Analytic approximation formulae for pricing forward‐starting Asian options

In this article we first identify a missing term in the Bouaziz, Briys, and Crouhy ( 1994 ) pricing formula for forward‐starting Asian options and derive the correct one. First, illustrate in certain cases that the missing term in their pricing formula could induce large pricing errors or unreasonable option prices. Second, we derive new analytic approximation formulae for valuing forward‐starting Asian options by adding the second‐order term in the Taylor series. We show that our formulae can accurately value forward‐starting Asian options with a large underlying asset's volatility or a longer time window for the average of the underlying asset prices, whereas the pricing errors for these options with the previously mentioned formula could be large. Third, we derive the hedge ratios for these options and compare their properties with those of plain vanilla options. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:487–516, 2003     

Recursive Formula for Arithmetic Asian Option Prices

I derive a recursive formula for arithmetic Asian option prices with finite observation times in semimartingale models. The method is based on the relationship between the risk‐neutral expectation of the quadratic variation of the return process and European option prices. The computation of arithmetic Asian option prices is straightforward whenever European option prices are available. Applications with numerical results under the Black–Scholes framework and the exponential Lévy model are proposed. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 34:220–234, 2014     

Pricing of moving‐average‐type options with applications

Moving‐average‐type options are complex path‐dependent derivatives whose payoff depends on the moving average of stock prices. This article concentrates on two such options traded in practice: the moving‐average‐lookback option and the moving‐average‐reset option. Both options were issued in Taiwan in 1999, for example. The moving‐average‐lookback option is an option struck at the minimum moving average of the underlying asset's prices. This article presents efficient algorithms for pricing geometric and arithmetic moving‐average‐lookback options. Monte Carlo simulation confirmed that our algorithms converge quickly to the option value. The price difference between geometric averaging and arithmetic averaging is small. Because it takes much less time to price the geometric‐moving‐average version, it serves as a practical approximation to the arithmetic moving‐average version. When applied to the moving‐average‐lookback options traded on Taiwan's stock exchange, our algorithm gave almost the exact issue prices. The numerical delta and gamma of the options revealed subtle behavior and had implications for hedging. The moving‐average‐reset option was struck at a series of decreasing contract‐specified prices on the basis of moving averages. Similar results were obtained for such options with the same methodology. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:415–440, 2003     

ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi‐ or single‐asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi‐asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single‐asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi‐asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi‐asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi‐asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.     

EXPLICIT IMPLIED VOLATILITIES FOR MULTIFACTOR LOCAL‐STOCHASTIC VOLATILITY MODELS

We consider an asset whose risk‐neutral dynamics are described by a general class of local‐stochastic volatility models and derive a family of asymptotic expansions for European‐style option prices and implied volatilities. We also establish rigorous error estimates for these quantities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under four different model dynamics: constant elasticity of variance local volatility, Heston stochastic volatility, three‐halves stochastic volatility, and SABR local‐stochastic volatility.     

Randomized Stopping Times and American Option Pricing with Transaction Costs

In a general discrete-time market model with proportional transaction costs, we derive new expectation representations of the range of arbitrage-free prices of an arbitrary American option. The upper bound of this range is called the upper hedging price, and is the smallest initial wealth needed to construct a self-financing portfolio whose value dominates the option payoff at all times. A surprising feature of our upper hedging price representation is that it requires the use of randomized stopping times (Baxter and Chacon 1977), just as ordinary stopping times are needed in the absence of transaction costs. We also represent the upper hedging price as the optimum value of a variety of optimization problems. Additionally, we show a two-player game where at Nash equilibrium the value to both players is the upper hedging price, and one of the players must in general choose a mixture of stopping times. We derive similar representations for the lower hedging price as well. Our results make use of strong duality in linear programming.     

A cointegrated commodity pricing model

We propose a commodity pricing model that extends the Gibson–Schwartz two‐factor model to incorporate the effect of linear relations among commodity spot prices, and provide a condition under which such linear relations represent cointegration. We derive futures and call option prices for the proposed model, and indicate that, unlike in Duan and Pliska (2004), the linear relations among commodity prices should affect commodity derivative prices, even when the volatilities of commodity returns are constant. Using crude oil and heating oil market data, we estimate the model and apply the results to the hedging of long‐term futures using short‐term ones.     

