The performance of traders' rules in options market

This study focuses on the usefulness of the traders' rules to predict future implied volatilities for pricing and hedging KOSPI 200 index options. There are two versions of this approach. In the “relative smile” approach, the implied volatility skew is treated as a fixed function of moneyness. In the “absolute smile” approach, the implied volatility skew is treated as a fixed function of the strike price. It is found that the “absolute smile” approach shows better performance than Black, F. and Scholes, L. ( 1973 ) model and the stochastic volatility model for both pricing and hedging options. Consistent with Jackwerth, J. C. and Rubinstein, M. (2001) and Li, M. and Pearson, N. D. (2007), the traders' rules dominate mathematically more sophisticated model, that is, the stochastic volatility model. The traders' rules can be an alternative to the sophisticated and complicated models for pricing and hedging options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:999–1020, 2009     

A NEW LOOK AT SHORT‐TERM IMPLIED VOLATILITY IN ASSET PRICE MODELS WITH JUMPS

We analyze the behavior of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalization of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal–Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short‐end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model‐independent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on short‐maturity option prices.     

THE NORMALIZING TRANSFORMATION OF THE IMPLIED VOLATILITY SMILE

We study specific nonlinear transformations of the Black–Scholes implied volatility to show remarkable properties of the volatility surface. No arbitrage bounds on the implied volatility skew are given. Pricing formulas for European payoffs are given in terms of the implied volatility smile.     

Why Doesn't the Black‐‐Scholes Model Fit Japanese Warrants and Convertible Bonds?

In this paper, we investigate the systematic departures of traded prices of Japanese equity warrants and convertible bonds from their theoretical Black–Scholes values. We briefly consider transactions costs and the dilution adjustment as potential explanations of the discrepancy. However, our major focus is on shifts in volatility of the prices of the underlying stocks as a function of the stock price changes; such shifts are not taken into account in the Black–Scholes values. We assume that the pseudo‐probability distributions of prices of stocks of cross‐sections of companies which are roughly similar in size are identical. This simple assumption, which can be generalized, enables us to infer the implied probability distribution and binomial tree for stock price changes using the Derman and Kani (1994), Rubinstein (1994) and Shimko (1993) approach. The cross‐section of warrant prices implies an inverse volatility smile and a positively skewed probability density for stock prices. Rubinstein's identifying assumptions generate an implied binomial tree in which the relative size of up‐steps and down‐steps, and thus volatility, changes systematically as stock prices change. We briefly consider potential explanations for the implied behaviour, and for the difference in the smile pattern between index options and the warrants and convertibles.     

Fitting and testing for the implied volatility curve using parametric models

Numerous issues have arisen over the past few decades relating to the implied volatility smile in the options market; however, the extant literature reveals that relatively little effort has thus far been placed into comparing the various implied volatility models, essentially as a result of the lack of any theoretical foundation on which to base such comparative analysis. In this study, we use a comprehensive options database and employ methods of combining the various hypothesis tests to compare the different implied volatility models. To the best of our knowledge, this is the first study of its kind to address this issue using combination tests. Our empirical results reveal that the linear piecewise model is the most appropriate model for capturing the implied volatility smile, with additional robustness checks confirming the validity of this finding.     

Pricing American Options Fitting the Smile

This paper is a compendium of results—theoretical and computational—from a series of recent papers developing a new American option valuation technique based on linear programming (LP). Some further computational results are included for completeness. A proof of the basic analytical theorem is given, as is the analysis needed to solve the inverse problem of determining local (one‐factor) volatility from market data. The ideas behind a fast accurate revised simplex method, whose performance is linear in time and space discretizations, are described and the practicalities of fitting the volatility smile are discussed. Numerical results are presented which show the LP valuation technique to be extremely fast—lattice speed with PDE accuracy. American options valued in the paper range from vanilla, through exotic with constant volatility, to exotic options fitting the volatility smile.     

On Feedback Effects from Hedging Derivatives

This paper proposes a new explanation for the smile and skewness effects in implied volatilities. Starting from a microeconomic equilibrium approach, we develop a diffusion model for stock prices explicitly incorporating the technical demand induced by hedging strategies. This leads to a stochastic volatility endogenously determined by agents' trading behavior. By using numerical methods for stochastic differential equations, we quantitatively substantiate the idea that option price distortions can be induced by feedback effects from hedging strategies.     

