# «Recovering Stochastic Processes from Option Prices by Jens Carsten Jackwerth and Mark Rubinstein* July 29, 2001 Abstract How do stock prices evolve ...»

Working paper, London Business School

Recovering Stochastic Processes from Option Prices

by

Jens Carsten Jackwerth and Mark Rubinstein*

July 29, 2001

Abstract

How do stock prices evolve over time? The standard assumption of geometric Brownian

motion, questionable as it has been right along, is even more doubtful in light of the stock market

crash of 1987 and the subsequent prices of U.S. index options. With the development of rich and

deep markets in these options, it is now possible to use options prices to make inferences about the risk-neutral stochastic process governing the underlying index. We compare the ability of models including Black-Scholes, naïve volatility smile predictions of traders, constant elasticity of variance, displaced diffusion, jump diffusion, stochastic volatility, and implied binomial trees to explain otherwise identical observed option prices that differ by strike prices, times-toexpiration, or times. The latter amounts to examining predictions of future implied volatilities.

Certain naïve predictive models used by traders seem to perform best, although some academic models are not far behind. We find that the better performing models all incorporate the negative correlation between index level and volatility. Further improvements to the models seem to require predicting the future at-the-money implied volatility. However, an “efficient markets result” makes these forecasts difficult, and improvements to the option pricing models might then be limited.

* Jens Carsten Jackwerth is an assistant professor of finance at the University of Wisconsin at Madison (jjackwerth@bus.wisc.edu) (http://instruction.bus.wisc.edu/jjackwerth/) and a professor of finance at the University of Konstanz. Mark Rubinstein is a professor of finance at the Haas School of Business, University of California at Berkeley (rubinste@haas.berkeley.edu). The authors gratefully acknowledge a research grant from the Q-Group. For helpful comments the authors would like to thank an anonymous referee, David Brown, Jim Hodder, David Modest and seminar participants at the AFA meetings 1996, Berkeley Program in Finance 1998, and at Erasmus, Konstanz, Warwick, Dartmouth, Iowa, UBC, Madison, and Oxford.

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2008/5437/ URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-54372 Recovering Stochastic Processes from Option Prices July 29, 2001

How do stock prices evolve over time? The standard assumption of geometric Brownian motion, questionable as it has been right along, is even more doubtful in light of the stock market crash of 1987 and the subsequent prices of U.S. index options. With the development of rich and deep markets in these options, it is now possible to use options prices to make inferences about the risk-neutral stochastic process governing the underlying index. We compare the ability of models including Black-Scholes, naïve volatility smile predictions of traders, constant elasticity of variance, displaced diffusion, jump diffusion, stochastic volatility, and implied binomial trees to explain otherwise identical observed option prices that differ by strike prices, times-toexpiration, or times. The latter amounts to examining predictions of future implied volatilities.

Certain naïve predictive models used by traders seem to perform best, although some academic models are not far behind. We find that the better performing models all incorporate the negative correlation between index level and volatility. Further improvements to the models seem to require predicting the future at-the-money implied volatility. However, an “efficient markets result” makes these forecasts difficult, and improvements to the option pricing models might then be limited.

** Recovering Stochastic Processes from Option Prices**

How do stock prices evolve over time? Ever since Osborne (1959), the standard view has been that stock prices follow a geometric Brownian motion. Merton (1973) uses this assumption as the basis for an intertemporal model of market equilibrium, and Black and Scholes (1973) uses it as the basis for their option pricing model. Tests of options on stock in the early years of exchange-traded options more or less supported the implications of Brownian motion, see, for example, Rubinstein (1985). While it has long been well known that empirical return distributions exhibit fatter tails than implied by Brownian motion, evidence that something is not all right with this world is that S&P 500 index options since the crash of 1987 exhibit pronounced volatility smiles, see Jackwerth and Rubinstein (1996). A volatility smile describes implied volatilities that are largely convex and monotonically decreasing functions of strike prices.1 Such volatilities contradict the assumptions of geometric Brownian motion and perfect markets, which would imply a flat line. Another way to describe this is that the implied risk-neutral probability densities are heavily skewed to the left and highly leptokurtic, unlike the lognormal assumption in BlackScholes. Like the equity premium puzzle, this option pricing puzzle may ultimately lead us to a better understanding of the determinants of security prices.

**There are three possibilities why option prices can spuriously exhibit volatility smiles:**

First, there are market imperfections, and observed option prices are always different from the true option prices at any time. The S&P 500 index option market is a rather deep and liquid market with rather unfettered access. Its daily notional volume is sizable, as reported in Table I for longer-term options. Even as the daily notional volume increased six-fold from $1.5 billion in 1989 to $8.5 billion in 1995, the volatility smile did not change. Most of our results are based on longer-term options, which account for about 4% of the total daily notional volume in all maturities. However, our results do not seem to be sensitive to our focus on the longer-term options.

** Table I about here**

Since the S&P 500 index is rather high (370 dollars on average from 1986 through 1995), the value of an option is high compared to the bid/ask spread, which for at-the-money options is only some 42 cents, decreasing to 33 cents for out-of-the-money options. Moreover, we expect the true option price to be close to the mid-point quote for most of the time. Thus, market imperfections are not likely candidates to explain the volatility smile.

The second possibility is that option prices are measured correctly but that the implied probabilities are calculated incorrectly. For example, the wrong interpolation or extrapolation method is used to obtain a dense set of option prices across strike prices.

