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Canonical Valuation of Options in the Presence of Stochastic Volatility

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Canonical Valuation of Options in the Presence of Stochastic Volatility CANONICAL VALUATION OF OPTIONS IN THE PRESENCE OF STOCHASTIC VOLATILITY PHILIP GRAY* SCOTT NEWMAN Proposed by M. Stutzer (1996), canonical valuation is a new method for valuingderivative securities under the risk-neutral framework. It is non- parametric, simple to apply, and, unlike many alternative approaches, does not require any option data. Although canonical valuation has great poten- tial, its applicability in realistic scenarios has not yet been widely tested. This article documents the ability of canonical valuation to price derivatives in a number of settings. In a constant-volatility world, canonical estimates of option prices struggle to match a Black-Scholes estimate based on his- torical volatility. However, in a more realistic stochastic-volatility setting, canonical valuation outperforms the Black-Scholes model. As the volatility generating process becomes further removed from the constant-volatility world, the relative performance edge of canonical valuation is more evident. In general, the results are encouraging that canonical valuation is a useful technique for valuingderivatives. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:1–19, 2005 The authors are grateful for the comments and suggestions of Jamie Alcock, Stephen Gray, Philip Hoang, Egon Kalotay, an anonymous referee, participants at the 16th Australian Finance and Banking conference, and funding from a UQ Business School summer research grant. *Correspondence author, UQ Business School, The University of Queensland, St. Lucia 4072, Australia; e-mail: [email protected] Received November 2003; Accepted March 2004 I Philip Gray is an Associate Professor at UQ Business School at the University of Queensland in Brisbane, Australia. I Scott Newman is with the UQ Business School at the University of Queensland in Brisbane, Australia. The Journal of Futures Markets, Vol. 25, No. 1, 1–19 (2005) © 2005 Wiley Periodicals, Inc. Published online in Wiley InterScience (www.interscience.wiley.com). DOI:10.1002/fut.20140 TLFeBOOK 2 Gray and Newman INTRODUCTION Thirty years after the seminal work of Black and Scholes (1973) and Merton (1973), the benchmark in option pricingcontinues to be the widely applied Black-Scholes formula. However, while the strongpara- metric assumptions underlyingthe Black-Scholes model allow a simple, closed-form solution to the price of a European call option, empirical tests suggest that the assumptions are violated in practice. For example, rather than beingconstant, implied volatilities from observed option prices are systematically related to moneyness and maturity (see Derman & Kani, 1994; MacBeth & Merville, 1979; Rubinstein, 1985). There is also considerable evidence that stock returns are not normally distributed (see Jackwerth & Rubinstein, 1996; Kon, 1994). In light of these problems, alternative methods have been developed to price options. One such example is canonical valuation. Developed by Stutzer (1996), canonical valuation is a nonparametric technique for valuing derivatives. Unlike the Black-Scholes model, it makes no restric- tive assumptions about the underlying asset’s return generating process; rather, the historical distribution of returns on the underlying asset is used to predict the distribution of future stock prices. A maximum- entropy principle is employed to transform this real-world distribution into its risk-neutral counterpart, from which option prices follow easily using the standard risk-neutral approach. In addition to being relatively simple to implement, a major advantage of canonical valuation is that option price data are not required as input.1 Despite its potential, canonical valuation has only been examined in a handful of papers to date. Usinga simulated Black-Scholes world, Stutzer (1996) reports that the accuracy of canonical estimates of option prices is comparable to Black-Scholes estimates usinghistorical volatility. Foster and Whiteman (1999) modify canonical valuation to incorporate a more sophisticated Bayesian predictive model. In an application to the soybean futures options market, the modified model performs well with reference to both the simple canonical valuation model usinghistorical returns and Black-Scholes. Finally, Stutzer and Chowdhury (1999) apply canonical valuation to bond futures options, with results also suggesting that the method performs well. This article explores the potential usefulness of canonical valuation in two directions. First, the analysis of Stutzer (1996) is extended to document the accuracy of canonical valuation across various levels of 1Although no option prices are strictly required, such data are easily incorporated into canonical