Error analysis of finite difference and Markov chain approximations for option pricing

Mijatovi? and Pistorius proposed an efficient Markov chain approximation method for pricing European and barrier options in general one‐dimensional Markovian models. However, sharp convergence rates of this method for realistic financial payoffs, which are nonsmooth, are rarely available. In this paper, we solve this problem for general one‐dimensional diffusion models, which play a fundamental role in financial applications. For such models, the Markov chain approximation method is equivalent to the method of lines using the central difference. Our analysis is based on the spectral representation of the exact solution and the approximate solution. By establishing the convergence rate for the eigenvalues and the eigenfunctions, we obtain sharp convergence rates for the transition density and the price of options with nonsmooth payoffs. In particular, we show that for call‐/put‐type payoffs, convergence is second order, while for digital‐type payoffs, convergence is generally only first order. Furthermore, we provide theoretical justification for two well‐known smoothing techniques that can restore second‐order convergence for digital‐type payoffs and explain oscillations observed in the convergence for options with nonsmooth payoffs. As an extension, we also establish sharp convergence rates for European options for a rich class of Markovian jump models constructed from diffusions via subordination. The theoretical estimates are confirmed using numerical examples.     

A Forward Monte Carlo Method for American Options Pricing

This study proposes a forward Monte Carlo method for the pricing of American options. The main advantage of this method is that it does not use backward induction as required by other methods. Instead, the proposed approach relies on a wise determination about whether a simulated stock price has entered the exercise region. The validity of the proposed method is supported by the mathematical proofs for the vanilla cases. With some adaption, it is shown that this forward method can be extended to price other American style options such as chooser and exchange options. This study demonstrates the effectiveness of the proposed approach using a series of numerical examples, revealing significant improvements in numerical efficiency and accuracy in contrast with the standard regression‐based method of Longstaff and Schwartz (2001). © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:369‐395, 2013     

A Continuity Correction for Discrete Barrier Options

The payoff of a barrier option depends on whether or not a specified asset price, index, or rate reaches a specified level during the life of the option. Most models for pricing barrier options assume continuous monitoring of the barrier; under this assumption, the option can often be priced in closed form. Many (if not most) real contracts with barrier provisions specify discrete monitoring instants; there are essentially no formulas for pricing these options, and even numerical pricing is difficult. We show, however, that discrete barrier options can be priced with remarkable accuracy using continuous barrier formulas by applying a simple continuity correction to the barrier. The correction shifts the barrier away from the underlying by a factor of exp(bet sig sqrt dt), where bet approx 0.5826, sig is the underlying volatility, and dt is the time between monitoring instants. The correction is justified both theoretically and experimentally.     

Pricing Discrete European Barrier Options Using Lattice Random Walks

This paper designs a numerical procedure to price discrete European barrier options in Black-Scholes model. The pricing problem is divided into a series of initial value problems, one for each monitoring time. Each initial value problem is solved by replacing the driving Brownian motion by a lattice random walk. Some results from the theory of Besov spaces show that the convergence rate of lattice methods for initial value problems depends on two factors, namely the smoothness of the initial value (or the value function) and the moments for the increments of the lattice random walk. This fact is used to obtain an efficient method to price discrete European barrier options. Numerical examples and comparisons with other methods are carried out to show that the proposed method yields fast and accurate results.     

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     

Richardson extrapolation techniques for the pricing of American‐style options

In this article, the authors reexamine the American‐style option pricing formula of R. Geske and H.E. Johnson (1984), and extend the analysis by deriving a modified formula that can overcome the possibility of nonuniform convergence (which is likely to occur for nonstandard American options whose exercise boundary is discontinuous) encountered in the original Geske–Johnson methodology. Furthermore, they propose a numerical method, the Repeated‐Richardson extrapolation, which allows the estimation of the interval of true option values and the determination of the number of options needed for an approximation to achieve a given desired accuracy. Using simulation results, our modified Geske–Johnson formula is shown to be more accurate than the original Geske–Johnson formula for pricing American options, especially for nonstandard American options. This study also illustrates that the Repeated‐Richardson extrapolation approach can estimate the interval of true American option values extremely well. Finally, the authors investigate the possibility of combining the binomial Black–Scholes method proposed by M. Broadie and J.B. Detemple (1996) with the Repeated‐Richardson extrapolation technique. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:791–817, 2007     

Useful Properties of Exchange Rate Forecasts for Risk Management and Derivative Pricing

We examine three useful properties of the yen/dollar exchange‐rate forecasts that are published in January and July in the Wall Street Journal. Those properties are the level of explanatory power, whether some forecasters are consistently better than others, and whether the dispersion of forecasts can predict the volatility of the exchange rates. The results show that the relative accuracy of the individual forecasts has not been random each period, and the evidence suggests that some forecasters are consistently better than others. The forecasts from the best forecasters explain about half of the variability in the semiannual exchange‐rate series. Finally, the dispersion of all the forecasts each period, as measured by the standard deviation, has predictive power with respect to the daily volatility of the forecasts for the three months following the survey. This final property has implications for the pricing and use of currency options. © 2001 John Wiley & Sons, Inc.     

