Pricing and hedging long-term options

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Abstract

Do long-term and short-term options contain differential information? If so, can long-term options better differentiate among alternative models? To answer these questions, we first demonstrate analytically that differences among alternative models usually may not surface when applied to short-term options, but do so when applied to long-term contracts. Using S&P 500 options and LEAPS, we find that short- and long-term contracts indeed contain different information. While the data suggest little gains from modeling stochastic interest rates or random jumps (beyond stochastic volatility) for pricing LEAPS, incorporating stochastic interest rates can nonetheless enhance hedging performance in certain cases involving long-term contracts.

Introduction

Option pricing has played a central role in the general theory of asset pricing. Its importance comes about because of the derivative nature of option contracts, that is, the value of an option is almost completely derived from, and hence closely tied to, the value of the underlying asset. Not surprisingly, option pricing has been proven to be the best place for exemplifying the power of the much celebrated arbitrage valuation approach. From an application perspective, however, option pricing formulas based on the arbitrage approach have not performed well empirically. Take the prominent Black and Scholes (1973) model as an example. When applied to European-style options, it produces pricing errors that are related to both moneyness and maturity in a U-shaped manner. Thus, the ‘implied volatility smiles’. The unsatisfactory performance by the Black–Scholes has led to a search for better alternatives that extend the classic model in one, or a combination, of three directions: (i) to allow for stochastic volatility; (ii) to allow for stochastic interest rates; and (iii) to allow for random jumps to occur in the underlying price process.1 Each of the alternatives in principle offers some flexibility to correct for the biases of the Black–Scholes. For example, the stochastic-volatility (SV) models rely on the correlation coefficient between volatility and underlying price changes to internalize the level of skewness, and the variation-of-volatility coefficient to generate a desired kurtosis level, necessary to correct the volatility smiles. But, since existing SV models typically let volatility follow a diffusion process, the extent to which high levels of kurtosis in the return distribution can be internalized is limited. This points out a special role to be played by random jumps in the modeled price process. Thus, one should expect an option pricing model allowing for stochastic volatility and jumps (SVJ) to further enhance performance. In addition, even casual empiricism suggests that modeling stochastic interest rates in any pricing formula should be of practical significance as it ensures proper discounting of future payoffs. A model with stochastic volatility and stochastic interest rates (SVSI) should also have promise to further improve pricing and hedging performance.

While each generalization beyond the Black–Scholes may be sound and justifiably appealing on normative grounds, given the application-oriented nature of the problem at hand it is ultimately an empirical issue whether a given generalization and its consequential model complication are justified by the additional performance benefits (if any at all). Motivated by this, Bakshi et al. (1997) conduct a comprehensive empirical study on the relative performance of the above alternative models, using regular S&P 500 index option prices. Their conclusions can be summarized as follows. First, while the SV model typically reduces the Black–Scholes pricing errors by about a half, adding jumps does not further improve the SV model's pricing performance, except for extremely short-term options; Neither does incorporating stochastic interest rates enhance the SV's pricing performance. Second, when hedging errors are used as a performance benchmark, the SV model still does better than the Black–Scholes, but the SVSI and the SVJ models do not show any improvement beyond the SV. These findings are somewhat surprising given that one would expect incorporating random jumps or stochastic interest rates to further enhance the performance.

Given that the options used in most existing studies are generally short-term (typically with less than one year to expiration), the purpose of this paper is to address two related questions. First, do long-term options contain different information than short-term options? Second, if so, can long-term options better differentiate among the Black–Scholes and its alternatives? Answering such questions is important not only because it will help better appreciate and learn more about existing parametric option pricing models, but also because it will enhance our understanding of long-term contracts in general. In recent years, such contracts as equity LEAPS (Long-term Equity Anticipation Securities) have attracted increasingly more attention in the investment community. Yet, at the same time the academic literature has paid, at most, limited attention to issues related to long-term contracts (see Ross, 1996, for a treatment on long-term versus short-term futures commitments). This paper thus serves to fill in this gap. Our study is also timely and feasible as closed-form option pricing formulas have recently become available even for the general cases. On the data front, we now have high-quality intradaily quote and transaction data available for such popular contracts as the S&P 500 LEAPS. Unlike the regular S&P 500 options, these LEAPS have up to three years to expiration and are hence ideal for the purpose of this study.

