Early Exercise Boundary Research Articles

Closed-form valuation of American call options on stocks paying multiple dividends

An American call option on a stock paying a single known dividend can be valued using the Roll–Geske–Whaley formula. This paper extends the Roll–Geske–Whaley model to the n dividends case by using the generalized n-fold compound option model. In this way this paper offers a closed-form solution for American options on stocks paying n known discrete dividends. Moreover, the model also offers the critical values of the early exercise boundaries at each ex-dividend date instant, making it easy to define an early exercise strategy. Numerical examples are included to illustrate this approach.

American Options in Regime-Switching Models

The pricing problem for American options in Markov-modulated Levy models is solved. The early exercise boundaries and prices are calculated using a generalization of Carr's randomization procedure for regime-switching models. The pricing procedure is efficient even if the number of states is large, provided the transition rates are not large w.r.t. the riskless rates. The payoffs and riskless rates may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modeled as finite-state Markov chains.

Perpetual American Options in Regime-Switching Models

In the paper, we solve the pricing problem for perpetual American options in Markov-modulated Levy models. The early exercise boundaries and prices are calculated using an iteration procedure. The pricing procedure is efficient even if the number of states is large provided the transition rates are not large w.r.t. the riskless rates. The payoffs and riskless rates may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modelled as finite-state Markov chains.

American Call Options on Jump-Diffusion Processes: A Fourier Transform Approach

This paper considers the Fourier transform approach to derive the implicit integral equation for the price of an American call option in the case where the underlying asset follows a jump-diffusion process. Using the method of Jamshidian (1992), we demonstrate that the call option price is given by the solution to an inhomogeneous integro-partial differential equation in an unbounded domain, and subsequently derive the solution using Fourier transforms. We also extend McKean’s incomplete Fourier transform approach to solve the free boundary problem under Merton’s framework, for a general jump size distribution. We show how the two methods are related to each other, and also to the Geske-Johnson compound option approach used by Gukhal (2001). The paper also derives results concerning the limit for the free boundary at expiry, and presents a numerical algorithm for solving the linked integral equation system for the American call price, delta and early exercise boundary. This scheme is applied to Merton’s jump-diffusion model, where the jumps are log-normally distributed.

American Options in Regime-Switching Models

In the paper, we solve the pricing problem for American options in Markov-modulated Levy models. The early exercise boundaries and prices are calculated using a generalization of Carr's randomization for regime-switching models. The pricing procedure is efficient even if the number of states is large provided the transition rates are not large w.r.t. the riskless rates. The payoffs and riskless rates may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modelled as finite-state Markov chains.

American and European Options Near Expiry, Under Markov Processes with Jumps

We derive an explicit formula for time decay, theta, for out-of-the-money European options at expiry, in terms of the density of jumps and payoff $g$. We use this formula to show that in the presence of jumps, the limit of the no-exercise region as time to expiry tends to 0 is typically larger than in the pure Gaussian case. In particular, for many families of non-Gaussian processes used in empirical studies of financial markets, the early exercise boundary for the American put without dividends is separated from the strike price by a non-vanishing margin on the interval [0, T).

Early exercise boundary and option prices in Lévy driven models

Pricing and hedging of European, American, barrier options and interest rate derivatives for wide classes of Lévy driven models is considered in situations where qualitative and quantitative differences between Gaussian and Lévy modelling are most prominent, and the dependence on the choice of a family of Lévy processes is analysed. Asymptotics of option prices near the barrier and expiry are calculated; for American options, two fast numerical methods are constructed. It is shown that for many classes of Lévy processes, the early exercise boundary of the American put is separated from the strike by a non-vanishing margin, and as the riskless rate vanishes, the early exercise boundary tends to 0 uniformly over the interval [0, T). Implications for fitting of parameters are discussed.

Early exercise boundary and option prices in Lévy driven models

Early exercise boundary and option prices in Lévy driven models

QUANTO LOOKBACK OPTIONS

The lookback feature in a quanto option refers to the payoff structure where the terminal payoff of the quanto option depends on the realized extreme value of either the stock price or the exchange rate. In this paper, we study the pricing models of European and American lookback options with the quanto feature. The analytic price formulas for two types of European‐style quanto lookback options are derived. The success of the analytic tractability of these quanto lookback options depends on the availability of a succinct analytic representation of the joint density function of the extreme value and terminal value of the stock price and exchange rate. We also analyze the early exercise policies and pricing behaviors of the quanto lookback options with the American feature. The early exercise boundaries of these American quanto lookback options exhibit properties that are distinctive from other two‐state American option models.

PRICING OF THE AMERICAN PUT UNDER LÉVY PROCESSES

We consider the American put with finite time horizon T, assuming that, under an EMM chosen by the market, the stock returns follow a regular Lévy process of exponential type. We formulate the free boundary value problem for the price of the American put, and develop the non-Gaussian analog of the method of lines and Carr's randomization method used in the Gaussian option pricing theory. The result is the (discretized) early exercise boundary and prices of the American put for all strikes and maturities from 0 to T. In the case of exponential jump-diffusion processes, a simple efficient pricing scheme is constructed. We show that for many classes of Lévy processes, the early exercise boundary is separated from the strike price by a non-vanishing margin on the interval [0, T), and that as the riskless rate vanishes, the optimal exercise price goes to zero uniformly over the interval [0, T), which is in the stark contrast with the Gaussian case.

