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     

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     

Pricing continuously sampled Asian options with perturbation method

This article explores the price of continuously sampled Asian options. For geometric Asian options, we present pricing formulas for both backward‐starting and forward‐starting cases. For arithmetic Asian options, we demonstrate that the governing partial differential equation (PDE) cannot be transformed into a heat equation with constant coefficients; therefore, these options do not have a closed‐form solution of the Black–Scholes type, that is, the solution is not given in terms of the cumulative normal distribution function. We then solve the PDE with a perturbation method and obtain an analytical solution in a series form. Numerical results show that as compared with Zhang's ( 2001 ) highly accurate numerical results, the series converges very quickly and gives a good approximate value that is more accurate than any other approximate method in the literature, at least for the options tested in this article. Graphical results determine that the solution converges globally very quickly especially near the origin, which is the area in which most of the traded Asian options fall. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:535–560, 2003     

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     

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.     

Laguerre Series for Asian and Other Options

This paper has four goals: (a) relate ladder height distributions to option values; (b) show how Laguerre expansions may be used in the computation of densities, distribution functions, and option prices; (c) derive some new results on the integral of geometric Brownian motion over a finite interval; and (d) apply the preceding results to the determination of the distribution of the integral of geometric Brownian motion and the computation of Asian option values. The usual fixed‐strike options on the average are treated, as well as options with payoffs expressed in terms of one over the average of the underlying security, which this author calls “reciprocal Asian options.” In all cases the underlying asset is represented by geometric Brownian motion, the averages are performed continuously, and the options are of European type.     

PRICING SWAPTIONS UNDER MULTIFACTOR GAUSSIAN HJM MODELS

Several approximations have been proposed in the literature for the pricing of European‐style swaptions under multifactor term structure models. However, none of them provides an estimate for the inherent approximation error. Until now, only the Edgeworth expansion technique of Collin‐Dufresne and Goldstein is able to characterize the order of the approximation error. Under a multifactor HJM Gaussian framework, this paper proposes a new approximation for European‐style swaptions, which is able to set bounds on the magnitude of the approximation error and is based on the conditioning approach initiated by Curran and Rogers and Shi. All the proposed pricing bounds will arise as a simple by‐product of the Nielsen and Sandmann setup, and will be shown to provide a better accuracy–efficiency trade‐off than all the approximations already proposed in the literature.     

Pricing basket and Asian options under the jump‐diffusion process

This study derives approximate valuation formulas for basket options and Asian options under the jump‐diffusion process. To obtain an approximation for options prices under the jump‐diffusion process, we extend the Taylor expansion method developed by Ju N. ( 2002 ) under the diffusion process. We show that the Taylor expansion method, suggested in this study, provides better pricing performance as compared to log‐normal or four‐moment methods. The performance improvement using the Taylor expansion method increases as the time to maturity increases. In addition, our numerical analysis shows that jump effects become significant when the expected jump sizes take large negative values. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 31:830–854, 2011     

Canonical Distribution,Implied Binomial Tree,and the Pricing of American Options

In this study, a new approach to pricing American options is proposed and termed the canonical implied binomial (CIB) tree method. CIB takes advantage of both canonical valuation (Stutzer, 1996) and the implied binomial tree method (Rubinstein, 1994). Using simulated returns from geometric Brownian motions (GBM), CIB produced very similar prices for calls and European puts as those of Black–Scholes (BS). Applied to a set of over 15,000 American‐style S&P 100 Index puts, CIB outperformed BS with historic volatility in pricing out‐of‐the‐money options; in addition, it outperformed the canonical least‐squares Monte Carlo (Liu, 2010) in the dynamic hedging of in‐the‐money options. Furthermore, CIB suggests that regular GBM‐based Monte Carlo can be extended to American options pricing by also utilizing the implied binomial tree. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

