Optimal No‐Arbitrage Bounds on S&P 500 Index Options and the Volatility Smile

This article shows that the volatility smile is not necessarily inconsistent with the Black–Scholes analysis. Specifically, when transaction costs are present, the absence of arbitrage opportunities does not dictate that there exists a unique price for an option. Rather, there exists a range of prices within which the option's price may fall and still be consistent with the Black–Scholes arbitrage pricing argument. This article uses a linear program (LP) cast in a binomial framework to determine the smallest possible range of prices for Standard & Poor's 500 Index options that are consistent with no arbitrage in the presence of transaction costs. The LP method employs dynamic trading in the underlying and risk‐free assets as well as fixed positions in other options that trade on the same underlying security. One‐way transaction‐cost levels on the index, inclusive of the bid–ask spread, would have to be below six basis points for deviations from Black–Scholes pricing to present an arbitrage opportunity. Monte Carlo simulations are employed to assess the hedging error induced with a 12‐period binomial model to approximate a continuous‐time geometric Brownian motion. Once the risk caused by the hedging error is accounted for, transaction costs have to be well below three basis points for the arbitrage opportunity to be profitable two times out of five. This analysis indicates that market prices that deviate from those given by a constant‐volatility option model, such as the Black–Scholes model, can be consistent with the absence of arbitrage in the presence of transaction costs. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:1151–1179, 2001     

Pricing FTSE 100 index options under stochastic volatility

The autoregressive conditional heteroscedasticity/generalized autoregressive conditional heteroscedasticity (ARCH/GARCH) literature and studies of implied volatility clearly show that volatility changes over time. This article investigates the improvement in the pricing of Financial Times‐Stock Exchange (FTSE) 100 index options when stochastic volatility is taken into account. The major tool for this analysis is Heston’s (1993) stochastic volatility option pricing formula, which allows for systematic volatility risk and arbitrary correlation between underlying returns and volatility. The results reveal significant evidence of stochastic volatility implicit in option prices, suggesting that this phenomenon is essential to improving the performance of the Black–Scholes model (Black & Scholes, 1973) for FTSE 100 index options. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:197–211, 2001     

A new simple square root option pricing model

This study derives a simple square root option pricing model using a general equilibrium approach in an economy where the representative agent has a generalized logarithmic utility function. Our option pricing formulae, like the Black–Scholes model, do not depend on the preference parameters of the utility function of the representative agent. Although the Black–Scholes model introduces limited liability in asset prices by assuming that the logarithm of the stock price has a normal distribution, our basic square root option pricing model introduces limited liability by assuming that the square root of the stock price has a normal distribution. The empirical tests on the S&P 500 index options market show that our model has smaller fitting errors than the Black–Scholes model, and that it generates volatility skews with similar shapes to those observed in the marketplace. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark     

Pricing options using implied trees: Evidence from FTSE‐100 options

Previously, few, if any, comparative tests of performance of Jackwerth's ( 1997 ) generalized binomial tree (GBT) and Derman and Kani ( 1994 ) implied volatility tree (IVT) models were done. In this paper, we propose five different weight functions in GBT and test them empirically compared to both the Black‐Scholes model and IVT. We use the daily settlement prices of FTSE‐100 index options from January to November 1999. With both American and European options traded on the FTSE‐100 index, we construct both GBT and IVT from European options and examine their performance in both the hedging of European option and the pricing of its American counterpart. IVT is found to produce least hedging errors and best results for American call options with earlier maturity than the maturity span of the implied trees. GBT appears to produce better results for American ATM put pricing for any maturity, and better in‐sample fit for options with maturity equal to the maturity span of the implied trees. Deltas calculated from IVT are consistently lower (higher) than Black‐Scholes deltas for both European and American calls (puts) in absolute term. The reverse holds true for GBT deltas. These empirical findings about the relative performance of GBT, IVT, and Standard Black‐Scholes models are important to practitioners as they indicate that different methods should be used for different applications, and some cautions should be exercised. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:601–626, 2002     