We begin our quest by demonstrating analytically that long-term options can distinguish among alternative models more dramatically than short-term options. Specifically, we first examine whether the Black–Scholes, the SV, the SVSI, and the SVJ models yield different hedge ratios, or option deltas, for a given option contract. When implemented using estimated parameters and implied volatility values, all the models assuming stochastic volatility produce stock price option deltas that are drastically different from those based on the Black–Scholes. This is true regardless of option maturity and whether it is a low-volatility day, an average-volatility day, or a high-volatility day. But, when the SV, the SVSI, and the SVJ are compared to each other, the hedge ratios for a given option are either similar or significantly different, depending on whether the option is short term or long term. For a 45-d put, for instance, the three models yield virtually the same hedge ratios when the option is at the money, and are thus indistinguishable from each other. For a two-year put, however, the model-specific hedge ratios are generally far apart across the three models, even for a low-volatility day.

In the theory of asset pricing it is well understood that each parametric pricing model is associated with an Arrow–Debreu state-price density (SPD). This SPD function completely embodies the pricing structure implicit in the pricing model (see, e.g., Ait-Sahalia and Lo, 1998). Therefore, another way to examine alternative option pricing models is to compare their implicit SPDs, as their differences in pricing performance must come from differences in their SPDs. Again, when implemented using estimated parameters and implied volatilities, the SPDs of the SV, the SVSI, and the SVJ differ significantly from that of the Black–Scholes (regardless of time horizon): the former all assign more risk-neutral probability mass to the far lower tail and less mass to the upper tail of the underlying asset's return distribution, which effectively corrects and flattens out the volatility smiles associated with the Black–Scholes. For short-term options, the SPDs of the SV, the SVSI and the SVJ models coincide almost everywhere, except in the far left and far right tails. For long-term options, however, the difference among the SPDs of these models becomes especially pronounced when parameters implied by option prices taken from a relatively volatile day are used as input. Thus, long-term options should distinguish among alternative models more effectively than short-term options.

In the empirical exercise, we apply the method of simulated moments (MSM) to estimate each model's structural parameters out of the following considerations. First, the unobservability of the stock volatility process precludes the estimation by maximum likelihood. The unavailability of the moments of option prices in closed form also rules out the use of the generalized method of moments (GMM). With the MSM, on the other hand, we can jointly simulate the sample paths for the stock price and its return volatility, to construct a time series of simulated option prices of different strikes and maturities. The structural parameters estimated via the MSM will then reflect both the cross-sectional and the time-series information contained in option prices.

Empirically, we find that the S&P 500 LEAPS and regular options do provide distinct information. First, on a typical day, the two sets of options imply different volatility values (for any given option pricing model). Second, the implied-volatility time-series implied by regular options follows a drastically different path than that by LEAPS. Finally, the LEAPS-implied volatility exhibits a much higher level of long memory than the short term options implied, which suggests that volatility innovations will persist relatively longer (also see the recent work by Bollerslev and Mikkelsen 1996, Bollerslev and Mikkelsen 1999).

In terms of out-of-sample pricing, the SVJ model performs the best among the four models in pricing short-term puts. In pricing medium-term options the SVSI model does better than the SV in certain categories, while the SV performs better in pricing other moneyness-maturity puts. In pricing long-term puts, however, the SV model performs the best. Overall, even for pricing long-term options, adding the stochastic interest rates feature does not lead to consistent improvement in pricing errors. Overall, all four models are still misspecified statistically. For example, they each have moneyness- and maturity-related biases, though to varying degrees.

For the hedging exercise, we divide the discussion into two parts. In the first, the goal is to evaluate the relative effectiveness of (i) short-, medium- and long-term options (as hedging instruments) in hedging the underlying stock portfolio and (ii) the alternative option models in devising the desired hedge. The main results from this part can be summarized as follows. First, irrespective of the option model used, deep in-the-money LEAPS puts yield the lowest hedging errors on average, but short-term deep in-the-money puts generate the most stable hedging errors over time. Second, among the four option models, the BS-based hedge is always the least effective, regardless of the hedging instrument. The SV and the SVJ models lead to similar hedging errors, and both perform better than the SVSI.

In the second part of our hedging exercise, we let a LEAPS put be the hedging target and evaluate the relative effectiveness of (i) the underlying asset, (ii) a short-term put, and (iii) a medium-term put, as the hedging instrument. In a common practice, users and underwriters of long-term contracts often have no other choice but rely on exchange-traded, relatively short-term contracts to hedge their long-term commitments. At least, such short-term contracts have high liquidity and relatively low trading costs. Therefore, it is important to address the question of how effectively can short-term contracts hedge their long-term counterparts. On the other hand, this exercise also allows us to examine the relative hedging performance of the alternative models. The overall conclusion from this part is that medium-term options are generally the best instrument for hedging LEAPS, partly because they are more similar to the hedging target than either the underlying asset or a short-term option. Next, between the underlying asset and the short-term option as a hedging instrument, the former dominates the latter in hedging out-of-the-money LEAPS puts. Short-term contracts are good instruments only for hedging in-the-money LEAPS. In terms of model choice, the SVSI generally dominates the other models. Therefore, at least for devising hedges of long-term options, modeling stochastic interest rates does help improve empirical performance.

