American Option Valuation Research Articles

American option pricing with imprecise risk-neutral probabilities

The aim of this paper is to price an American option in a multiperiod binomial model, when there is uncertainty on the volatility of the underlying asset. American option valuation is usually performed, under the risk-neutral valuation paradigm, by using numerical procedures such as the binomial option pricing model of Cox et al. [J.C. Cox, S.A. Ross, S. Rubinstein, Option pricing, a simplified approach, Journal of Financial Economics 7 (1979) 229–263]. A key input of the multiperiod binomial model is the volatility of the underlying asset, that is an unobservable parameter. As it is hard to give a precise estimate for the volatility, in this paper we use a possibility distribution in order to model the uncertainty on the volatility. Possibility distributions are one of the most popular mathematical tools for modelling uncertainty. The standard risk-neutral valuation paradigm requires the derivation of the risk-neutral probabilities, that in a one-period binomial model boils down to the solution of a linear system of equations. As a consequence of the uncertainty in the volatility, we obtain a possibility distribution on the risk-neutral probabilities. Under these measures, we perform the risk-neutral valuation of the American option.

Valuation of vulnerable American options with correlated credit risk

This article evaluates vulnerable American options based on the two-point Geske and Johnson method. In accordance with the Martingale approach, we provide analytical pricing formulas for European and multi-exercisable options under risk-neutral measures. Employing Richardson’s extrapolation gets the values of vulnerable American options. To demonstrate the accuracy of our proposed method, we use numerical examples to compare the values of vulnerable American options from our proposed method with the benchmark values from the least-square Monte Carlo simulation method. We also perform sensitivity analyses for vulnerable American options and show how the prices of vulnerable American options vary with the correlation between the underlying assets and the option writer’s assets.

On the Quadratic Valuation of American Options: A Functional Guess

This paper extends the framework of analytical approximations to the valuation of American options by suggesting alternative forms for the value function of the American option. We model the value function of the early exercise premium as a product of two functions, one of which being a function of time and the other being a function of price and test the accuracy of an alternative functional forms for the time-function. The tested functional form leads to fast and accurate valuation of options with early exercise opportunities.

Richardson extrapolation techniques for the pricing of American‐style options

AbstractIn 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

Robust numerical valuation of European and American options under the CGMY process

We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Lévy process. For processes of finite variation, quadratic convergence is obtained as the mesh and timestep are refined. For infinite variation processes, better than first-order accuracy is achieved. The jump component in the neighborhood of log jump size zero is specially treated by using a Taylor expansion approximation and the drift term is dealt with using a semi-Lagrangian scheme. The resulting partial integro-differential equation is then solved using a preconditioned BiCGSTAB method coupled with a fast Fourier transform. Proofs of fully implicit timestepping stability and monotonicity are provided. The convergence properties of the BiCGSTAB scheme are discussed and compared with a fixed point iteration. Numerical tests showing the convergence and performance of this method for European and American options under processes of finite and infinite variation are presented.

OPTIMAL TIMING FOR INVESTMENT DECISIONS

The net present value (NVP) is an important concept in investment decisions. As Ingersoll and Ross [7] have pointed out, the future fluctuation of interest rates is expected to have significant effects on the present value (PV) of the project concerned. If interest rates are expected to fall off in the next year, deferring an investment for yet another year is likely to be more gainful even if its current NPV is positive. The effects of deferment can be valued from its corresponding American option value. Berk [1] proposed a simple criterion for investment decisions which incorporate this American option value of investment. The simplicity of this model is obtained from the appropriate usage of a callable bond. It is admirable that this model does not postulate any assumptions on the behavior of interest rates. But this construction of the model has the pros and cons. It is easy to implement this model in business because the only adjustment required in this model is to replace the interest rate in NPV with the callable rate. On the other hand, the properties of this criterion have not been clarified. In this paper we analyze Berk's model under the assumption that interest rates follow the geometric Brownian motion (GBM). By assuming the movement of interest rates, we can derive an analytical solution for the optimal timing for the investments in terms of the parameters of the GBM. This enables us to perform comparative statics and simulation. These results extract some properties of Berk's model and help the decision makers in implementing Berk's model.

