American Option Valuation Research Articles

Hedging housing price risks: some empirical evidence from the US

We analyze household hedging costs and market liquidity of exchange traded options on a set of well-developed U.S. home price indexes allowing homeowners to hedge the downside risk of housing prices. We estimate empirically the impact of hedging costs on market liquidity of housing derivatives using prices from Case–Shiller Home Price Index (CSI) futures options and Barone-Adesi and Whaley [Efficient analytic approximation of American option values. J. Finance, 1987, 42, 301–320] simulations. We find that hedging costs significantly affect household savings resulting from hedging. We propose a new cost-based illiquidity measure for housing derivatives and link it with traditional contract-based liquidity measures in thinly traded derivatives markets. We document a negative relation between savings from hedging and our cost-based illiquidity measure. We further perform a series of robustness checks. Overall we suggest that the liquidity of exchange traded housing derivatives could benefit U.S. homeowners.

REFLECTED BSDES WITH STOCHASTIC MONOTONE GENERATOR AND APPLICATION TO VALUING AMERICAN OPTIONS

In this paper, we prove the existence and uniqueness of the solution to backward stochastic differential equations with lower reflecting barrier in a Brownian setting under stochastic monotonicity and general increasing growth conditions. As an application, we study the fair valuation of American options.

Dynamic programming for valuing American options under a variance‐gamma process

AbstractLévy processes provide a solution to overcome the shortcomings of the lognormal hypothesis. A growing literature proposes the use of pure‐jump Lévy processes, such as the variance‐gamma (VG) model. In this setting, explicit solutions for derivative prices are unavailable, for instance, for the valuation of American options. We propose a dynamic programming approach coupled with finite elements for valuing American‐style options under an extended VG model. Our numerical experiments confirm the convergence and show the efficiency of the proposed methodology. We also conduct a numerical investigation that focuses on American options on S&P 500 futures contracts.

A new method of valuing American options based on Brownian models

This paper investigates the valuation of American option by developing a new technique for solving the optimal stopping problem modeled with Brownian motion. In particular, applying a martingale technique, the value function of option is approximated by specific basis functions. Compared with the traditional approaches, our method is more efficient without losing the accuracy. We show both the theoretical analysis and numerical implements.

Option Investor Rationality Revisited: The Role of Exercise Boundary Violations

Do option investors rationally exercise their options? Numerous studies report evidence of irrational behavior. In this paper, we pay careful attention to intraday option quotes and reach the opposite conclusion. An exercise boundary violation (EBV) occurs when the best bid price for an American option is below the option’s intrinsic value. Far from being unusual, we show that EBVs occur very frequently. Under these conditions, the rational response of an investor liquidating an option is to exercise the option rather than sell it. Empirically, we find that the likelihood of early exercise is strongly influenced by the existence and duration of EBVs. Not only do these results reverse standard theory on American option valuation and optimal exercise strategy, but they also suggest that the ability to avoid selling at an EBV price creates an additional source of value for American options that is unrelated, and in addition to, dividend payments. This additional value may help explain why American options appear overpriced relative to European options.

American Option Pricing: Optimal Lattice Models and Multidimensional Efficiency Tests

This paper introduces a set of lattice techniques with a view to accelerating computation time and improving the accuracy of American Option valuation. Estimation speed can be enhanced through developing a parsimonious early exercise boundary search routine combined with reliance on dynamic memory and lattice truncation. Furthermore, Black-Scholes and Richardson extrapolation modifications to the lattices can also be applied individually and/or together to improve the accuracy of lattices. In this paper, we investigate the improvement introduced by obtaining the best combination of varying features. By introducing these techniques to the Leisen-Reimer and Tian binomial model, we can achieve a level of accuracy and efficiency combined that surpass analytical analogues prominent in the literature. Significantly, the Leisen-Reimer and Tian structure can accommodate arbitrary improvements in accuracy by simply increasing the density of their own mesh. Analytical methods generally do not afford much scope for optimising speed and efficiency in a granular fashion. We also compare efficient lattice models with analytical formulae for pricing different groups of options according to the deepness of American quality and the moneyness of the options. The appropriate model is recommended for pricing particular types of the options. Lattices importantly afford an explicit trade-off locus between accuracy and speed that can be navigated according to predetermined precision tolerance levels and option types. This should have practical relevance to trading platforms that require real-time estimates of implied volatility.

A semi-analytic valuation of American options under a two-state regime-switching economy

In this study, we develop a semi-analytic method to evaluate American options under a two-state regime-switching economy. The two free boundaries corresponding to the states divide the pricing domain into two regions: a common continuation region and a transition region. Non-linear partial differential equation (PDE) systems are derived under the Black–Scholes framework for each region. The Laplace transform method is used to solve the PDE systems. Equations for determining the optimal exercise prices are obtained analytically and solved numerically in the Laplace space. A numerical inversion technique is then used to obtain the free boundaries and the option prices in the original time space. The results of various examples show that our technique is efficient and accurate.

