DISCRETE TIME HEDGING OF THE AMERICAN OPTION

In a complete financial market we consider the discrete time hedging of the American option with a convex payoff. It is well known that for the perfect hedging the writer of the option must trade continuously in time, which is impossible in practice. In reality, the writer hedges only at some discrete time instants. The perfect hedging requires the knowledge of the partial derivative of the value function of the American option in the underlying asset, the explicit form of which is unknown in most cases of practical importance. Several approximation methods have been developed for the calculation of the value function of the American option. We claim in this paper that having at hand any uniform approximation of the American option value function at equidistant discrete rebalancing times it is possible to construct a discrete time hedging portfolio, the value process of which uniformly approximates the value process of the continuous time perfect delta‐hedging portfolio. We are able to estimate the corresponding discrete time hedging error that leads to a complete justification of our hedging method for nonincreasing convex payoff functions including the important case of the American put. This method is essentially based on a new type square integral estimate for the derivative of an arbitrary convex function recently found by Shashiashvili.     

A MODEL‐FREE VERSION OF THE FUNDAMENTAL THEOREM OF ASSET PRICING AND THE SUPER‐REPLICATION THEOREM

We propose a Fundamental Theorem of Asset Pricing and a Super‐Replication Theorem in a model‐independent framework. We prove these theorems in the setting of finite, discrete time and a market consisting of a risky asset S as well as options written on this risky asset. As a technical condition, we assume the existence of a traded option with a superlinearly growing payoff‐function, e.g., a power option. This condition is not needed when sufficiently many vanilla options maturing at the horizon T are traded in the market.     

MONTE CARLO METHODS FOR THE VALUATION OF MULTIPLE-EXERCISE OPTIONS

We discuss Monte Carlo methods for valuing options with multiple-exercise features in discrete time. By extending the recently developed duality ideas for American option pricing, we show how to obtain estimates on the prices of such options using Monte Carlo techniques. We prove convergence of our approach and estimate the error. The methods are applied to options in the energy and interest rate derivative markets.     

FAST SWAPTION PRICING IN GAUSSIAN TERM STRUCTURE MODELS

We propose a fast and accurate numerical method for pricing European swaptions in multifactor Gaussian term structure models. Our method can be used to accelerate the calibration of such models to the volatility surface. The pricing of an interest rate option in such a model involves evaluating a multidimensional integral of the payoff of the claim on a domain where the payoff is positive. In our method, we approximate the exercise boundary of the state space by a hyperplane tangent to the maximum probability point on the boundary and simplify the multidimensional integration into an analytical form. The maximum probability point can be determined using the gradient descent method. We demonstrate that our method is superior to previous methods by comparing the results to the price obtained by numerical integration.     

THE TRACKING ERROR RATE OF THE DELTA‐GAMMA HEDGING STRATEGY

We analyze the convergence rate of the quadratic tracking error, when a Delta‐Gamma hedging strategy is used at N discrete times. The fractional regularity of the payoff function plays a crucial role in the choice of the trading dates, in order to achieve optimal rates of convergence.     

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.     

On the errors and comparison of Vega estimation methods

This article discusses convergence problems when calculating Vega (option sensitivity to volatility) that arise from discretization errors embedded in the lattice approach. Four alternative improvements to the traditional binomial method are discussed and investigated for performance. We also propose a new Modified Binomial (MB) Method to calculate Vegas. Numerical results show that although the MB is not the most price accurate of the models, due to its error structure as a function of volatility, it produces the most accurate and fastest Vega estimates. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:21–38, 2005     