Empirical performance of alternative pricing models of currency options

This article examines the out‐of‐sample pricing performance and biases of the Heston’s stochastic volatility and modified Black‐Scholes option pricing models in valuing European currency call options written on British pound. The modified Black‐Scholes model with daily‐revised implied volatilities performs as well as the stochastic volatility model in the aggregate sample. Both models provide close and similar correspondence to actual prices for options trading near‐ or at‐the‐money. The prices generated from the stochastic volatility model are subject to fewer and weaker aggregate pricing biases than are the prices from the modified Black‐Scholes model. Thus, the stochastic volatility model may provide improved estimates of the measures of option price sensitivities to key option parameters that may lead to more effective hedging and speculative strategies using currency options. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:265–291, 2000     

Option pricing under extended normal distribution

This article proposes a closed pricing formula for European options when the return of the underlying asset follows extended normal distribution, that is, any different degrees of skewness and kurtosis relative to the normal distribution induced by the Black‐Scholes model. The moment restriction is suggested, so that the pricing model under any arbitrary distribution for an underlying asset must satisfy the arbitrage‐free condition. Numerical experiments and comparison of empirical performance of the proposed model with the Black‐Scholes, ad hoc Black‐Scholes, and Gram‐Charlier distribution models are carried out. In particular, an estimation of implied parameters such as standard deviation, skewness, and kurtosis of the return on the underlying asset from the market prices of the KOSPI 200 index options is made, and in‐sample and out‐of‐sample tests are performed. These results not only support the previous finding that the actual density of the underlying asset shows skewness to the left and high peaks, but also demonstrate that the present model has good explanatory power for option prices. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:845–871, 2005     

Does model fit matter for hedging? Evidence from FTSE 100 options

This study implements a variety of different calibration methods applied to the Heston model and examines their effect on the performance of standard and minimum‐variance hedging of vanilla options on the FTSE 100 index. Simple adjustments to the Black–Scholes–Merton model are used as a benchmark. Our empirical findings apply to delta, delta‐gamma, or delta‐vega hedging and they are robust to varying the option maturities and moneyness, and to different market regimes. On the methodological side, an efficient technique for simultaneous calibration to option price and implied volatility index data is introduced. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 32:609–638, 2012     

Black‐Scholes‐Merton revisited under stochastic dividend yields

European options are priced in a framework à la Black‐Scholes‐Merton, which is extended to incorporate stochastic dividend yield under a stochastic mean–reverting market price of risk. Explicit formulas are obtained for call and put prices and their Greek parameters. Some well‐known properties of the Black‐Scholes‐Merton formula fail to hold in this setting. For example, the delta of the call can be negative and even greater than one in absolute terms. Moreover, call prices can be a decreasing function of the underlying volatility although the latter is constant. Finally, and most importantly, option prices highly depend on the features of the market price of risk, which does not need to be specified at all in the standard Black‐Scholes‐Merton setting. The results are simulated in order to assess the economic impact of assuming that the dividend yield is deterministic when it is actually stochastic, as well as to assess the economic importance of the features of the market price of risk. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:703–732, 2006     

Canonical valuation and hedging of index options

Canonical valuation is a nonparametric method for valuing derivatives proposed by M. Stutzer (1996). Although the properties of canonical estimates of option price and hedge ratio have been studied in simulation settings, applications of the methodology to traded derivative data are rare. This study explores the practical usefulness of canonical valuation using a large sample of index options. The basic unconstrained canonical estimator fails to outperform the traditional Black–Scholes model; however, a constrained canonical estimator that incorporates a small amount of conditioning information produces dramatic reductions in mean pricing errors. Similarly, the canonical approach generates hedge ratios that result in superior hedging effectiveness compared to Black–Scholes‐based deltas. The results encourage further exploration and application of the canonical approach to pricing and hedging derivatives. © 2007 Wiley Periodicals, Inc. Jnl Fut Mark 27: 771–790, 2007     

Pricing and hedging S&P 500 index options with Hermite polynomial approximation: empirical tests of Madan and Milne's model