A Method-of-lines Approach for Solving American Option Problems

The early exercise property of American option changes the original Black-Scholes equation to an inequality that cannot be solved via traditional finite difference method. Therefore, finding the early exercise boundary prior to spatial discretization is a must in each time step. This overhead slows down the computation and the accuracy of solution relies on if the early exercise boundary can be accurately located. A simple numerical method based on finite difference and method of lines is proposed here to overcome this difficulty in American option valuation. Our method averts the otherwise necessary procedure of locating the optimal exercise boundary before applying finite difference discretization. The method is efficient and flexible to all kinds of pay-off. Computations of American put, American call with dividend, American strangle, two-factor American basket put option, and two-factor convertible bond with embedded call and put options are demonstrated to show the efficiency of the current method.

Derivative pricing as a transport problem: MPDATA solutions to Black–Scholes-type equations

We discuss applications of the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) to numerically solve partial differential equations arising from stochastic models in quantitative finance. In particular, we develop a framework for solving Black–Scholes-type equations by transforming them into advection–diffusion problems. The equations are then numerically integrated backward in time using an iterative explicit finite-difference approach, in which the Fickian term is represented as an additional advective term. We discuss the correspondence between transport phenomena and financial models, uncovering the possibility of expressing the no-arbitrage principle as a conservation law. We show second-order accuracy in time and space of the embraced numerical scheme. This is done via a convergence analysis comparing MPDATA numerical solutions with classic Black–Scholes analytical formulæ for the valuation of European options. We demonstrate in addition a way of applying MPDATA to solve the free boundary problem (leading to a linear complementarity problem) for the valuation of American options. We finally comment on the potential of MPDATA methods with respect to more complex models typically used in quantitative finance.

A finite volume–alternating direction implicit method for the valuation of American options under the Heston model

ABSTRACTA finite volume–alternating direction implicit method is proposed for numerical valuation of the American options under the Heston model. It is based on decoupling correlated stock price process and volatility process so that corresponding partial differential operator does not contain the mixed partial derivative term. Hence, the proposed method is numerically simple and fast. Numerical results are presented to examine the accuracy of the proposed method and to compare it with the others.

Deep Learning-Based Least Square Forward-Backward Stochastic Differential Equation Solver for High-Dimensional Derivative Pricing

We propose a new forward-backward stochastic differential equation solver for highdimensional derivative pricing problems by combining deep learning solver with least square regression technique widely used in the least square Monte Carlo method for the valuation of American options. Our numerical experiments demonstrate the accuracy of our least square backward deep neural network solver and its capability to produce accurate prices for complex early exercise derivatives, such as callable yield notes. Our method can serve as a generic numerical solver for pricing derivatives across various asset groups, in particular, as an accurate means for pricing high-dimensional derivatives with early exercise features.

Optimal Seismic Upgrade Timing in Seaports with Increasing Throughput Demand via Real Options

A real options (RO) formulation is proposed for decision-making on the timing to upgrade the seismic performance of existing seaports with increasing throughput demand in earthquake prone areas. The pay-off of the seismic upgrade investment option is estimated based on projected net earnings, repair cost, and downtime for a damaging reference seismic event having a pre-specified annual probability of occurrence. These projections inform a discrete-time RO binomial tree, following the American option valuation framework, which propagates the probability of the reference seismic event assuming Poisson temporal distribution of earthquake occurrence. The net present value of the expected annual payoff of the considered investment is used as an index supporting risk-informed decision-making discounted by the weighted average cost of capital (WACC). Numerical examples pertaining to decision makers with different capital cost, namely port authorities and terminal operators, operating in different economic environments typical of developed and developing countries are furnished to illustrate the applicability of the proposed RO formulation. It is found that high WACC and/or low throughput growth bring the optimal seismic upgrade timing forward, while earthquake consequences and upgrade cost have almost no influence on this timing.