The paper is organized as follows. Section 2 develops an option formula that takes into account stochastic volatility, stochastic interest rate and random jumps. A description of the regular and LEAPS S&P 500 option data is provided in Section 3. Section 4 discusses the implementation of each option model and the MSM estimation of the structural parameters. Section 5 contrasts the information in short-term versus long-term options. The out-of-sample pricing exercise is conducted in Section 6. Section 7 addresses issues related to hedging the underlying stock portfolio, and Section 8 evaluates the relative effectiveness of the underlying asset, short-term and medium-term options in hedging LEAPS. Concluding remarks are offered in Section 9. Proofs to all pricing formulas and hedging strategies can be found in the Appendix.

Section snippets

Valuation of European options

In this section we derive a closed-form option pricing model that incorporates stochastic volatility, stochastic interest rates, and random jumps. The model is sufficiently general to include as special cases all the models which we investigate in the empirical sections. As in Bakshi et al. (1997), we take a risk-neutral probability measure as given and specify from the outset risk-neutral dynamics for the spot interest rate, the spot stock price, and the stock return volatility.2

The S&P 500 options and LEAPS

Two sets of option contracts on the S&P 500 index are used in our empirical exercise.

  • Regular S&P 500 index options (SPX). These options have up to one year to expiration.

  • S&P 500 Long-term Equity Anticipation Securities (LEAPS), which usually expire two to three years from the date of listing.

Both types of option are European in nature and share the same trading hours and the same settlement arrangements. They are both listed on the Chicago Board Options Exchange and hence subject to the same

Estimating structural parameters and spot volatility

In implementing each candidate model, we estimate its relevant structural parameters and the spot volatility in two separate steps. First, we estimate the structural parameters by using the method of simulated moments (MSM) (e.g., Duffie and Singleton, 1993; Gouriéroux and Monfort, 1996).8 The MSM is chosen over the maximum likelihood method or the GMM because (i) volatility is unobservable, which hinders the use of

Differential information in regular options versus LEAPS

The purpose of this section is to investigate the differential information embedded in short-term options versus LEAPS. We start with Table 3, which lists the average daily BS implied volatility across moneyness and maturity. Two patterns can be observed in the table. First, for short-term puts the implied volatility is U-shaped, whereas for medium- and long-term puts the implied volatility is declining as the put goes from OTM to ITM. Among the three maturity groups, the LEAPS’ implied

Out-of-sample pricing of regular and LEAPS puts

The preceding evidence suggests that each generalization to the BS model helps to improve the in-sample fit of option prices. Still, the question is: how large is the improvement in terms of out-of-sample pricing and hedging? Note that any model with good in-sample fit does not necessarily perform well out of sample. Our first step in answering this question is to examine the relative magnitude of out-of-sample option pricing errors. As before, our emphasis is not only on investigating the

Hedging the underlying asset

Having studied the out-of-sample pricing performance, we now examine each model's hedging performance as hedging reflects a model's dynamic fit of option prices. For our first exercise, we assume that a manager wants to hedge a long position in the underlying asset (the S&P 500). In choosing an instrument for such a hedge, the manager can select from a large set of put options: short term, medium term, and long term. The goal in this section is to examine (i) the relative effectiveness of

Hedging long-term options

As noted before, most exchange-traded derivatives have, even to this date, relatively short terms to expiration. Commitments taken over the counter, on the other hand, are often long term. As made clear by Ross (1996) using the Metallgesellschaft's oil-contracts case, users and underwriters of illiquid long-term contracts usually have no other choice but rely on the underlying asset or exchange-traded short-term contracts to hedge their commitments. At least, these short-term contracts are

Conclusions

This paper has studied the differential information in, and the pricing and hedging of, short-term versus long-term equity options, by comparing four alternative option pricing models. Theoretically, we have shown that long-term options should be able to differentiate the alternative models more effectively than short-term options. This has been illustrated by the differences among the option hedge ratios and the state-price densities of the models. Empirically, short-term and long-term options

Acknowledgements

For helpful comments and suggestions, we would like to thank Kerry Back, David Bates, Steve Buser, Phil Dybvig, Stephen Figlewski, Gerald Gay, Jason Greene, Frank Hatheway, Eric Chang, Steve Heston, Bill Kracaw, David Nachman, Steve Smith, René Stulz, Guofu Zhou, and especially Eric Ghysels and the anonymous referee. The paper has also benefited from the comments of seminar participants at the Federal Reserve Board, Georgia State University, Ohio State University and Washington University at

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