Step Options

Motivated by risk management problems with barrier options, we propose a flexible modification of the standard knock‐out and knock‐in provisions and introduce a family of path‐dependent options: step options . They are parametrized by a finite knock‐out (knock‐in) rate , ρ. For a down‐and‐out step option, its payoff at expiration is defined as the payoff of an otherwise identical vanilla option discounted by the knock‐out factor exp(-ρτ_B^‐) or max(1‐ρτ^-B,0), where &\tau;_B^‐ is the total time during the contract life that the underlying price was lower than a prespecified barrier level ( occupation time ). We derive closed‐form pricing formulas for step options with any knock‐out rate in the range $[0,∞). For any finite knock‐out rate both the step option's value and delta are continuous functions of the underlying price at the barrier. As a result, they can be continuously hedged by trading the underlying asset and borrowing. Their risk management properties make step options attractive "no‐regrets" alternatives to standard barrier options. As a by‐product, we derive a dynamic almost‐replicating trading strategy for standard barrier options by considering a replicating strategy for a step option with high but finite knock‐out rate. Finally, a general class of derivatives contingent on occupation times is considered and closed‐form pricing formulas are derived.     

The accuracy and efficiency of alternative option pricing approaches relative to a log‐transformed trinomial model

This article presents a log‐transformed trinomial approach to option pricing and finds that various numerical procedures in the option pricing literature are embedded in this approach with choices of different parameters. The unified view also facilitates comparisons of computational efficiency among numerous lattice approaches and explicit finite difference methods. We use the root‐mean‐squared relative error and the minimum convergence step to evaluate the accuracy and efficiency for alternative option pricing approaches. The numerical results show that the equal‐probability trinomial specification of He ( 12 ) and Tian ( 25 ) and the sharpened trinomial specification of Omberg ( 21 ) outperform other lattice approaches and explicit finite difference methods. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:557–577, 2002     

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.     

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.     

From Discrete to Continuous Financial Models: New Convergence Results For Option Pricing

In this paper we develop a new notion of convergence for discussing the relationship between discrete and continuous financial models, D ²-convergence. This is stronger than weak convergence, the commonly used mode of convergence in the finance literature. We show that D ²-convergence, unlike weak convergence, yields a number of important convergence preservation results, including the convergence of contingent claims, derivative asset prices and hedge portfolios in the discrete Cox-Ross-Rubinstein option pricing models to their continuous counterparts in the Black-Scholes model. Our results show that D ²-convergence is characterized by a natural lifting condition from nonstandard analysis (NSA), and we demonstrate how this condition can be reformulated in standard terms, i.e., in language that only involves notions from standard analysis. From a practical point of view, our approach suggests procedures for constructing good (i.e., convergent) approximate discrete claims, prices, hedge portfolios, etc. This paper builds on earlier work by the authors, who introduced methods from NSA to study problems arising in the theory of option pricing.     

MARTINGALE APPROACH TO PRICING PERPETUAL AMERICAN OPTIONS ON TWO STOCKS

We study the pricing of American options on two stocks without expiration date and with payoff functions which are positively homogeneous with respect to the two stock prices. Examples of such options are the perpetuai Margrabe option, whose payoff is the amount by which one stock outperforms the other, and the perpetual maximum option, whose payoff is the maximum of the two stock prices Our approach to pricing such options is to take advantage of their stationary nature and apply the optional sampling theorem to two martingales constructed with respect to the risk-neutral measure the optimal exercise boundaries, which do not vary with respect to the time variable, are determined by the smooth pasting or high contact condition the martingale approach avoids the use of differential equations.     

CRITICAL PRICE NEAR MATURITY FOR AN AMERICAN OPTION ON A DIVIDEND-PAYING STOCK IN A LOCAL VOLATILITY MODEL

We consider an American put option on a dividend-paying stock whose volatility is a function of the stock value. Near the maturity of this option, an expansion of the critical stock price is given. If the stock dividend rate is greater than the market interest rate, the payoff function is smooth near the limit of the critical price. We deduce an expansion of the critical price near maturity from an expansion of the value function of an optimal stopping problem. It turns out that the behavior of the critical price is parabolic. In the other case, we are in a less regular situation and an extra logarithmic factor appears. To prove this result, we show that the American and European critical prices have the same first-order behavior near maturity. Finally, in order to get an expansion of the European critical price, we use a parity formula for exchanging the strike price and the spot price in the value functions of European puts.     