PRICING AND SEMIMARTINGALE REPRESENTATIONS OF VULNERABLE CONTINGENT CLAIMS IN REGIME‐SWITCHING MARKETS

Using a suitable change of probability measure, we obtain a Poisson series representation for the arbitrage‐free price process of vulnerable contingent claims in a regime‐switching market driven by an underlying continuous‐time Markov process. As a result of this representation, along with a short‐time asymptotic expansion of the claim's price process, we develop an efficient novel method for pricing claims whose payoffs may depend on the full path of the underlying Markov chain. The proposed approach is applied to price not only simple European claims such as defaultable bonds, but also a new type of path‐dependent claims that we term self‐decomposable, as well as the important class of vulnerable call and put options on a stock. We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the predefault price function of European vulnerable claims, which enables us to rigorously deduce Feynman‐Ka? representations for the predefault pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk‐neutral and objective probability measures.     

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 and hedging of quanto range accrual notes under Gaussian HJM with cross‐currency Levy processes

This study analyzes the pricing and hedging problems for quanto range accrual notes (RANs) under the Heath‐Jarrow‐Morton (HJM) framework with Levy processes for instantaneous domestic and foreign forward interest rates. We consider the effects of jump risk on both interest rates and exchange rates in the pricing of the notes. We first derive the pricing formula for quanto double interest rate digital options and quanto contingent payoff options; then we apply the method proposed by Turnbull (Journal of Derivatives, 1995, 3, 92–101) to replicate the quanto RAN by a combination of the quanto double interest rate digital options and the quanto contingent payoff options. Using the pricing formulas derived in this study, we obtain the hedging position for each issue of quanto RANs. In addition, by simulation and assuming the jump risk to follow a compound Poisson process, we further analyze the effects of jump risk and exchange rate risk on the coupons receivable in holding a RAN. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:973–998, 2009     

PRICING AND HEDGING AMERICAN OPTIONS ANALYTICALLY: A PERTURBATION METHOD

This paper studies the critical stock price of American options with continuous dividend yield. We solve the integral equation and derive a new analytical formula in a series form for the critical stock price. American options can be priced and hedged analytically with the help of our critical-stock-price formula. Numerical tests show that our formula gives very accurate prices. With the error well controlled, our formula is now ready for traders to use in pricing and hedging the S&P 100 index options and for the Chicago Board Options Exchange to use in computing the VXO volatility index.     

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.     

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     

Pricing Eurodollar futures options with the Ho and Lee and Black,Derman, and Toy models: An empirical comparison

This article compares empirically the Ho and Lee (1986) and Black, Derman, and Toy (1990) discrete-time debt option pricing models in the pricing of Eurodollar futures options over the period from March 1997 through February 1998 using daily data. The results indicate that both models performed well. The average absolute pricing errors over the sample period were less than one tick (0.01) in every case. The Black, Derman, and Toy model slightly outperformed the Ho and Lee model in the pricing of in-the-money call options and out-of-the-money put options over the period studied. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 291–306, 1999     

DENSITY OF SKEW BROWNIAN MOTION AND ITS FUNCTIONALS WITH APPLICATION IN FINANCE

We derive the joint density of a Skew Brownian motion, its last visit to the origin, its local and occupation times. The result enables us to obtain explicit analytical formulas for pricing European options under both a two‐valued local volatility model and a displaced diffusion model with constrained volatility.     

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     

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     

VALUATION OF CONTINUOUSLY MONITORED DOUBLE BARRIER OPTIONS AND RELATED SECURITIES

In this paper, we apply Carr's randomization approximation and the operator form of the Wiener‐Hopf method to double barrier options in continuous time. Each step in the resulting backward induction algorithm is solved using a simple iterative procedure that reduces the problem of pricing options with two barriers to pricing a sequence of certain perpetual contingent claims with first‐touch single barrier features. This procedure admits a clear financial interpretation that can be formulated in the language of embedded options. Our approach results in a fast and accurate pricing method that can be used in a rather wide class of Lévy‐driven models including Variance Gamma processes, Normal Inverse Gaussian processes, KoBoL processes, CGMY model, and Kuznetsov's β ‐class. Our method can be applied to double barrier options with arbitrary bounded terminal payoff functions, which, in particular, allows us to price knock‐out double barrier put/call options as well as double‐no‐touch options.