SERIES EXPANSION OF THE SABR JOINT DENSITY

Under the SABR stochastic volatility model, pricing and hedging contracts that are sensitive to forward smile risk (e.g., forward starting options, barrier options) require the joint transition density. In this paper, we address this problem by providing closed‐form representations, asymptotically, of the joint transition density. Specifically, we construct an expansion of the joint density through a hierarchy of parabolic equations after applying total volatility‐of‐volatility scaling and a near‐Gaussian coordinate transformation. We then establish an existence result to characterize the truncation error and provide explicit joint density formulas for the first three orders. Our approach inherits the same spirit of a small total volatility‐of‐volatility assumption as in the original SABR analysis. Our results for the joint transition density serve as a basis for managing forward smile risk. Through numerical experiments, we illustrate the accuracy of our expansion in terms of joint density, marginal density, probability mass, and implied volatilities for European call options.     

OPTION HEDGING AND IMPLIED VOLATILITIES IN A STOCHASTIC VOLATILITY MODEL1

In the stochastic volatility framework of Hull and White (1987), we characterize the so-called Black and Scholes implied volatility as a function of two arguments the ratio of the strike to the underlying asset price and the instantaneous value of the volatility By studying the variation m the first argument, we show that the usual hedging methods, through the Black and Scholes model, lead to an underhedged (resp. overhedged) position for in-the-money (resp out-of the-money) options, and a perfect partial hedged position for at the-money options These results are shown to be closely related to the smile effect, which is proved to be a natural consequence of the stochastic volatility feature the deterministic dependence of the implied volatility on the underlying volatility process suggests the use of implied volatility data for the estimation of the parameters of interest A statistical procedure of filtering (of the latent volatility process) and estimation (of its parameters) is shown to be strongly consistent and asymptotically normal.     

Distributions implied by American currency futures options: A ghost's smile?

A new and easily applicable method for estimating risk‐neutral distributions (RND) implied by American futures options is proposed. It amounts to inverting the Barone‐Adesi and Whaley method (BAW method) to get the BAW implied volatility smile. Extensive empirical tests show that the BAW smile is equivalent to the volatility smile implied by corresponding European options. Therefore, the procedure leads to a legitimate RND estimation method. Further, the investigation of the currency options traded on the Chicago Mercantile Exchange and OTC markets in parallel provides us with insights on the structure and interaction of the two markets. Unequally distributed liquidity in the OTC market seems to lead to price distortions and an ensuing interesting “ghost‐like” shape of the RND density implied by CME options. Finally, using the empirical results, we propose a parsimonious generalization of the existing methods for estimating volatility smiles from OTC options. A single free parameter significantly improves the fit. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:147–178, 2004     

The Quanto Adjustment and the Smile

European quanto derivatives are usually priced using the well‐known quanto adjustment corresponding to the forward of the quantoed asset under the assumptions of the Black–Scholes model. In this article, I present the quanto adjustment corresponding to the local volatility model that allows pricing quanto derivatives consistently with the observed market equity skew and exchange rate smile. I then examine the model risk arising in the standard quanto adjustment by fitting the local volatility model to market data and then comparing the prices of European quanto euro derivatives on the Nikkei 225 index with those generated by the standard quanto adjustment. The results show that the standard quanto adjustment can be subject to significant pricing errors when compared with the local volatility model. I also compare the pricing performance of the local volatility model with a multivariate stochastic volatility model. The results show that when the correlation between the instantaneous variances associated with the underlying asset and the exchange rate is close to one, as it is the case when we consider historical data, there is little evidence of model risk for the local volatility model in the pricing of European quanto euro derivatives on the Nikkei 225 index. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 32:877–908, 2012     

Smiling less at LIFFE

This study investigates the structure of the implied volatility smile, using the prices of equity options traded on the LIFFE. First, the slope of the implied volatility curve is significantly negative for both individual stocks and index options, and the slope is less negative for longer‐term options. The implied volatility skew can be described by risk‐neutral skewness and kurtosis, with the former having the first‐order effect. Moreover, the implied volatility skew for individual stock options is less severe than for index options. Finally, the relationship between the real and risk‐neutral moments implied in option prices is significant. The results indicate that, for equity options traded on the LIFFE, the slope of the implied volatility skew is flatter than that on the Chicago Board of Exchange (CBOE). © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:57–81, 2008     

Capped equity swaps under the double‐jump stochastic volatility model with stochastic interest rates

