Testing Alternative Measure Changes in Nonparametric Pricing and Hedging of European Options

Abstract

Haley and Walker [Haley, M.R., & Walker, T. (2010). Journal of Futures Markets, 30, 983–1006] present the Euclidean and Empirical Likelihood nonparametric option pricing models as alternative tilts to Stutzer's [Stutzer, M. (1996). Journal of Finance, 51, 1633–1652] Canonical pricing method. We empirically test the comparative strengths of each of these methods using a large sample of traded options on the S&P100 Index. Furthermore, we explore an additional tilt based on Pearson's chi‐square, and derive and empirically test nonparametric delta hedges for each of these approaches. Differences in the pricing performance of the various tilts are a function of differences between the sample distribution and the real distribution of the underlying. When the sample distribution displays fatter (thinner) tails and/or higher (lower) volatility than the true distribution, the Euclidean (Pearson's chi‐square) model outperforms. Significantly, when these nonparametric methods utilize information contained in a small number of observed option prices they often outperform the implied volatility Black and Scholes [Black, F., & Scholes, M. (1973). Journal of Political Economy, 81, 637–654] model. These pricing performance differences do not translate into static and dynamic hedging performance differences. However, each of the nonparametric models induce an implied volatility smile and term structure that generally agree in form with the smile and term structure embedded in market prices. © 2013 Wiley Periodicals, Inc. Jrl Fut Mark 34:320–345, 2014

Non-parametric American option valuation using Cressie–Read divergences

“…We also find that the static and dynamic delta-hedging performance of these non-parametric schemes are comparable to those of the common parametric schemes. Similar to the findings relating to hedging of European-style options (Alcock and Smith, 2014;Smith, 2013), the differences in pricing performance between the various methods do not necessarily translate into differences in static and dynamic delta-hedging performance. We also find that this suite of non-parametric methods nicely captures the volatility-smile and term-structure of traded options.…”

“…For a small sample-size N , the bootstrapped sample return process R with maximum-entropy probabilities π  may not accurately represent the true asset price dynamics. If the sample is believed to have fatter tails and/or more outliers, then Cressie-Read members derived from α < 0 price European options more accurately, and vice versa (Alcock and Smith, 2014;Haley and Walker, 2010). When pricing European options, Haley and Walker (2010) and Alcock and Smith (2014) show that the pricing accuracy of the Kullback-Leibler divergence is improved upon by other members of the Cressie-Read family, including the Euclidean and Empirical Likelihood divergences.…”

“…In finite samples, where N < ∞, each member of the Cressie-Read family yields a different riskneutral measure in an incomplete market. For example, in the case of pricing European-style options the finite-sample Euclidean-based scheme ( = 2) α − outperforms all other methods whenever the historical sample distribution is heavier tailed than the real distribution of the underlying (Alcock and Smith, 2014;Haley and Walker, 2010).…”

“…Haley and Walker (2010) and Alcock and Smith (2014) find that the EL-divergence-based transformation generates the most accurate option values (compared to other members of the Cressie-Read family) when the sample asset price process correctly models the true dynamics, but not when the market dynamics are in a state of flux. Furthermore, Alcock and Smith (2014) demonstrate that these divergence functions generate option values that closely approximate the volatility smile and term structure observed in options markets.…”

“…The Cressie-Read family is a general class of divergence functions that includes many wellknown and commonly-used members, including the Kullback-Leibler (KL), Empirical Likelihood (EL), Euclidean (EU) and the Pearson's χ 2 (CHI) estimators (Bera and Bilias, 2002). Haley and Walker (2010) and Alcock and Smith (2014) find that the EL-divergence-based transformation generates the most accurate option values (compared to other members of the Cressie-Read family) when the sample asset price process correctly models the true dynamics, but not when the market dynamics are in a state of flux. Furthermore, Alcock and Smith (2014) demonstrate that these divergence functions generate option values that closely approximate the volatility smile and term structure observed in options markets.…”