B‐Spline‐Based Monotone Multigrid Methods

For the efficient numerical solution of elliptic variational inequalities on closed convex sets, multigrid methods based on piecewise linear finite elements have been investigated over the past decades. Essential to their success is the appropriate approximation of the constraint set on coarser grids which is based on function values for piecewise linear finite elements. On the other hand, there are a number of problems which profit from higher order approximations. Among these are the problem of pricing American options, formulated as a parabolic boundary value problem involving Black–Scholes’ equation with a free boundary. In addition to computing the free boundary (the optimal exercise price of the option) of particular importance are accurate pointwise derivatives of the value of the stock option up to order two, the so‐called Greek letters. In this paper, we propose a monotone multigrid method for discretizations in terms of B‐splines of arbitrary order to solve elliptic variational inequalities on a closed convex set. In order to maintain monotonicity (upper bound) and quasi optimality (lower bound) of the coarse grid corrections, we propose an optimized coarse grid correction (OCGC) algorithm which is based on B‐spline expansion coefficients. We prove that the OCGC algorithm is of optimal complexity of the degrees of freedom of the coarse grid and, therefore, the resulting monotone multigrid method is of asymptotically optimal multigrid complexity. Finally, the method is applied to a standard model for the valuation of American options. In particular, it is shown that a discretization based on B‐splines of order four enables us to compute the second derivative of the value of the stock option to high precision.

Valuing Employee Stock Options Through Simulations

This article propose a new and flexible approach to obtain the fair value of Employee Stock Options (ESOs) plans using simulations. We adapt the algorithm of Longstaff and Schwartz (2001) to value American options in order to consider common characteristics of ESOs plans like vesting period, departure risk or voluntary early exercise. Moreover, as an example of the flexibility of the simulation approach, we show how to introduce GARCH effects of the ESO in the valuation method and compute the price bias for a hypothetical ESO.

On Stability and Convergence of a Finite Difference Approximation to a Parabolic Variational Inequality Arising From American Option Valuation

In this article we study the numerical solution of a degenerate variational inequality of parabolic type arising from the valuation of American options by a finite difference method. Stability and convergence of the finite difference method are established. Numerical results are also presented to show the effectiveness and usefulness of the discretization scheme.

Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation

In this paper, we present a power penalty function approach to the linear complementarity problem arising from pricing American options. The problem is first reformulated as a variational inequality problem; the resulting variational inequality problem is then transformed into a nonlinear parabolic partial differential equation (PDE) by adding a power penalty term. It is shown that the solution to the penalized equation converges to that of the variational inequality problem with an arbitrary order. This arbitrary-order convergence rate allows us to achieve the required accuracy of the solution with a small penalty parameter. A numerical scheme for solving the penalized nonlinear PDE is also proposed. Numerical results are given to illustrate the theoretical findings and to show the effectiveness and usefulness of the method.

A NEW ANALYTICAL APPROXIMATION FORMULA FOR THE OPTIMAL EXERCISE BOUNDARY OF AMERICAN PUT OPTIONS

In this paper, a new analytical formula as an approximation to the value of American put options and their optimal exercise boundary is presented. A transform is first introduced to better deal with the terminal condition and, most importantly, the optimal exercise price which is an unknown moving boundary and the key reason that valuing American options is much harder than valuing its European counterparts. The pseudo-steady-state approximation is then used in the performance of the Laplace transform, to convert the systems of partial differential equations to systems of ordinary differential equations in the Laplace space. A simple and elegant formula is found for the optimal exercise boundary as well as the option price of the American put with constant interest rate and volatility. Other hedge parameters as the derivatives of this solution are also presented.

Analytical Approximations to the Valuation of American Options: A Boundary-Optimality Approach

The quadratic approximation to the valuation of american options on stocks is revisited, constructing a pricing approach based on the fact that the early exercise policy should be chosen to maximize the value of the option. At the first part of the paper, we apply this approach (boundary-optimality) in the setting of the pricing model suggested in Barone-Adesi and Whaley (1987). We enrich their original valuation setting with an additional parameter, computed with the help of a boundary-optimality boundary condition. This approach enhances the accuracy performance of the Barone-Adesi and Whaley (1987) approximation. In the second part of the paper we introduce a novel approximation approach, where option value is the product of two functions, one of the being a function of time and the other one being a function of the stock price. Applying the principle that the early exercise policy should maximize option value, this alternative option pricing technique provides accurate results for american call and put options.