Density forecasts of crude‐oil prices using option‐implied and ARCH‐type models

The predictive accuracy of competing crude‐oil price forecast densities is investigated for the 1994–2006 period. Moving beyond standard ARCH type models that rely exclusively on past returns, we examine the benefits of utilizing the forward‐looking information that is embedded in the prices of derivative contracts. Risk‐neutral densities, obtained from panels of crude‐oil option prices, are adjusted to reflect real‐world risks using either a parametric or a non‐parametric calibration approach. The relative performance of the models is evaluated for the entire support of the density, as well as for regions and intervals that are of special interest for the economic agent. We find that non‐parametric adjustments of risk‐neutral density forecasts perform significantly better than their parametric counterparts. Goodness‐of‐fit tests and out‐of‐sample likelihood comparisons favor forecast densities obtained by option prices and non‐parametric calibration methods over those constructed using historical returns and simulated ARCH processes. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:727–754, 2011     

Pricing arithmetic Asian and Amerasian options: A diffusion operator integral expansion approach

In this paper, we propose a new explicit series expansion formula for the price of an arithmetic Asian option under the Black–Scholes model and Merton's jump-diffusion model. The method is based on an equivalence in law relation together with the diffusion operator integral method proposed by Heath and Platen. The method yields explicit series expansion formula for the Asian options' prices. The theoretical convergence of the expansion to the true value is established. We also consider the American Asian option (i.e., Amerasian option) and derive the corresponding expansion formula through the early exercise premium representation. Numerical results illustrate the accuracy and efficiency of the method as compared with benchmarks in the literature.     

Lévy betas: Static hedging with index futures

This study considers calibration to forward‐looking betas by extracting information on equity and index options from prices using Lévy models. The resulting calibrated betas are called Lévy betas. The objective of the proposed approach is to capture market expectations for future betas through option prices, as betas estimated from historical data may fail to reflect structural change in the market. By assuming a continuous‐time capital asset pricing model (CAPM) with Lévy processes, we derive an analytical solution to index and stock options, thus permitting the betas to be implied from observed option prices. One application of Lévy betas is to construct a static hedging strategy using index futures. Employing Hong Kong equity and index option data from September 16, 2008 to October 15, 2009, we show empirically that the Lévy betas during the sub‐prime mortgage crisis period were much more volatile than those during the recovery period. We also find evidence to suggest that the Lévy betas improve static hedging performance relative to historical betas and the forward‐looking betas implied by a stochastic volatility model.     

Effects of rollover strategies and information stability on the performance measures in options markets: An examination of the KOSPI 200 index options market

One of the most widely used option valuation models among practitioners is the ad hoc Black–Scholes (AHBS) model. The main contribution of this study is methodological. We carefully consider two rollover strategies (nearest‐to‐next strategy and next‐to‐next) used in the AHBS model to investigate their effect on pricing errors. We suggest a new rollover strategy, next‐to‐next strategy, and demonstrate that our rollover strategy produces more consistent estimates between in‐sample market and model option prices. Probably even more important is that our new rollover strategy makes more accurate out‐of‐sample forecasts for 1‐day or 1‐week ahead prices. Prior literature has documented some anomalies associated with the use of AHBS model, for example, an overfitting problem. A secondary contribution is that our new rollover strategy does not suffer from this overfitting critique. Third, this study uses the mean square error for out‐of‐sample pricing and price changes to determine how the options investors are influenced by moneyness. The results indicate that underpricing (or overpricing) by the AHBS model for the near‐the‐money category is more likely to be maintained for the next several trading days but that such a phenomenon is disappeared for the deep out‐of‐the‐money category. Finally, we suggest the ratio of the number of option contracts to differences in strike prices available for trading between the current day and the previous day(s) as a good categorizing factor for options, such as moneyness. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

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     

A simple method for extracting the probability of default from American put option prices

We present a novel method for extracting the risk-neutral probability of default (PD) of a firm from American put option prices. Building on the idea of a default corridor proposed by Carr and Wu, we derive a parsimonious closed-form formula for American put option prices from which the PD can be inferred. The method is easy to implement. Our empirical results based on seven large US firms for the period 2002–2010 show that, in some cases, our option-implied PD can provide a more accurate estimate of default probability than the estimates implied from credit default swaps.     

