American Option Valuation Research Articles

The Valuation of American Options with Stochastic Stopping Time Constraints

This paper concerns the pricing of American options with stochastic stopping time constraints expressed in terms of the states of a Markov process. Following the ideas of Menaldi et al., we transform the constrained into an unconstrained optimal stopping problem. The transformation replaces the original payoff by the value of a generalized barrier option. We also provide a Monte Carlo method to numerically calculate the option value for multidimensional Markov processes. We adapt the Longstaff–Schwartz algorithm to solve the stochastic Cauchy–Dirichlet problem related to the valuation problem of the barrier option along a set of simulated trajectories of the underlying Markov process.

American Option Valuation: Implied Calibration of GARCH Pricing-Models

This article analyzes the issue of American option valuation when the underlying exhibits a GARCH-type volatility process. We propose the usage of Rubinstein's Edgeworth binomial tree (EBT) in contrast to simulation-based methods being considered in previous studies. The EBT-based valuation approach makes an implied calibration of the pricing model feasible. By empirically analyzing the pricing performance of American index and equity options, we illustrate the superiority of the proposed approach.

Tight bounds on American option prices

In contrast to the constant exercise boundary assumed by Broadie and Detemple (1996) [Broadie, M., Detemple, J., 1996. American option valuation: New bounds, approximations, and comparison of existing methods. Review of Financial Studies 9, 1211–1250], we use an exponential function to approximate the early exercise boundary. Then, we obtain lower bounds for American option prices and the optimal exercise boundary which improve the bounds of Broadie and Detemple (1996). With the tight lower bound for the optimal exercise boundary, we further derive a tight upper bound for the American option price using the early exercise premium integral of Kim (1990) [Kim, I.J., 1990. The analytic valuation of American options. Review of Financial Studies 3, 547–572]. The numerical results show that our lower and upper bounds are very tight and can improve the pricing errors of the lower bound and upper bound of Broadie and Detemple (1996) by 83.0% and 87.5%, respectively. The tightness of our upper bounds is comparable to some best accurate/efficient methods in the literature for pricing American options. Moreover, the results also indicate that the hedge ratios (deltas and gammas) of our bounds are close to the accurate values of American options.

European Put-Call Parity and the Early Exercise Premium for American Currency Options

The European put-call parity condition is used to estimate the early exercise premium for American currency options traded on the Philadelphia Stock Exchange. Using a sample of 331 pairs of call and put options with the same exercise price and time to expiration, evidence is provided for early exercise premiums that average 5.03% for put options and 4.60% for call options. The premiums for both call and put options are strongly related to the interest rate differential and time to expiration. These results have implications for the use of European option pricing models in the valuation of American options.

Static hedging and pricing American options

This paper utilizes the static hedge portfolio (SHP) approach of Derman et al. [Derman, E., Ergener, D., Kani, I., 1995. Static options replication. Journal of Derivatives 2, 78–95] and Carr et al. [Carr, P., Ellis, K., Gupta, V., 1998. Static hedging of exotic options. Journal of Finance 53, 1165–1190] to price and hedge American options under the Black-Scholes (1973) model and the constant elasticity of variance (CEV) model of Cox [Cox, J., 1975. Notes on option pricing I: Constant elasticity of variance diffusion. Working Paper, Stanford University]. The static hedge portfolio of an American option is formulated by applying the value-matching and smooth-pasting conditions on the early exercise boundary. The results indicate that the numerical efficiency of our static hedge portfolio approach is comparable to some recent advanced numerical methods such as Broadie and Detemple [Broadie, M., Detemple, J., 1996. American option valuation: New bounds, approximations, and a comparison of existing methods. Review of Financial Studies 9, 1211–1250] binomial Black-Scholes method with Richardson extrapolation (BBSR). The accuracy of the SHP method for the calculation of deltas and gammas is especially notable. Moreover, when the stock price changes, the recalculation of the prices and hedge ratios of the American options under the SHP method is quick because there is no need to solve the static hedge portfolio again. Finally, our static hedging approach also provides an intuitive derivation of the early exercise boundary near expiration.

