models like Black-Scholes would hold. Option pricing models would then require expected returns on options.

In Conclusion

When the Black-Scholes model was discovered, researchers were excited that the model did not require the expected return on the stock. Expected returns are difficult to estimate and notoriously unstable. Moreover, they are influenced by investors’ preferences, which are even more difficult to estimate, are unstable, and subject to irrational behavior. Hence, researchers would prefer to salvage the Black-Scholes model than consider the alternatives. But the cost of this approach is the implied volatility smile/skew, which leads to the irrational result that there is no unique volatility for the underlying stock. We may be at a great crossroads in the history of options.

We can continue to find ways to contort the Black-Scholes model so that it will be consistent with the volatility smile. Or we can look to general equilibrium models to give us option prices based on expectations and preferences. As reluctant as we are to turn back the pages of history, we may have to. To do otherwise is to argue that there is more than one volatility of the underlying. We might as well argue that there is more than one sun in our solar system. About the Author Don Chance is a professor of finance at Louisiana State University. He can be contact [email protected]

References

Bachelier, L. 1964. “Theory of Speculation (English translation),” in P. Cootner, ed. The Random Character of Stock Market Prices. Cambridge: MIT Press. 17-78.

Black, F. 1989. “How to Use the Holes in Black-Scholes.” Journal of Applied Corporate Finance. 1: 67-73.

Black, F. and M. Scholes. 1973. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy. 81: 637-659.

Blau, G. 1944-45. “Some Aspects of the Theory of Futures Trading.” The Review of Economic Studies. XII: 1-30.

Boness, A.J. 1964. “Elements of a Theory of Stock-Option Value.” Journal of Political Economy. 72: 163-175.

Lintner, J. 1965. “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets.” Review of Economics and Statistics. 47: 13-37.

Merton. R.C. 1973. “Theory of Rational Option Pricing.” Bell Journal of Economics and Management Science. 4: 141-183.

Samuelson, P. 1965. “Rational Theory of Warrant Pricing.” Industrial Management Review. 6: 13-31.

Sharpe, W.F. 1964. “Capital Asset Prices: A Theory of Asset Equilibrium Under Conditions of Risk.” The Journal of Finance. XIX: 425-442.

February 15, 2010 - Posted by | Advance, Financial Engineering

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