On the rate of convergence of binomial Greeks

This study investigates the convergence patterns and the rates of convergence of binomial Greeks for the CRR model and several smooth price convergence models in the literature, including the binomial Black–Scholes (BBS) model of Broadie M and Detemple J ( 1996 ), the flexible binomial model (FB) of Tian YS ( 1999 ), the smoothed payoff (SPF) approach of Heston S and Zhou G ( 2000 ), the GCRR‐XPC models of Chung SL and Shih PT ( 2007 ), the modified FB‐XPC model, and the modified GCRR‐FT model. We prove that the rate of convergence of the CRR model for computing deltas and gammas is of order O(1/n), with a quadratic error term relating to the position of the final nodes around the strike price. Moreover, most smooth price convergence models generate deltas and gammas with monotonic and smooth convergence with order O(1/n). Thus, one can apply an extrapolation formula to enhance their accuracy. The numerical results show that placing the strike price at the center of the tree seems to enhance the accuracy substantially. Among all the binomial models considered in this study, the FB‐XPC and the GCRR‐XPC model with a two‐point extrapolation are the most efficient methods to compute Greeks. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark     

Limits to labels: The role of eco‐labels in the assessment of product sustainability and routes to sustainable consumption

There has been rapid development in the methods, data and protocols for the assessment of product sustainability over the past decade. Notwithstanding this welcome development, the widespread provision of sustainable products has not occurred. Moreover, indications from a myriad of surveys suggest that consumers remain full of intent to purchase sustainably, yet these stated preferences have not translated into a widespread uptake in the purchase of more sustainable products. Heightened interest in climate change over the past couple of years has led to rising calls for labelling to allow consumers to differentiate between more or less sustainable options. Such calls apparently assume that if consumers are presented with appropriate label information their purchases will change and more sustainable purchasing will result. For many observers these calls bring more than a ring of déjà vu as the failures (or at least unfulfilled expectations) of environmental labelling schemes of the past spring to mind. A review and assessment of eco‐labelling schemes is presented. Discussion focuses on the history, successes and failures of such schemes, and consideration of their potential role (or not) in future shifts towards sustainable consumption. Behavioural, social practice, institutional and infrastructure factors are considered and labelling, legislation and other options are explored. Conclusions are drawn regarding potential routes to sustainable consumption, with particular reference to eco‐labels.     

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     

Liquidity Considerations in Estimating Implied Volatility

Option markets have significant variation in liquidity across different option series. Illiquidity reduces the informativeness of the price. Price information for illiquid options is more noisy, and thus the implied volatilities (IVs) based on these prices are more noisy. In this study, we propose weighting schemes to estimate IV, which reduce the importance attached to illiquid options. The two indexes using liquidity weights are SVIX, which is a spread‐adjusted volatility index, and TVVIX, which is a traded volume weighted VIX. We find SVIX outperforms TVVIX, the conventional schemes such as the traditional VXO, or vega weights, and volatility elasticity weights. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 32:714‐742, 2012     

The optimal method for pricing Bermudan options by simulation

Least‐squares methods enable us to price Bermudan‐style options by Monte Carlo simulation. They are based on estimating the option continuation value by least‐squares. We show that the Bermudan price is maximized when this continuation value is estimated near the exercise boundary, which is equivalent to implicitly estimating the optimal exercise boundary by using the value‐matching condition. Localization is the key difference with respect to global regression methods, but is fundamental for optimal exercise decisions and requires estimation of the continuation value by iterating local least‐squares (because we estimate and localize the exercise boundary at the same time). In the numerical example, in agreement with this optimality, the new prices or lower bounds (i) improve upon the prices reported by other methods and (ii) are very close to the associated dual upper bounds. We also study the method's convergence.     

The short‐time behavior of VIX‐implied volatilities in a multifactor stochastic volatility framework

We consider a modeling setup where the volatility index (VIX) dynamics are explicitly computable as a smooth transformation of a purely diffusive, multidimensional Markov process. The framework is general enough to embed many popular stochastic volatility models. We develop closed‐form expansions and sharp error bounds for VIX futures, options, and implied volatilities. In particular, we derive exact asymptotic results for VIX‐implied volatilities, and their sensitivities, in the joint limit of short time‐to‐maturity and small log‐moneyness. The expansions obtained are explicit based on elementary functions and they neatly uncover how the VIX skew depends on the specific choice of the volatility and the vol‐of‐vol processes. Our results are based on perturbation techniques applied to the infinitesimal generator of the underlying process. This methodology has previously been adopted to derive approximations of equity (SPX) options. However, the generalizations needed to cover the case of VIX options are by no means straightforward as the dynamics of the underlying VIX futures are not explicitly known. To illustrate the accuracy of our technique, we provide numerical implementations for a selection of model specifications.     