Analytical Approximations for American Options: The Binary Power Option Approach

This study proposes an innovative approach to value American options. Using a portfolio of binary power options to replicate the early exercise premium, we modify Medvedev and Scaillet (2010) to derive an analytical approximation of American option values under the Black-Scholes framework. Compared with Medvedev and Scaillet (2010), our approach provides a much simpler functional form of the early exercise premium that can be easily extended to high-order series expansions. The numerical results show that the pricing performance of our method is closely comparable to that of Medvedev and Scaillet (2010) and superior to that of Barone-Adesi and Whaley (1987). Key words: American option, binary power option, early exercise premium

An Efficient Algorithm for Options Under Merton’s Jump-Diffusion Model on Nonuniform Grids

In this paper, we consider the fast numerical valuation of European and American options under Merton’s jump-diffusion model, which is given by a partial integro-differential equations. Due to the singularities and discontinuities of the model, the time-space grids are nonuniform with refinement near the strike price and expiry. On such nonuniform grids, the spatial differential operators are discretized by finite difference methods, and time stepping is performed using the discontinuous Galerkin finite element method. Owing to the nonuniform grids, algebraic multigrid method is used for solving the dense algebraical system resulting from the discretization of the integral term associated with jumps in models, which is more challenging. Numerical comparison of algebraic multigrid, the generalized minimal residual method, and the incomplete LU preconditioner shows that algebraic multigrid method is superior to and more effective than the other two methods in solving such dense algebraical system.

Power Penalty Approach to American Options Pricing Under Regime Switching

This work aims at studying a power penalty approach to the coupled system of differential complementarity problems arising from the valuation of American options under regime switching. We introduce a power penalty method to approximate the differential complementarity problems, which results in a set of coupled nonlinear partial differential equations. By virtue of variational inequality theory, we establish the unique solvability of the system of differential complementarity problems. Moreover, the convergence property of this power penalty method in an appropriate infinite-dimensional space is explored, where an exponential convergence rate of the power penalty method is established and the monotonic convergence of the penalty method with respect to the penalty parameter is shown. Finally, some numerical experiments are presented to verify the convergence property of the power penalty method.

The valuation of American options in a multidimensional exponential Lévy model

AbstractWe 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.

American GARCH Employee Stock Option Valuation

We implement a new and flexible simulation based approach for the fair value of Employee Stock Option (ESO) plans to value American options in order to consider the common ESO characteristics of vesting period, departure risk and voluntary suboptimal early exercise. We introduce GARCH effects on the underlying asset and we analyze the price bias with respect to the constant volatility case. We also find a significant bias if we value the ESO using a short-memory GARCH model instead of a long-memory one. We also develop a sensitivity analysis with respect to changes in several ESO characteristics. We compare this valuation with FAS 123 method using the implicit expected life of the ESO revealing a FAS overvaluation. Finally, we value a real ESO plan providing the confidence intervals for the estimated ESO prices.

A generalized Brennan–Rubinstein approach for valuing options with stochastic interest rates

We construct a discrete-time option valuation model capable of taking into consideration multiple exercises and stochastic interests rates under a generalized Brennan–Rubinstein framework, and further apply the Geske and Johnson (1984) method for the valuation of American options. For implementation, we use only the once- and twice-exercisable option values to approximate American option values. Our numerical results show that the effects of stochastic interest rates are very large, particularly for out-of-the-money American put options with a high-risk underlying asset price. For options with a high-risk return on asset as the underlying assets, the effects of stochastic interest rates are negligible, even for options with long-term maturity. Finally, the sign of correlation between asset prices and bond prices plays an important role in determining whether the values of American put options under stochastic interest rates are larger or smaller than those under constant interest rates.

Gaussian Quadrature Method for Pricing American and Exotic Options in a Jump-Diffusion Process

In this paper we propose a Gaussian quadrature method to study American and exotic option pricing under the jump-diffusion model of Merton (1976). Our numerical experiments show that the Gaussian quadrature method, compared to several existing methods in the literature, including the fast Gauss transform method (Broadie and Yamamoto, 2003), the bivariate tree approach (Hilliard and Schwartz, 2005), and the extrapolation approach (Feng and Linetsky, 2008), is accurate for valuing American options. In addition to American options, we also show that the Gaussian quadrature method performs well for the valuation of exotic options under the jump-diffusion model. Overall, the Gaussian quadrature method is highly accurate and suitable for the valuation of price options with early exercise features under a jump-diffusion process.

A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation

A fast preconditioned policy iteration method is proposed for the Hamilton–Jacobi–Bellman (HJB) equation involving tempered fractional order partial derivatives, governing the valuation of American options whose underlying asset follows exponential Lévy processes. An unconditionally stable upwind finite difference scheme with shifted Grünwald approximation is first developed to discretize the established HJB equation under the tempered fractional diffusion models. Next, the policy iteration method as an outer iterative method is utilized to solve the discretized HJB equation and proven to be convergent in finite steps to its numerical solution. Given the Toeplitz-like structure of the coefficient matrix in each policy iteration, the resulting linear system can be fast solved by the Krylov subspace method as an inner iterative method via fast Fourier transform (FFT). Furthermore, a novel preconditioner is proposed to speed up the convergence rate of the inner Krylov subspace iteration with theoretical analysis to ensure the linear system can be solved in O(NlogN) operations under some mild conditions, where N is the number of spatial node points. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed fast preconditioned policy method.