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     

MCMC ESTIMATION OF LÉVY JUMP MODELS USING STOCK AND OPTION PRICES

We examine the performances of several popular Lévy jump models and some of the most sophisticated affine jump‐diffusion models in capturing the joint dynamics of stock and option prices. We develop efficient Markov chain Monte Carlo methods for estimating parameters and latent volatility/jump variables of the Lévy jump models using stock and option prices. We show that models with infinite‐activity Lévy jumps in returns significantly outperform affine jump‐diffusion models with compound Poisson jumps in returns and volatility in capturing both the physical and risk‐neutral dynamics of the S&P 500 index. We also find that the variance gamma model of Madan, Carr, and Chang with stochastic volatility has the best performance among all the models we consider.     

Who knows more about future currency volatility?

We use four currency pairs from October 1, 2001 to September 29, 2006 to compare the predictive power of the implied volatility derived from currency option prices that are traded on the Philadelphia Stock Exchange (PHLX), Chicago Mercantile Exchange (CME), and over‐the‐counter market (OTC). Among the competing implied volatility forecasts, OTC‐implied volatility subsumes the information content of PHLX‐ and CME‐implied volatility. Consistent with extant studies our result also shows that the implied volatility provides more information about future volatility–regardless of whether it is from the OTC, PHLX, or CME markets–than time series based volatility. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:270–295, 2009     

Dividend‐Rollover Effect and the Ad Hoc Black‐Scholes Model

One of the most widely used option‐valuation models among practitioners is the ad hoc Black‐Scholes (AHBS) model. The main contribution of this study is methodological. We carefully consider three dividend strategies (No dividend, Implied‐forward dividend, and Actual dividend) for the AHBS model to investigate their effect on pricing errors. We suggest a new dividend strategy, implied‐forward dividend, which incorporates expectational information on dividends embedded in option prices. We demonstrate that our implied‐forward dividend strategy produces more consistent estimates between in‐sample market and model option prices. More importantly our new implied‐forward dividend strategy makes more accurate out‐of‐sample forecasts for one‐day or one‐week ahead prices. Second, we document that both a “Return‐volatility” Smile and a “Return‐pricing Error” Smile exist. From these return characteristics, we make two conclusions: (1) the return dependency of implied volatility is an important explanatory variable and should be controlled to reduce the pricing error of an AHBS model, and (2) it is important for the hedging horizon to be based on return size, that is, the larger the contemporaneous return, the more frequent an option issuer must rebalance the option's hedge. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 32:742‐772, 2012     

Option pricing in the moderate deviations regime

We consider call option prices close to expiry in diffusion models, in an asymptotic regime (“moderately out of the money”) that interpolates between the well‐studied cases of at‐the‐money and out‐of‐the‐money regimes. First and higher order small‐time moderate deviation estimates of call prices and implied volatilities are obtained. The expansions involve only simple expressions of the model parameters, and we show how to calculate them for generic local and stochastic volatility models. Some numerical computations for the Heston model illustrate the accuracy of our results.     

Information content of cross‐sectional option prices: A comparison of alternative currency option pricing models on the Japanese yen

This article implements a currency option pricing model for the general case of stochastic volatility, stochastic interest rates, and jumps in an attempt to reconcile levels of risk‐neutral skewness and kurtosis with observed option prices on the Japanese yen and to analyze the information content of the cross section of option prices by investigating the hedging and pricing performance of various currency option pricing models. The study makes use of both a method of moments and a more traditional generalized‐least‐squares (GLS) estimation technique, taking advantage of the fact that methods of moments do not specifically require the use of cross‐sectional option prices, whereas GLS does. Results centered around the Asia economic crisis of 1997 and 1998 indicate that the cross section of option prices surprisingly does not appear to contain superior information as the two estimation techniques yield relatively similar results once idiosyncratic differences between them are acknowledged. Extensions of the G. Bakshi, C. Cao, and Z. Chen (1997) results to currencies are also provided. © 2006Wiley Periodicals, Inc. Jrl Fut Mark 26:33–59, 2006     

SWAPTION PRICING IN AFFINE AND OTHER MODELS

This paper shows that Singleton and Umantsev's method for swaption pricing in affine models can be simplified and extended to other models. Two alternative methods for approximating the option exercise boundary are introduced: one based on the multivariate Taylor series expansion, and the other based on duration‐matched zero‐coupon bond approximation. Applied to affine models and quadratic‐Gaussian models, these methods are found to give accurate swaption prices.