Currency barrier option pricing with mean reversion

This article develops a barrier option pricing model in which the exchange rate follows a mean‐reverting lognormal process. The corresponding closed‐form solutions for the barrier options with time‐dependent barriers are derived. The numerical results show that barrier option values and the corresponding hedge parameters under the proposed model are different from those based on the Black‐Scholes model. For an up‐and‐out call, the mean‐reverting process keeps the exchange rate in a small range around the mean level. When the mean level is below the barrier but above the strike price, the risk of the call to be knocked out is reduced and its option value is enhanced compared with the value under the Black‐Scholes model. The parameters of the mean‐reverting lognormal process therefore have a material impact on the valuation of currency barrier options and their hedge parameters. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:939–958, 2006     

Path‐dependent currency options with mean reversion

This paper develops a path‐dependent currency option pricing framework in which the exchange rate follows a mean‐reverting lognormal process. Analytical solutions are derived for barrier options with a constant barrier, lookback options, and turbo warrants. As the analytical solutions are obtained using a Laplace transform, this study numerically shows that the solution implemented with a numerical Laplace inversion is efficient and accurate. The pricing behavior of path‐dependent options with mean reversion is contrasted with the Black‐Scholes model. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:275–293, 2008     

Pricing Eurodollar Futures Options with the Heath—Jarrow—Morton Model

This article uses the algorithm developed by Ritchken and Sankarasubramanian (1995) to make comparisons among the Heath—Jarrow—Morton (HJM) models (Heath, Jarrow, & Morton, 1992) with different volatility structures in pricing the Eurodollar futures options. We show that the differences among the HJM models as well as the difference between the HJM models and Black's model can be insignificant when the volatility of the forward rate is relatively small. Moreover, our findings imply that the difference between the American‐style and European‐style options is insignificant for options with a life of less than 1 year. However, the difference can be significant for options with a 1‐year maturity, the difference depending on the exercise price. Finally, our tests indicate that the difference between the forward price and the futures price is insignificant if the volatility parameter is low enough and when the volatility of the spot rate is proportional to the spot rate. A higher volatility parameter can lead to a significant difference between the forward price and the futures price, although its impact on the price of the options will still be trivial. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21: 655–680, 2001     

HEDGING STRATEGIES AND MINIMAL VARIANCE PORTFOLIOS FOR EUROPEAN AND EXOTIC OPTIONS IN A LÉVY MARKET

This paper presents hedging strategies for European and exotic options in a Lévy market. By applying Taylor’s theorem, dynamic hedging portfolios are constructed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importance as such other derivatives might be readily available. Moment swaps or power jump assets are not typically liquidly traded. It is shown how minimal variance portfolios can be used to hedge the higher order terms in a Taylor expansion of the pricing function, investing only in a risk‐free bank account, the underlying asset, and potentially variance swaps. The numerical algorithms and performance of the hedging strategies are presented, showing the practical utility of the derived results.     

The Cross‐Currency Hedging Performance of Implied Versus Statistical Forecasting Models

This article examines the ability of several models to generate optimal hedge ratios. Statistical models employed include univariate and multivariate generalized autoregressive conditionally heteroscedastic (GARCH) models, and exponentially weighted and simple moving averages. The variances of the hedged portfolios derived using these hedge ratios are compared with those based on market expectations implied by the prices of traded options. One‐month and three‐month hedging horizons are considered for four currency pairs. Overall, it has been found that an exponentially weighted moving‐average model leads to lower portfolio variances than any of the GARCH‐based, implied or time‐invariant approaches. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:1043–1069, 2001     

On approximating deep in‐the‐money Asian options under exponential Lévy processes

This note proposes a new approach of valuing deep in‐the‐money fixed strike and discretely monitoring arithmetic Asian options. This new approach prices Asian options whose underlying asset price evolves according to the exponential of a Lévy process as a weighted sum of European options. Numerical results from experimenting on three different types of Lévy processes—a diffusion process, a jump diffusion process, and a pure jump process—illustrate the accuracy of the approach. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark     

On the Rate of Convergence of Discrete‐Time Contingent Claims

This paper characterizes the rate of convergence of discrete‐time multinomial option prices. We show that the rate of convergence depends on the smoothness of option payoff functions, and is much lower than commonly believed because option payoff functions are often of all‐or‐nothing type and are not continuously differentiable. To improve the accuracy, we propose two simple methods, an adjustment of the discrete‐time solution prior to maturity and smoothing of the payoff function, which yield solutions that converge to their continuous‐time limit at the maximum possible rate enjoyed by smooth payoff functions. We also propose an intuitive approach that systematically derives multinomial models by matching the moments of a normal distribution. A highly accurate trinomial model also is provided for interest rate derivatives. Numerical examples are carried out to show that the proposed methods yield fast and accurate results.