NEW NUMERICAL SCHEME FOR PRICING AMERICAN OPTION WITH REGIME-SWITCHING

This paper is concerned with regime-switching American option pricing. We develop new numerical schemes by extending the penalty method approach and by employing the θ-method. With regime-switching, American option prices satisfy a system of m free boundary value problems, where m is the number of regimes considered for the market. An (optimal) early exercise boundary is associated with each regime. Straightforward implementation of the θ-method would result in a system of nonlinear equations requiring a time-consuming iterative procedure at each time step. To avoid such complications, we implement an implicit approach by explicitly treating the nonlinear terms and/or the linear terms from other regimes, resulting in computationally efficient algorithms. We establish an upper bound condition for the time step size and prove that under the condition the implicit schemes satisfy a discrete version of the positivity constraint for American option values. We compare the implicit schemes with a tree model that generalizes the Cox-Ross-Rubinstein (CRR) binomial tree model, and with an analytical approximation solution for two-regime case due to Buffington and Elliott. Numerical examples demonstrate the accuracy and stability of the new implicit schemes.

Optimal convergence rate of the explicit finite difference scheme for American option valuation

An optimal convergence rate O ( Δ x ) for an explicit finite difference scheme for a variational inequality problem is obtained under the stability condition σ 2 Δ t Δ x 2 ⩽ 1 using completely PDE methods. As a corollary, a binomial tree scheme of an American put option (where σ 2 Δ t Δ x 2 = 1 ) is convergent unconditionally with the rate O ( ( Δ t ) 1 / 2 ) .

Lagrange Multiplier Approach with Optimized Finite Difference Stencils for Pricing American Options under Stochastic Volatility

The deterministic numerical valuation of American options under Heston's stochastic volatility model is considered. The prices are given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. A new truncation of the domain is described for small asset values, while for large asset values and variance a standard truncation is used. The finite difference discretization is constructed by numerically solving a quadratic optimization problem aiming to minimize the truncation error at each grid point. A Lagrange approach is used to treat the linear complementarity problems. Numerical examples demonstrate the accuracy and effectiveness of the proposed approach.

On the quadratic valuation of American call options: challenging the functional form

This paper extends the framework of semi-analytical approximations to the valuation of American options by exploring the performance of competing functional forms for the value of an American option. The value function of the early exercise premium is modelled as a product of two functions, one being a function of time and the other being a function of price. The accuracy of the suggested functional form is verified through numerical tests.

Improved lower and upper bound algorithms for pricing American options by simulation

This paper introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American-style options. For simulation algorithms that compute lower bounds of American option values, we apply martingale control variates and introduce the local policy enhancement, which adopts a local simulation to improve the exercise policy. For duality-based upper bound methods, specifically the primal–dual simulation algorithm, we have developed two improvements. One is sub-optimality checking, which saves unnecessary computation when it is sub-optimal to exercise the option along the sample path; the second is boundary distance grouping, which reduces computational time by skipping computation on selected sample paths based on the distance to the exercise boundary. Numerical results are given for single asset Bermudan options, moving window Asian options and Bermudan max options. In some examples the computational time is reduced by a factor of several hundred, while the confidence interval of the true option value is considerably tighter than before the improvements.

Valuation of American options by the gradient projection method

We study an equivalent optimization problem with an inequality constraint and boundary conditions, whose necessary condition for optimality is the variational inequality presentation of American options. To solve the problem, we use the gradient projection method, with discretizations both in time and space. We tested the algorithm and compared with the projective successive over-relaxation method.

Convergence analysis of a monotonic penalty method for American option pricing

This paper is devoted to study the convergence analysis of a monotonic penalty method for pricing American options. A monotonic penalty method is first proposed to solve the complementarity problem arising from the valuation of American options, which produces a nonlinear degenerated parabolic PDE with Black–Scholes operator. Based on the variational theory, the solvability and convergence properties of this penalty approach are established in a proper infinite dimensional space. Moreover, the convergence rate of the combination of two power penalty functions is obtained.