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     

LIQUIDITY IN A BINOMIAL MARKET

We study the binomial version of the illiquid market model introduced by Çetin, Jarrow, and Protter for continuous time and develop efficient numerical methods for its analysis. In particular, we characterize the liquidity premium that results from the model. In Çetin, Jarrow, and Protter, the arbitrage free price of a European option traded in this illiquid market is equal to the classical value. However, the corresponding hedge does not exist and the price is obtained only in L² ‐approximating sense. Çetin, Soner, and Touzi investigated the super‐replication problem using the same supply curve model but under some restrictions on the trading strategies. They showed that the super‐replicating cost differs from the Black–Scholes value of the claim, thus proving the existence of liquidity premium. In this paper, we study the super‐replication problem in discrete time but with no assumptions on the portfolio process. We recover the same liquidity premium as in the continuous‐time limit. This is an independent justification of the restrictions introduced in Çetin, Soner, and Touzi. Moreover, we also propose an algorithm to calculate the option’s price for a binomial market.     

The valuation of American options in a multidimensional exponential Lévy model

We consider the problem of valuation of American options written on dividend‐paying assets whose price dynamics follow a multidimensional exponential Lévy model. We carefully examine the relation between the option prices, related partial integro‐differential variational inequalities, and reflected backward stochastic differential equations. In particular, we prove regularity results for the value function and obtain the early exercise premium formula for a broad class of payoff functions.     

PRICING OF HIGH‐DIMENSIONAL AMERICAN OPTIONS BY NEURAL NETWORKS

Pricing of American options in discrete time is considered, where the option is allowed to be based on several underlyings. It is assumed that the price processes of the underlyings are given Markov processes. We use the Monte Carlo approach to generate artificial sample paths of these price processes, and then we use the least squares neural networks regression estimates to estimate from this data the so‐called continuation values, which are defined as mean values of the American options for given values of the underlyings at time t subject to the constraint that the options are not exercised at time t. Results concerning consistency and rate of convergence of the estimates are presented, and the pricing of American options is illustrated by simulated data.     

Risk Management of Nonstandard Basket Options with Different Underlying Assets

Basket options are among the most popular products of the new generation of exotic options. They are particularly attractive because they can efficiently and simultaneously hedge a wide variety of intrinsically different financial risks and are flexible enough to cover all the risks faced by firms. Oddly, the existing literature on basket options considers only standard baskets where all underlying assets are of the same type and hedge the same kind of risk. Moreover, the empirical implementation of basket‐option models remains in its early stages, particularly when the baskets contain different underlying assets. This study focuses on various steps for developing sound risk management of basket options. We first propose a theoretical model of a nonstandard basket option on commodity price with stochastic convenience yield, exchange rate, and domestic and foreign zero‐coupon bonds in a stochastic interest rate setting. We compare the hedging performance of the extended basket option containing different underlying assets with that of a portfolio of individual options. The results show that the basket strategy is more efficient. We apply the maximum likelihood method to estimate the parameters of the basket model and the correlations between variables. Monte Carlo simulations are conducted to examine the performance of the maximum likelihood estimator in finite samples of simulated data. A real‐data study for a nonfinancial firm is presented to illustrate ways practitioners could use the extended basket option. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:299‐326, 2013     

PRICING ASIAN OPTIONS FOR JUMP DIFFUSION

We construct a sequence of functions that uniformly converge (on compact sets) to the price of an Asian option, which is written on a stock whose dynamics follow a jump diffusion. The convergence is exponentially fast. We show that each element in this sequence is the unique classical solution of a parabolic partial differential equation (not an integro‐differential equation). As a result we obtain a fast numerical approximation scheme whose accuracy versus speed characteristics can be controlled. We analyze the performance of our numerical algorithm on several examples.