This study proposes a double‐jump stochastic volatility model with stochastic interest rates to price capped equity swaps and other multi‐period derivative securities. Closed‐form solutions for capped equity swaps with a fixed or variable notional principle are derived. In addition, numerical examples are employed to analyze comparative statics properties, counterparty risks, and the dynamics of the forward smile. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:340–370, 2011     

A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES

We propose a model which can be jointly calibrated to the corporate bond term structure and equity option volatility surface of the same company. Our purpose is to obtain explicit bond and equity option pricing formulas that can be calibrated to find a risk neutral model that matches a set of observed market prices. This risk neutral model can then be used to price more exotic, illiquid, or over‐the‐counter derivatives. We observe that our model matches the equity option implied volatility surface well since we properly account for the default risk in the implied volatility surface. We demonstrate the importance of accounting for the default risk and stochastic interest rate in equity option pricing by comparing our results to Fouque et al., which only accounts for stochastic volatility.     

Existence of a calibrated regime switching local volatility model

By Gyöngy's theorem, a local and stochastic volatility model is calibrated to the market prices of all European call options with positive maturities and strikes if its local volatility (LV) function is equal to the ratio of the Dupire LV function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a stochastic differential equation (SDE) nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented in Guyon and Henry‐Labordère [Risk Magazine, 25, 92–97], provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no global existence result is available for the SDE nonlinear in the sense of McKean. When the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level, we prove existence of a solution to the associated Fokker–Planck equation under the condition that the range of the squared stochastic volatility factor is not too large. We then deduce existence to the calibrated model by extending the results in Figalli [Journal of Functional Analysis, 254(1), 109–153].     

The quality of volatility traded on the over‐the‐counter currency market: A multiple horizons study

Previous studies of the quality of market‐forecasted volatility have used the volatility that is implied by exchange‐traded option prices. The use of implied volatility in estimating the market view of future volatility has suffered from variable measurement errors, such as the non‐synchronization of option and underlying asset prices, the expiration‐day effect, and the volatility smile effect. This study circumvents these problems by using the quoted implied volatility from the over‐the‐counter (OTC) currency option market, in which traders quote prices in terms of volatility. Furthermore, the OTC currency options have daily quotes for standard maturities, which allows the study to look at the market's ability to forecast future volatility for different horizons. The study finds that quoted implied volatility subsumes the information content of historically based forecasts at shorter horizons, and the former is as good as the latter at longer horizons. These results are consistent with the argument that measurement errors have a substantial effect on the implied volatility estimator and the quality of the inferences that are based on it. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:261–285, 2003     

MOMENT EXPLOSIONS AND LONG‐TERM BEHAVIOR OF AFFINE STOCHASTIC VOLATILITY MODELS

We consider a class of asset pricing models, where the risk‐neutral joint process of log‐price and its stochastic variance is an affine process in the sense of Duffie, Filipovic, and Schachermayer. First we obtain conditions for the price process to be conservative and a martingale. Then we present some results on the long‐term behavior of the model, including an expression for the invariant distribution of the stochastic variance process. We study moment explosions of the price process, and provide explicit expressions for the time at which a moment of given order becomes infinite. We discuss applications of these results, in particular to the asymptotics of the implied volatility smile, and conclude with some calculations for the Heston model, a model of Bates and the Barndorff‐Nielsen–Shephard model.     

Regime‐dependent smile‐adjusted delta hedging

We introduce several regime‐dependent smile‐adjusted deltas and compare their efficiency with the smile‐adjusted deltas that are popular with option traders. Using years of daily option prices, out‐of‐sample hedging performance tests for options of all moneyness and maturities and daily, weekly, or fortnightly rebalancing show that even the simplest regime‐dependent smile‐adjustment consistently outperforms implied BSM delta hedging and local volatility and minimum variance smile‐adjustments. Markov‐switching deltas offer the best performance, with delta‐hedging errors often half the size of implied BSM hedging errors. During volatile markets risk reduction from regime‐dependent delta hedging is much greater than during tranquil periods.     

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     

Options on bond futures: Isolating the risk premium

The introduction of unspanned sources of risk (and frictions) implies that option prices include a risk premium. Prima facie evidence of the existence of risk premia in option prices is contained in the implied volatility smile patterns reported in the literature. This article isolates the risk premium (defined as the simple difference between estimated and observed option prices) on options on U.K. Gilts, German Bunds, and U.S. Treasury bond futures using models that include price jumps and stochastic volatility. This study finds that single and multi‐factor stochastic volatility models with jumps may explain the empirical regularities observed in bond futures. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:169–215, 2003