American option valuation under stochastic interest rates

By applying Ho, Stapleton and Subrahmanyam's (1997, hereafter HSS) generalised Geske–Johnson (1984, hereafter GJ) method, this paper provides analytic solutions for the valuation and hedging of American options in a stochastic interest rate economy. The proposed method simplifies HSS's three-dimensional solution to a one-dimensional solution. The simulations verify that the proposed method is more efficient and accurate than the HSS (1997) method. We illustrate how the price, the delta, and the rho of an American option vary between the stochastic and non-stochastic interest rate models. The magnitude of this effect depends on the moneyness of the option, interest rates, volatilities of the underlying asset price and the bond price, as well as the correlation between them. This revised version was published online in June 2006 with corrections to the Cover Date.     

A new method for generating approximation algorithms for financial mathematics applications

This paper describes a new technique that can be used in financial mathematics for a wide range of situations where the calculation of complicated integrals is required. The numerical schemes proposed here are deterministic in nature but their proof relies on known results from probability theory regarding the weak convergence of probability measures. We adapt those results to unbounded payoffs under certain mild assumptions that are satisfied in finance. Because our approximation schemes avoid repeated simulations and provide computational savings, they can potentially be used when calculating simultaneously the price of several derivatives contingent on the same underlying. We show how to apply the new methods to calculate the price of spread options and American call options on a stock paying a known dividend. The method proves useful for calculations related to the log-Weibull model proposed recently for empirical asset pricing.     

American option pricing under the double Heston model based on asymptotic expansion

This paper focuses on pricing American put options under the double Heston model proposed by Christoffersen et al. By introducing an explicit exercise rule, we obtain the asymptotic expansion of the solution to the partial differential equation for pricing American put options. We calculate American option price by the sum of the European option price and the early exercise premium. The early exercise premium is calculated by the difference between the American and European option prices based on asymptotic expansions. The European option price is obtained by the efficient COS method. Based on the obtained American option price, the double Heston model is calibrated by minimizing the distance between model and market prices, which yields an optimization problem that is solved by a differential evolution algorithm combined with the Matlab function fmincon.m. Numerical results show that the pricing approach is fast and accurate. Empirical results show that the double Heston model has better performance in pricing short-maturity American put options and capturing the volatility term structure of American put options than the Heston model.     

Long term spread option valuation and hedging

This paper investigates the valuation and hedging of spread options on two commodity prices which in the long run are in dynamic equilibrium (i.e., cointegrated). The spread exhibits properties different from its two underlying commodity prices and should therefore be modelled directly. This approach offers significant advantages relative to the traditional two price methods since the correlation between two asset returns is notoriously hard to model. In this paper, we propose a two factor model for the spot spread and develop pricing and hedging formulae for options on spot and futures spreads. Two examples of spreads in energy markets – the crack spread between heating oil and WTI crude oil and the location spread between Brent blend and WTI crude oil – are analyzed to illustrate the results.     

On perpetual American put valuation and first-passage in a regime-switching model with jumps

In this paper we consider the problem of pricing a perpetual American put option in an exponential regime-switching Lévy model. For the case of the (dense) class of phase-type jumps and finitely many regimes we derive an explicit expression for the value function. The solution of the corresponding first-passage problem under a state-dependent level rests on a path transformation and a new matrix Wiener–Hopf factorization result for this class of processes. Research supported by the Nuffield Foundation, grant NAL/00761/G, and EPSRC grant EP/D039053/1.     

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.     

Truncation and acceleration of the Tian tree for the pricing of American put options

We present a new method for truncating binomial trees based on using a tolerance to control truncation errors and apply it to the Tian tree together with acceleration techniques of smoothing and Richardson extrapolation. For both the current (based on standard deviations) and the new (based on tolerance) truncation methods, we test different truncation criteria, levels and replacement values to obtain the best combination for each required level of accuracy. We also provide numerical results demonstrating that the new method can be 50% faster than previously presented methods when pricing American put options in the Black–Scholes model.     

On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives

This paper analyses the robustness of Least-Squares Monte Carlo, a technique proposed by Longstaff and Schwartz (2001) for pricing American options. This method is based on least-squares regressions in which the explanatory variables are certain polynomial functions. We analyze the impact of different basis functions on option prices. Numerical results for American put options show that this approach is quite robust to the choice of basis functions. For more complex derivatives, this choice can slightly affect option prices. This revised version was published online in June 2006 with corrections to the Cover Date.     

American bond option pricing in one-factor dynamic term structure models

Arbitrage-tree pricing of American options on bonds in one-factor dynamic term structure models is investigated. We re-derive a general decomposition result which states that the American bond option premium can be split into the value of an otherwise equivalent European option and anearly exercise premium. This extends earlier work on American equity options by e.g. Kim (1990), Jamshidian (1992) and Carr, Jarrow, and Myneni (1992) and parallels recent work by Jamshidian (1991, 1992, 1993) and Chesney, Elliott, and Gibson (1993). We examine a Gaussian class of special cases in some detail and provide a variety of numerical valuation results.An earlier version of the paper was entitled American Bond Option Pricing in One-Factor Spot Interest Rate Models.I am grateful for many helpful comments from two anonymous referees, the participants of the Second Nordic Symposium on Contingent Claims Analysis in Finance held in Bergen, Norway in May of 1994 and from the participants of the EIASM Doctoral Tutorial held in connection with the 1994 EFA annual meeting in Bruxelles. I am particularly indebted to Krishna Ramaswamy for his help and advice during my stay as visiting doctoral fellow at the Wharton School of the University of Pennsylvania. Financial support from the Aarhus University Research Foundation (Grants # E-1994-SAM-1-1-72 & E-1995-SAM-1-59), the Danish Social Science Research Council, and the Danish Research Academy is gratefully acknowledged. All errors and omissions are my own.     

