Pricing American exchange options in a jump‐diffusion model

A way to estimate the value of an American exchange option when the underlying assets follow jump‐diffusion processes is presented. The estimate is based on combining a European exchange option and a Bermudan exchange option with two exercise dates by using Richardson extrapolation as proposed by R. Geske and H. Johnson (1984). Closed‐form solutions for the values of European and Bermudan exchange options are derived. Several numerical examples are presented, illustrating that the early exercise feature may have a significant economic value. The results presented should have potential for pricing over‐the‐counter options and in particular for pricing real options. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:257–273, 2007     

Trinomial or binomial: Accelerating American put option price on trees

We investigate the pricing performance of eight trinomial trees and one binomial tree, which was found to be most effective in an earlier study, under 20 different implementation methodologies for pricing American put options. We conclude that the binomial tree, the Tian third‐order moment‐matching tree with truncation, Richardson extrapolation, and smoothing, performs better than the trinomial trees. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:826–839, 2009     

An efficient approximation method for American exotic options

The authors suggest a modified quadratic approximation scheme, and apply this scheme to American barrier (knock‐out) and floating‐strike lookback options. This modified scheme introduces an additional parameter into the quadratic approximation method, originally suggested by G. Barone‐Adesi and R. Whaley (1987), to reduce pricing errors. When the barrier is close to the underlying asset's current price, the approximation formula is more accurate than lattice methods because the optimal exercise boundary is independent of the underlying asset's current price. That is, the proposed method overcomes the “near‐barrier” problem that occurs in lattice methods. In addition, the pricing error decreases when the underlying asset's volatility is high. This approximation scheme is more efficient than B. Gao, J. Huang, and M. Subrahmanyam's (2000) method. As a second application of the modified approximation scheme, the authors provide an approximation formula for American floating‐strike lookback options which is the first approximation formula ever suggested in the literature. Compared to S. Babbs' (2000) binomial approach, our approximation method is more efficient after controlling for pricing errors, and is more accurate after controlling for computing time. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:29–59, 2007     

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     

A Closer Look at Barrier Exchange Options

A barrier exchange option is an exchange option that is knocked out the first time the prices of two underlying assets become equal. Lindset, S., & Persson, S.‐A. (2006) present a simple dynamic replication argument to show that, in the absence of arbitrage, the current value of the barrier exchange option is equal to the difference in the current prices of the underlying assets and that this pricing formula applies irrespective of whether the option is European or American. In this study, we take a closer look at barrier exchange options and show, despite the simplicity of the pricing formula presented by Lindset, S., & Persson, S.‐A. (2006), that the barrier exchange option in fact involves a surprising array of key concepts associated with the pricing of derivative securities including: put–call parity, barrier in–out parity, static vs. dynamic replication, martingale pricing, continuous vs. discontinuous price processes, and numeraires. We provide valuable intuition behind the pricing formula which explains its apparent simplicity. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 33:29–43, 2013     

A non‐lattice pricing model of American options under stochastic volatility

In this article, an analytical approach to American option pricing under stochastic volatility is provided. Under stochastic volatility, the American option value can be computed as the sum of a corresponding European option price and an early exercise premium. By considering the analytical property of the optimal exercise boundary, the formula allows for recursive computation of the American option value. Simulation results show that a nonlattice method performs better than the lattice‐based interpolation methods. The stochastic volatility model is also empirically tested using S&P 500 futures options intraday transactions data. Incorporating stochastic volatility is shown to improve pricing, hedging, and profitability in actual trading. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:417–448, 2006     

Hedging under the influence of transaction costs: An empirical investigation on FTSE 100 index options

The Black–Scholes (BS; F. Black & M. Scholes, 1973) option pricing model, and modern parametric option pricing models in general, assume that a single unique price for the underlying instrument exists, and that it is the mid‐ (the average of the ask and the bid) price. In this article the authors consider the Financial Times and London Stock Exchange (FTSE) 100 Index Options for the time period 1992–1997. They estimate the ask and bid prices for the index, and show that, when substituted for the mid‐price in the BS formula, they provide superior option price predictors, for call and put options, respectively. This result is reinforced further when they .t a non‐parametric neural network model to market prices of liquid options. The empirical .ndings in this article suggest that the ask and bid prices of the underlying asset provide a superior fit to the mid/closing price because they include market maker's, compensation for providing liquidity in the market for constituent stocks of the FTSE 100 index. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:471–494, 2007     

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     

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     

A generalization of the Barone‐Adesi and Whaley approach for the analytic approximation of American options