The American Put and European Options Near Expiry, Under Levy Processes

We derive explicit formulas for time decay, for the European call and put options at expiry, and use them to calculate analytical approximations to the price of the American put and early exercise boundary near expiry. We show that for many families of non-Gaussian processes used in empirical studies of financial markets, the early exercise boundary for the American put without dividends is separated from the strike price by a non-vanishing margin on the time interval up to expiry, the expiry date excluding. As the riskless rate vanishes and the drift decreases accordingly so that the stock remains a martingale, the optimal exercise price goes to zero uniformly over the time interval up to expiry, the expiry date excluding. The implications for parameters' fitting are discussed.

Exercise Regions And Efficient Valuation Of American Lookback Options

This paper presents an efficient method to compute the values and early exercise boundaries of American fixed strike lookback options. The method reduces option valuation to a single optimal stopping problem for standard Brownian motion and an associated path‐dependent functional, indexed by one parameter in the absence of dividends and by two parameters in the presence of a dividend rate. Numerical results obtained by this method show that, after a space‐time transformation, the stopping boundaries are well approximated by certain piecewise linear functions with a few pieces, leading to fast and accurate approximations for American lookback option values. An explicit decomposition formula for American lookback options is derived and applied not only to the development of these approximations but also to the asymptotic analysis of the early exercise boundary near the expiration date.

Pricing American Options under Stochastic Volatility: A New Method Using Chebyshev Polynomials to Approximate the Early Exercise Boundary

This paper presents a new numerical method for pricing American call options when the volatility of the price of the underlying stock is stochastic. By exploiting a log-linear relationship of the optimal exercise boundary with respect to volatility changes, we derive an integral representation of an American call price and the early exercise premium which holds under stochastic volatility. This representation is used to develop a numerical method for pricing the American options based on an approximation of the optimal exercise boundary by Chebyshev polynomials. Numerical results show that our numerical approach can quickly and accurately price American call options both under stochastic and/or constant volatility.

Differential quadrature method for pricing American options

AbstractIn this article, differential quadrature method (DQM), a highly accurate and efficient numerical method for solving nonlinear problems, is used to overcome the difficulty in determining the optimal exercise boundary of American option. The following three parts of the problem in pricing American options are solved. The first part is how to treat the uncertainty of the early exercise boundary, or free boundary in the language of the PDE treatment of the American option, because American options can be exercised before the date of expiration. The second part is how to solve the nonlinear problem, because the problem of pricing American options is nonlinear. And the third part is how to treat the initial value condition with the singularity and the boundary conditions in the DQM. Numerical results for the free boundary of American option obtained by both DQM and finite difference method (FDM) are given and from which it can be seen the computational efficiency is greatly improved by DQM. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 711–725, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10028.

Pricing American Option Based on a Chebyshev Approximation of the Early Exercise Boundary

In this paper we introduce a new method of pricing an American call option by approximating its early exercise boundary based on Chebyshev polynomial functions. We implement the method to price options under the standard assumptions of the Black-Scholes model, for European options, and under stochastic volatility. For the latter, we provide an integral representation which unbundles the American call option price to Arrow-Debreu security prices. Numerical results indicate that our method is an effective alternative to other exercise boundary approximation methods under both the standard assumptions of the Black-Scholes and stochastic volatility.

On the Early Exercise Boundary of the American Put Option

We study the short time behavior of the early exercise boundary for American style put options in the Black--Scholes theory. We develop an asymptotic expansion which shows that the simple lower bound of Barles et al. is a more accurate approximation to the actual boundary than the more complex upper bound. Our expansion is obtained through iteration using a boundary integral equation. This integral equation is derived from the time derivative of the option value function, which closely resembles the classical Stefan free boundary value problem for melting ice. Our analytical results are supported by numerical computations designed for very short times.

Exercise boundaries and efficient approximations to American option prices and hedge parameters

This paper presents a new numerical method to solve the integral equation defining the early exercise boundary of an American option. It is shown that the early exercise boundaries of standard American options are well approximated by linear splines with a few knots, implying that the new solution method can actually be carried out on a coarse grid of time points with reasonable accuracy. This leads to a fast and reasonably accurate method to compute the early exercise boundaries, values, and hedge parameters of American options. In this connection, a brief survey of recent developments in approximations to American option prices and hedge parameters are also given.

A simple approach to the pricing of Bermudan swaptions in the multifactor LIBOR market model

This paper considers the pricing of Bermuda-style swaptions in the LIBOR market model and its extensions. Application of lattice methods to this model class is generally not feasible because of the large number of state variables, so instead a simple technique to incorporate early exercise features into the Monte Carlo method is considered. The approach here involves a direct search for an early exercise boundary parametrized in intrinsic value and the values of still-alive swaptions. The results of the proposed algorithm are compared against prices obtained from Markov chain approximations, nonrecombining trees, and finite difference methods. The proposed algorithm is fast and robust, and produces a lower bound on Bermudan swaption prices that appears to be very tight for many realistic structures. The paper contains several numerical results against which other methods can be tested.

A new integral representation of the early exercise boundary for American put options

A new integral representation of the early exercise boundary for American put options

Evaluation of American option prices in a path integral framework using Fourier–Hermite series expansions

Abstract In this paper we review the path integral technique which has wide applications in statistical physics and relate it to the backward recursion technique which is widely used for the evaluation of derivative securities. We formulate the pricing of equity options, both European and American, using the path integral framework. Discretising in the time variable and using expansions in Fourier–Hermite series for the continuous representation of the underlying asset price, we show how these options can be evaluated in the path integral framework. For American options, the solution technique facilitates the accurate determination of the early exercise boundary as part of the solution. Additionally, the continuous representation of the state variable allows the relatively accurate and efficient evaluation of the option prices and the delta hedge ratio.