The universal use of the Black and Scholes option pricing model to value a wide range of option contracts partly accounts for the almost systematic use of Gaussian distributions in finance. Empirical studies, however, suggest that there is an information content beyond the second moment of the distribution that must be taken into consideration.This article applies a Hermite polynomial-based model developed by Madan and Milne (1994) to an investigation of S&P 500 index option prices from the CBOE when the distribution of the underlying index is unknown. The model enables us to incorporate the non-normal skewness and kurtosis effects empirically observed in option-implied distributions of index returns. Out-of-sample tests confirm that the model outperforms Black and Scholes in terms of pricing and hedging. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 735–758, 1999     

A modified static hedging method for continuous barrier options

This study modifies the static replication approach of Derman, E., Ergener, D., and Kani, I. (1995, DEK) to hedge continuous barrier options under the Black, F. and Scholes, M. (1973) model. In the DEK method, the value of the static replication portfolio, consisting of standard options with varying maturities, matches the zero value of the barrier option at n evenly spaced time points when the stock price equals the barrier. In contrast, our modified DEK method constructs a portfolio of standard options and binary options with varying maturities to match not only the zero value but also zero theta on the barrier. Our numerical results indicate that the modified DEK approach improves performance of static hedges significantly for an up‐and‐out call option under the BS model even if the bid–ask spreads are considered. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark     

Knock‐in American options

A knock‐in American option under a trigger clause is an option contract in which the option holder receives an American option conditional on the underlying stock price breaching a certain trigger level (also called barrier level). We present analytic valuation formulas for knock‐in American options under the Black‐Scholes pricing framework. The price formulas possess different analytic representations, depending on the relation between the trigger stock price level and the critical stock price of the underlying American option. We also performed numerical valuation of several knock‐in American options to illustrate the efficacy of the price formulas. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:179–192, 2004     

The performance of VIX option pricing models: Empirical evidence beyond simulation

We examine the pricing performance of VIX option models. Such models possess a wide‐range of underlying characteristics regarding the behavior of both the S&P500 index and the underlying VIX. Our tests employ three representative models for VIX options: Whaley ( 1993 ), Grunbichler and Longstaff ( 1996 ), Carr and Lee ( 2007 ), Lin and Chang ( 2009 ), who test four stochastic volatility models, as well as to previous simulation results of VIX option models. We find that no model has small pricing errors over the entire range of strike prices and times to expiration. In particular, out‐of‐the‐money VIX options are difficult to price, with Grunbichler and Longstaff's mean‐reverting model producing the smallest dollar errors in this category. Whaley's Black‐like option model produces the best results for in‐the‐money VIX options. However, the Whaley model does under/overprice out‐of‐the‐money call/put VIX options, which is opposite the behavior of stock index option pricing models. VIX options exhibit a volatility skew opposite the skew of index options. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark31:251–281, 2011     

Empirical tests of canonical nonparametric American option‐pricing methods

Alcock and Carmichael (2008, The Journal of Futures Markets, 28, 717–748) introduce a nonparametric method for pricing American‐style options, that is derived from the canonical valuation developed by Stutzer (1996, The Journal of Finance, 51, 1633–1652). Although the statistical properties of this nonparametric pricing methodology have been studied in a controlled simulation environment, no study has yet examined the empirical validity of this method. We introduce an extension to this method that incorporates information contained in a small number of observed option prices. We explore the applicability of both the original method and our extension using a large sample of OEX American index options traded on the S&P100 index. Although the Alcock and Carmichael method fails to outperform a traditional implied‐volatility‐based Black–Scholes valuation or a binomial tree approach, our extension generates significantly lower pricing errors and performs comparably well to the implied‐volatility Black–Scholes pricing, in particular for out‐of‐the‐money American put options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:509–532, 2010     

Option pricing with a non‐zero lower bound on stock price

Black, F. and Scholes, M. (1973) assume a geometric Brownian motion for stock prices and therefore a normal distribution for stock returns. In this article a simple alternative model to Black and Scholes (1973) is presented by assuming a non‐zero lower bound on stock prices. The proposed stock price dynamics simultaneously accommodate skewness and excess kurtosis in stock returns. The feasibility of the proposed model is assessed by simulation and maximum likelihood estimation of the return probability density. The proposed model is easily applicable to existing option pricing models and may provide improved precision in option pricing. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:775–794, 2005     