**Jackwerth and Rubinstein (1996) show however, that the choice of method does not really matter much because 1 The implied volatility (σ*) causes the Black-Scholes formula to accurately price the option in the market:**

C = Sd − t N ( x ) − Kr − t N ( x − σ * t ), where S is the index level, d the dividend yield, t the time, N(•) the ln( Sd − t Kr − t ) 1 * + 2 σ t, and K the strike price.

x= cumulative normal distribution, σ* t most methods back out virtually the same risk-neutral distribution, as long as there are a sufficient number of strike prices, say, about 15.2 The third possibility is that the observed option prices are systematically distorted, and that one can make money in the options market by exploiting such mispricing. Jackwerth (2000) takes this view to some extent.

We assume instead that we see correctly measured option prices that yield meaningful implied risk-neutral probability distributions. The volatility smile is then a way of describing the relation of option prices at the same time, with the same underlying asset and the same time-toexpiration, but with different strike prices. Option prices also provide three other types of comparisons that can be windows into an understanding of the stochastic process of the

**underlying assets:**

(1) Option prices at the same time, with the same underlying asset, and the same strike price, but with different times-to-expiration.

(2) Option prices with the same underlying asset, the same expiration date, and the same strike price, but observed at different times.

(3) Option prices at the same time, with the same time-to-expiration and with the same strike price, but with different underlying assets.

Jackwerth and Rubinstein (1996) consider relationships among option prices at the same time and with the same underlying and time-to-expiration, but with different strike prices. The ultimate objective is to discover a single model that can explain all four relations simultaneously. For example, the post-crash smile of index options and the implied binomial tree model of Rubinstein (1994) strongly suggest that a key aspect of the “correct” model will be one that builds in a negative correlation between index level and at-the-money implied volatility. This can explain the relation in Jackwerth and Rubinstein (1996) and turns out in the post-crash period to be an empirical regularity of relation (2).

While we focus here on the smile in the S&P 500 data for the U.S., Tompkins (1998) documents that similar smiles, albeit not as steep as the U.S. smile, are seen in the UK, Japan, and Germany. In addition, Dennis and Mayhew (2000) show that individual option smiles in the U.S.

are not as steep as the index smile, a finding that likely holds for the other markets as well but that has not been documented.

There are several rational economic reasons why the post-crash smile effect might obtain.

First, corporate leverage effects imply that as stock prices fall, debt-equity ratios (in market values) rise, causing stock volatility to increase. Second, Kelly (1994) notes that equity prices have become more highly correlated in down markets, again causing an increase in volatility.

Third, risk aversion effects can cause investors who are poorer after a downturn in the market to react more dramatically to news events. This would lead to increased volatility after a downturn.

Fourth, the market could be more likely to jump down rather than up. Indeed, since the stock market crash period of 1987 until the end of 1998, the five greatest moves in the S&P 500 index have been down. Finally, as the volatility of the market increases, the required risk premium rises, 2 The methods differ most in the tails, where they tend to agree on the total tail probability but distribute this probability differently. We avoid this difficulty by focusing on the center of the distribution and not using far-awayfrom-the-money option prices. Further evidence on the performance of different methods is surveyed in Jackwerth (1999).

too. A higher risk premium will in turn depress stock prices. We do not try to provide an economic explanation for observed smile patterns, but rather have the more limited objective of comparing alternative models that purport to explain relations (1) and (2). We leave to subsequent research an investigation of relation (3). A comparison of smile patterns for index options and individual stock options, as in Dennis and Mayhew (2000), provides a way to distinguish between leverage and wealth effects as explanations of the inverse correlation between at-the-money option implied volatilities and index levels. If leverage is the force behind the scenes, the downward slope of the smiles for index and stock options should be about the same. If the wealth effect is predominant, the downward slope of the smile would be highest for index options and become less sloped the lower the ratio of a stock’s systematic variance to its total variance.

To investigate the empirical problems, we suggest two main tests. Our first test investigates relation (1), using options prices at the same time and with the same underlying and strike price, but with different times-to-expiration. Here we find out how well different option pricing models are capable of simultaneously explaining option prices of different times-to-expiration. For this, we deduce shorter-term option prices from longer-term option prices. The volatility smile for the longer-term options is assumed known, and the volatility smile for the shorter-term options is unknown. The problem of relation (1) is to fit alternative option pricing models to the longer-term option prices. We can then compare the model values with the observed market prices for the shorter-term options and calculate pricing errors. To help understand the source of remaining errors, we also conduct a related experiment. We assume in addition that we also know the at-themoney implied volatility of the shorter-term options.

The second test investigates relation (2), using option prices with the same underlying asset, expiration date, and strike price, but observed at different times. In this case, we use option valuation models to forecast future option prices conditional on the future underlying asset price.

We calibrate alternative models on current longer-term option prices. Then, we wait 10 and 30 days, observe the underlying asset price, and assess the errors in our forecasts. A related test extends the forecasting procedure by incorporating information from both current longer-term and current shorter-term option prices. Again, to decompose the source of any remaining errors, we also assume in addition that we know in advance the future at-the-money option price.

For all tests, we evaluate five kinds of option valuation models (nine models altogether). We compare deterministic models and stochastic models and naïve trader rules. Related empirical work is in Dumas, Fleming, and Whaley (1998), Bates (2000), and Bakshi, Cao, and Chen (1997).

The first paper investigates only different deterministic volatility models while the other two compare only different stochastic models.

The five categories of models are: first, mostly for reference, the Black-Scholes formula;

second, two naïve smile-based predictions that use today’s observed smile directly for prediction;

third, two versions of Cox’s (1996) constant elasticity of variance (CEV) formula; fourth, an implied binomial tree model; fifth, three parametric models that specify the stochastic process of the underlying, namely, displaced diffusion, jump diffusion, and stochastic volatility.