The Valuation of Volatility Options

This paper examines the valuation of European- and American-style volatilityoptions based on a general equilibrium stochastic volatility framework.Properties of the optimal exercise region and of the option price areprovided when volatility follows a general diffusion process. Explicitvaluation formulas are derived in four particular cases. Emphasis is placedon the MRLP (mean-reverting in the log) volatility model which has receivedconsiderable empirical support. In this context we examine the propertiesand hedging behavior of volatility options. Unlike American options,European call options on volatility are found to display concavity at highlevels of volatility.     

Sequential American Exchange Property Options

Property development activities often occur in stages, which are appropriately modeled as sequential American exchange property options, where there are interim expenditures required in order to keep the property development options “alive”. Normally American exchange options require a numerical solution, but herein there is a new closed-form approximate solution, which is computationally efficient and accurate. This method combines repeats of Margrabe European exchange and Geske compound option solutions with tight upper boundaries of either American perpetuities or European exchange options with a high volatility. Illustrations are provided of the sensitivity of the real sequential options and optimal timing to changes in several parameters, which provide a framework for property policy (tax, subsidy and regulatory) guidelines and for property development strategy evaluation. There are several plausible applications of these real option models in commercial and residential property development, within commercial property leases, with regard to switching tenants, and agricultural alternatives.     

American Parisian options

In this article, we describe the various sorts of American Parisian options and propose valuation formulae. Although there is no closed-form valuation for these products in the non-perpetual case, we have been able to reformulate their price as a function of the exercise frontier. In the perpetual case, closed-form solutions or approximations are obtained by relying on excursion theory. We derive the Laplace transform of the first instant Brownian motion reaches a positive level or, without interruption, spends a given amount of time below zero. We perform a detailed comparison of perpetual standard, barrier and Parisian options.     

American GARCH employee stock option valuation

We implement a flexible simulation-based approach for the fair value of employee stock option (ESO) that accounts for the 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 perform a sensitivity analysis with respect to changes in several ESO characteristics. We compare this valuation with FAS 123 method revealing a FAS overvaluation. Finally, we value a real ESO plan providing the confidence intervals for the estimated ESO prices.     

Bivariate normal mixture spread option valuation

We discuss the pricing and hedging of European spread options on correlated assets when the marginal distribution of each asset return is assumed to be a mixture of normal distributions. Being a straightforward two-dimensional generalization of a normal mixture diffusion model, the prices and hedge ratios have a firm behavioural and theoretical foundation. In this ‘bivariate normal mixture’ (BNM) model no-arbitrage option values are just weighted sums of different ‘2GBM’ option values that are based on the assumption of two correlated lognormal diffusions, and likewise for their sensitivities. The main advantage of this approach is that BNM option values are consistent with both volatility smiles and with the implied correlation ‘frown’. No other ‘frown consistent’ spread option valuation model has such straightforward implementation. We apply analytic approximations to compare BNM valuations of European spread options with those based on the 2GBM assumption and explain the differences between the two as a weighted sum of six second-order 2GBM sensitivities. We also examine BNM option sensitivities, finding that these, like the option values, can sometimes differ substantially from those obtained under the 2GBM model. Finally, we show how the correlation frown that is implied by the BNM model is affected as we change (a) the correlation structure and (b) the tail probabilities in the joint density of the asset returns.     

American option pricing with discrete and continuous time models: An empirical comparison

This paper considers discrete time GARCH and continuous time SV models and uses these for American option pricing. We first of all show that with a particular choice of framework the parameters of the SV models can be estimated using simple maximum likelihood techniques. We then perform a Monte Carlo study to examine their differences in terms of option pricing, and we study the convergence of the discrete time option prices to their implied continuous time values. Finally, a large scale empirical analysis using individual stock options and options on an index is performed comparing the estimated prices from discrete time models to the corresponding continuous time model prices. The results show that, while the overall differences in performance are small, for the in the money put options on individual stocks the continuous time SV models do generally perform better than the discrete time GARCH specifications.     

Some calculations for Israeli options

Recently Kifer (2000) introduced the concept of an Israeli (or Game) option. That is a general American-type option with the added possibility that the writer may terminate the contract early inducing a payment exceeding the holders claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating a saddle point problem associated with Dynkin games. In this short text we give two examples of perpetual Israeli options where the solutions are explicit.Received: December 2002, Mathematics Subject Classification: 90A09, 60J40, 90D15JEL Classification: G13, C73I would like to express thanks to Chris Rogers for a valuable conversation.     

Market switching in shipping — A real option model applied to the valuation of combination carriers

This paper derives a real options model of flexibility and applies it to shipping, valuing the option to switch between the dry bulk market and wet bulk market for a combination carrier, a ship type that is capable of operating in both markets but that has fallen out of favor due to high price tags. The model is a mean-reverting (Ornstein–Uhlenbeck) version of a standard entry–exit model with stochastic prices. A closed form solution for the value of flexibility is derived, expressed in terms of Kummer functions. The estimated value of flexibility is related to historical price differentials between combination carriers and oil tankers of comparable size. Based on numerical experiments it is concluded that new combination carriers may enter the market in the near future.     

Real option valuation of abandoned farmland

This paper investigates the decision-making process of an owner of abandoned farmland that is currently restricted to agricultural use but will be available for nonagricultural use in the future. I make a real option valuation of the abandoned farmland taking prospective land conversion into consideration. Results from the model calibration demonstrate that a slight probability of land conversion greatly increases the land value and discourages the owner from cultivating the land. Further, the results show that a small gap in the anticipation of land conversion prevents the owner from selling or leasing the land to a more efficient farmer.