BLACK–SCHOLES REPRESENTATION FOR ASIAN OPTIONS

Asian options are securities with a payoff that depends on the average of the underlying stock price over a certain time interval. We identify three natural assets that appear in pricing of the Asian options, namely a stock S, a zero coupon bond B^T with maturity T, and an abstract asset A (an “average asset”) that pays off a weighted average of the stock price number of units of a dollar at time T. It turns out that each of these assets has its own martingale measure, allowing us to obtain Black–Scholes type formulas for the fixed strike and the floating strike Asian options. The model independent formulas are analogous to the Black–Scholes formula for the plain vanilla options; they are expressed in terms of probabilities under the corresponding martingale measures that the Asian option will end up in the money. Computation of these probabilities is relevant for hedging. In contrast to the plain vanilla options, the probabilities for the Asian options do not admit a simple closed form solution. However, we show that it is possible to obtain the numerical values in the geometric Brownian motion model efficiently, either by solving a partial differential equation numerically, or by computing the Laplace transform. Models with stochastic volatility or pure jump models can be also priced within the Black–Scholes framework for the Asian options.     

The Statistical Properties of the Black–Scholes Option Price

This paper investigates the statistical properties of the Black–Scholes option price, considered as a random variable. The option is conditioned on the current price and/or the estimated volatility of the underlying security. In both cases, some exact results for the distribution functions of the true option price and the predicted option price are derived. Extensions to puts and American contracts are considered. Numerical results are presented for option prices based on parameters appropriate for the FTSE 100 Index.     

The effect of multiple listings on the bid–ask spread in option markets: The case of Montreal Exchange

In this article, we examine the effect of multiple listings of options on their bid–ask spread, by comparing options contracts listed only on the Montreal Exchange with those interlisted on that exchange and on a U.S. exchange as well. Using a statistical procedure adapted to panel data and two models for the determination of the bid–ask spread, we find that the bid–ask spreads of Montreal options interlisted in U.S. markets are narrower than those of non‐interlisted options. That advantage tends to disappear, however, with an increase in option price and to increase with its volatility, but is not affected by the volume of transactions in the option market. The analysis also shows that interlisting may result in time lags in the convergence of quotes between Montreal and the U.S. markets. Moreover, our evidence shows that with interlisting, volume shifts to the option market where trading in the underlying security is concentrated, irrespective of the location where the option was first introduced. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:939–957, 2002     

Nonparametric American option pricing

A nonparametric method is introduced to accurately price American-style contingent claims. This method uses only historical stock price data, not option price data, to generate the American option price. The accuracy of this method is tested in a controlled experimental environment under both Black, F and Scholes, M (1973) and Heston, S (1993) assumptions, and an error-metric analysis is performed. These numerical experiments demonstrate that this method is an accurate and precise method of pricing American options under a variety of market conditions. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:717–748, 2008     

THE WIENER–HOPF TECHNIQUE AND DISCRETELY MONITORED PATH‐DEPENDENT OPTION PRICING

Fusai, Abrahams, and Sgarra (2006) employed the Wiener–Hopf technique to obtain an exact analytic expression for discretely monitored barrier option prices as the solution to the Black–Scholes partial differential equation. The present work reformulates this in the language of random walks and extends it to price a variety of other discretely monitored path‐dependent options. Analytic arguments familiar in the applied mathematics literature are used to obtain fluctuation identities. This includes casting the famous identities of Baxter and Spitzer in a form convenient to price barrier, first‐touch, and hindsight options. Analyzing random walks killed by two absorbing barriers with a modified Wiener–Hopf technique yields a novel formula for double‐barrier option prices. Continuum limits and continuity correction approximations are considered. Numerically, efficient results are obtained by implementing Padé approximation. A Gaussian Black–Scholes framework is used as a simple model to exemplify the techniques, but the analysis applies to Lévy processes generally.