This article introduces a general quadratic approximation scheme for pricing American options based on stochastic volatility and double jump processes. This quadratic approximation scheme is a generalization of the Barone‐Adesi and Whaley approach and nests several option models. Numerical results show that this quadratic approximation scheme is efficient and useful in pricing American options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:478–493, 2009     

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     

Valuing real options using implied binomial trees and commodity futures options

A real option on a commodity is valued using an implied binomial tree (IBT) calibrated using commodity futures options prices. Estimating an IBT in the absence of spot options (the norm for commodities) allows real option models to be calibrated for the first time to market‐implied probability distributions for commodity prices. In addition, the existence of long‐dated futures options means that good volatility estimates may now be incorporated into capital budgeting evaluations of real options projects with long planning horizons. An example is given using gold futures options and a real option to extract gold from a mine. A detailed out‐of‐sample test is included that shows how IBT option pricing errors evolve on subtrees emanating from future levels of the underlying asset. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:203–226, 2007     

The valuation of European options when asset returns are autocorrelated

This article derives the closed‐form formula for a European option on an asset with returns following a continuous‐time type of first‐order moving average process, which is called an MA(1)‐type option. The pricing formula of these options is similar to that of Black and Scholes, except for the total volatility input. Specifically, the total volatility input of MA(1)‐type options is the conditional standard deviation of continuous‐compounded returns over the option's remaining life, whereas the total volatility input of Black and Scholes is indeed the diffusion coefficient of a geometric Brownian motion times the square root of an option's time to maturity. Based on the result of numerical analyses, the impact of autocorrelation induced by the MA(1)‐type process is significant to option values even when the autocorrelation between asset returns is weak. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:85–102, 2006     

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     

ON THE CONSISTENCY OF REGRESSION‐BASED MONTE CARLO METHODS FOR PRICING BERMUDAN OPTIONS IN CASE OF ESTIMATED FINANCIAL MODELS

In many applications of regression‐based Monte Carlo methods for pricing, American options in discrete time parameters of the underlying financial model have to be estimated from observed data. In this paper suitably defined nonparametric regression‐based Monte Carlo methods are applied to paths of financial models where the parameters converge toward true values of the parameters. For various Black–Scholes, GARCH, and Levy models it is shown that in this case the price estimated from the approximate model converges to the true price.     

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     

The robust pricing–hedging duality for American options in discrete time financial markets

We investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the first part of the paper, we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a nondominated family of probability measures. Our first insight is that, by considering an enlargement of the space, we can see American options as European options and recover the pricing–hedging duality, which may fail in the original formulation. This can be seen as a weak formulation of the original problem. Our second insight is that a duality gap arises from the lack of dynamic consistency, and hence that a different enlargement, which reintroduces dynamic consistency is sufficient to recover the pricing–hedging duality: It is enough to consider fictitious extensions of the market in which all the assets are traded dynamically. In the second part of the paper, we study two important examples of the robust framework: the setup of Bouchard and Nutz and the martingale optimal transport setup of Beiglböck, Henry‐Labordère, and Penkner, and show that our general results apply in both cases and enable us to obtain the pricing–hedging duality for American options.     

Pricing average options on commodities

This study proposes a new approximation formula for pricing average options on commodities under a stochastic volatility environment. In particular, it derives an option pricing formula under Heston and an extended λ‐SABR stochastic volatility models (which includes an extended SABR model as a special case). Moreover, numerical examples support the accuracy of the proposed average option pricing formula. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark Mark 31:407–439, 2011     

Pricing American options on foreign currency with stochastic volatility,jumps, and stochastic interest rates

By applying the Heath–Jarrow–Morton (HJM) framework, an analytical approximation for pricing American options on foreign currency under stochastic volatility and double jump is derived. This approximation is also applied to other existing models for the purpose of comparison. There is evidence that such types of jumps can have a critical impact on earlyexercise premiums that will be significant for deep out‐of‐the‐money options with short maturities. Moreover, the importance of the term structure of interest rates to early‐exercise premiums is demonstrated as is the sensitivity of these premiums to correlation‐related parameters. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:867–891, 2007     

Volatility options: Hedging effectiveness,pricing, and model error

Motivated by the growing literature on volatility options and their imminent introduction in major exchanges, this article addresses two issues. First, the question of whether volatility options are superior to standard options in terms of hedging volatility risk is examined. Second, the comparative pricing and hedging performance of various volatility option pricing models in the presence of model error is investigated. Monte Carlo simulations within a stochastic volatility setup are employed to address these questions. Alternative dynamic hedging schemes are compared, and various option‐pricing models are considered. It is found that volatility options are not better hedging instruments than plain‐vanilla options. Furthermore, the most naïve volatility option‐pricing model can be reliably used for pricing and hedging purposes. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:1–31, 2006