An exact and explicit solution for the valuation of American put options

In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, is the optimal exercise boundary being elegantly and temporarily removed in the solution process of each order, and, consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical and exact series-expansion solution for the optimal exercise boundary and the option price of American put options.

Monte Carlo Estimation of a Joint Density Using Malliavin Calculus, and Application to American Options

We use the Malliavin integration by parts formula in order to provide a family of representations of the joint density (which does not involve Dirac measures) of (X_?, X ? + ?), where X is a d-dimensional Markov diffusion (d ? 1), ? > 0 and ? > 0. Following Bouchard et al. (2004), the different representations are determined by a pair of localizing functions. We discuss the problem of variance reduction within the family of separable localizing functions: We characterize a pair of exponential functions as the unique integrated-variance minimizer among this class of separable localizing functions. We test our method on the d-dimensional Brownian motion and provide an application to the problem of American options valuation by the quantization tree method introduced by Bally et al. (2002).

A Simple Iterative Method for the Valuation of American Options

We introduce a simple iterative method to determine the optimal exercise boundary for American options, allowing us to compute the values of American options and their Greeks quickly and accurately. Following Little, Pant, and Hou’s (2000) idea, we derive a new equation for the optimal exercise boundary containing a single integral. The proposed method is an iterative numerical method for finding its solution. Using it, we can calculate the entire optimal exercise boundary in a non-time-recursive way, while conventional methods do not. Extensive numerical results indicate that our method is computationally more efficient than the methods currently available, particularly when finding hedge ratios.

Optimal Early Retirement Near the Expiration of a Pension Plan

In a recent paper, Friedman and Shen (Finance Stoch 6: 273–302, 2002) have considered a pension plan with the option of early retirement. They showed that the financial value V of the retirement benefits is the solution of a variational inequality and have studied the associated free boundary. A description of the free boundary near maturity is given, thanks to integral equation methods. However, V is also the solution of an optimal stopping problem very close to the American option valuation problem. Comparing V to specific options, we derive an expansion of the free boundary near the expiration of the pension plan.

Finite volume methods for the valuation of American options

We consider the use of finite volume methods for the approximation of a parabolic variational inequality arising in financial mathematics. We show, under some regularity conditions, the convergence of the upwind implicit finite volume scheme to a weak solution of the variational inequality in a bounded domain. Some results, obtained in comparison with other methods on two dimensional cases, show that finite volume schemes can be accurate and efficient.

A non-lattice pricing model of American options under stochastic volatility

In this article, an analytical approach to American option pricing under stochastic volatility is provided. Under stochastic volatility, the American option value can be computed as the sum of a corresponding European option price and an early exercise premium. By considering the analytical property of the optimal exercise boundary, the formula allows for recursive computation of the American option value. Simulation results show that a nonlattice method performs better than the lattice-based interpolation methods. The stochastic volatility model is also empirically tested using S&P 500 futures options intraday transactions data. Incorporating stochastic volatility is shown to improve pricing, hedging, and profitability in actual trading. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:417–448, 2006

On the Quadratic Approximation to the Value of American Options: An Alternative

The valuation of put options with early-exercise opportunities constitutes a major challenge of asset pricing. The option-theoretic response to this challenge relies on the estimation of the optimal exercise boundary. This paper introduces a novel quadratic approximation for the valuation of American options on common stock. The paper’s contribution lies in the tradition of semi-analytical approximation of American put options, which was put forward in Barone-Adesi and Whaley (1987). Assuming that the interest rate and the volatility are constant, the early-exercise premium is modeled as a product of two functions, one being a function of time and the other being a function of the stock price. The numerical results demonstrate the accuracy of the method, over competing alternatives such as the Barone-Adesi and Whaley (1987) algorithm.

A posteriori error analysis for FEM of American options

In this paper, we present the a posteriori error analysis for thefinite element approximation of American option valuation problems. We introduce an efficient and reliable error estimator both for the semi discrete andfully discrete backward Euler scheme.