A FAST, STABLE AND ACCURATE NUMERICAL METHOD FOR THE BLACK–SCHOLES EQUATION OF AMERICAN OPTIONS

In this work we improve the algorithm of Han and Wu [SIAM J. Numer. Anal. 41 (2003), 2081–2095] for American Options with respect to stability, accuracy and order of computational effort. We derive an exact discrete artificial boundary condition (ABC) for the Crank–Nicolson scheme for solving the Black–Scholes equation for the valuation of American options. To ensure stability and to avoid any numerical reflections we derive the ABC on a purely discrete level. Since the exact discrete ABC includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate ABCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove a simple stability criteria for the approximated artificial boundary conditions. Finally, we illustrate the efficiency and accuracy of the proposed method on several benchmark examples and compare it to previously obtained discretized ABCs of Mayfield and Han and Wu.

Exercisability Randomization of the American Option

The valuation of American options is an optimal stopping time problem which typically leads to a free boundary problem. We introduce here the randomization of the exercisability of the option. This method considerably simplifies the problematic by transforming the free boundary problem into an evolution equation. This evolution equation can be transformed in a way that decomposes the value of the randomized option into a European option and the present value of continuously paid benefits. This yields a new binomial approximation for American options. We prove that the method is accurate and numerical results illustrate that it is computationally efficient.

Perpetual American options within CTRWs

Continuous-time random walks are a well suited tool for the description of market behaviour at the smallest scale: the tick-to-tick evolution. We will apply this kind of market model to the valuation of perpetual American options: derivatives with no maturity that can be exercised at any time. Our approach leads to option prices that fulfill financial formulas when canonical assumptions on the dynamics governing the process are made, but it is still suitable for more exotic market conditions.

Numerical Valuation of European and American Options under Kou's Jump-Diffusion Model

Numerical methods are developed for pricing European and American options under Kou's jump-diffusion model, which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is log-double-exponentially distributed. The price of a European option is given by a partial integro-differential equation (PIDE), while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differences on nonuniform grids, and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy-to-implement recursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a stationary iteration. For American options two ways to solve the LCPs are described: an operator slitting method and a penalty method. Numerical experiments confirm that the developed methods are very efficient, as fairly accurate option prices can be computed in a few milliseconds on a PC.

A Front-Fixing Finite Element Method for the Valuation of American Options

A front-fixing finite element method is developed for the valuation of American options on stocks. Stability and solution nonnegativity are established under some appropriate assumptions. Numerical results are presented to examine our method and to compare it with the other methods.

On the dangers of a simplistic American option simulation valuation method

Chen and Shen [Chen, A.-S., and P.-F. Shen. 2003. Computational complexity analysis of least-squares Monte Carlo (LSM) for pricing US derivatives. Applied Economics Letters 10: 223–9] argue that we can improve the least squares Monte Carlo method (LSMC) to value American options by removing the least squares regression module. This would make it not only faster but also more accurate. We demonstrate, using a large sample of 2500 put options, that the proposed algorithm – the perfect foresight method (PFM) – is, as argued by the authors, faster than the LSMC algorithm but, contrary to what they state, it is not more accurate than the LSMC. In fact, the PFM algorithm incorrectly prices American options. We therefore, do not recommend the use of the PFM.

On the Dangers of a Simplistic American Option Simulation Valuation Method

Chen and Shen (2003) argue that it is possible to improve the Least Squares Monte Carlo Method (LSMC) of Longstaff and Schwartz (2001) to value American options by removing the least squares regression module. This would make not only faster but also more accurate. We demonstrate, using a large sample of 2500 put options that the proposed algorithm - the Perfect Foresight Method (PFM) - is, as argued by the authors, faster than the LSMC algorithm but, contrary to what they state, it is not more accurate than the LSMC. In fact, the PFM algorithm incorrectly prices American options, resulting in an upward biased value of the option. We therefore, do not recommend the use of the PFM.

Valuation of life insurance surrender and exchange options

In this paper I analyze two American-type options related to life and pension insurance contract. I use Monte Carlo simulations combined with the Longstaff and Schwartz approach for the valuation of American options to find the value of a typical surrender option. I find that the values may be much lower than previously indicated. This reduction of value is due to a different treatment of bonuses, limiting the customers’ ability to forecast the return of their policies. The numerical results show that the value may be higher than the corresponding surrender option.