valuation if desired. TLFeBOOK Canonical Valuation of Options 3 maturity and moneyness. Working in a constant-volatility Black-Scholes world, stock prices are simulated under a geometric Brownian motion so that true option prices are known. Prices are then estimated using both canonical (CAN) and historical-volatility-based Black-Scholes (HBS) methods, and the properties of pricing errors are documented. We also examine the potential to reduce pricing errors by incorporating a token amount of option data. The canonical estimator is modified such that the risk-neutral density is estimated subject to the constraint that a single at-the-money option is correctly priced. The performance of the constrained canonical estimator (CON) is compared to CAN and HBS estimators. The sensitivity of all findings to the number of returns used to estimate the risk-neutral density is also examined. Because there is considerable evidence that volatility is noncon- stant, Stutzer (1996) foreshadows that the usefulness of canonical valu- ation will be most apparent when we move beyond the Black-Scholes world. This issue has not previously been examined. The second contri- bution of this article, therefore, is to evaluate the performance of canon- ical valuation in the presence of stochastic volatility. Heston’s (1993) model provides an ideal environment to test this conjecture as it admits a closed-form solution to the price of a call option under stochastic volatility. Stock prices are simulated under Heston’s stochastic volatility model, and the performance of CAN and HBS estimates is assessed rel- ative to the true price. Simulation results are reported for a range of moneyness and time to maturity. Finally, the sensitivity of results to key parameters in the stochastic volatility model is examined. There are several key findings in this article. Not surprisingly, the Black-Scholes model outperforms canonical valuation in a constant- volatility world. HBS estimates are less biased than standard canonical estimates, and this performance edge persists even as sample size increases. The practice of incorporatingminimal option data into the estimation of the risk-neutral density under canonical valuation produces significant improvements in pricing performance. The con- strained canonical estimator arguably outperforms HBS estimates, particularly for deep out-of-the-money options which can be difficult to price. Moving to simulations assuming stochastic volatility, the magnitude of pricing errors under both CAN and HBS estimators rises markedly highlighting the difficulty in pricing real-world options. The out-of-the- money superiority of the HBS estimator over CAN documented under constant volatility disappears. Plain-vanilla canonical estimation produces less biased estimates for out-of-the-money options, and the TLFeBOOK 4 Gray and Newman constrained canonical estimator performs admirably regardless of moneyness. Sensitivity analysis identifies the volatility of the volatility dynamics as the key parameter impacting on the success of alternative pricing methods. The remainder of the article is structured as follows. The second section, An Overview of Canonical Valuation, reviews other popular non- parametric approaches to pricing derivatives, outlines the potential advantages of canonical valuation, and provides a brief overview of the canonical valuation approach. The next two sections, Canonical Valuation in a Black-Scholes World and Canonical Valuation in a Stochastic Volatility World, conduct simulation experiments to docu- ment the properties of alternative pricing methods in constant-volatility and stochastic-volatility worlds, respectively. The last section is the Conclusion. AN OVERVIEW OF CANONICAL VALUATION A risk-neutral approach is often adopted to price derivatives. The pri- mary task is to estimate the risk-neutral probability distribution of the underlying asset, from which the expected payoff to the derivative can be calculated. There are, however, different ways to estimate the required risk-neutral density. Black and Scholes (1973) typify the parametric approach by specifying the dynamics of the underlying asset from which the risk-neutral density (lognormal in this case) is derived. In contrast, Rubinstein (1994), Hutchinson, Lo, and Poggio (1994), Jackwerth and Rubinstein (1996), and Aït-Sahalia and Lo (1998) propose nonparamet- ric methods of estimating the risk-neutral density. Although they make fewer restrictive assumptions over the data- generating process, these nonparametric methods require as input large quantities of market option prices across a range of strike prices. An advantage of canonical valuation is that option prices are unnecessary; the method can be implemented merely using a time-series of data for the underlying asset. Note also that the ability to price options without using observed market
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