American Exchange Options Research Articles

Primal-Dual Active-Set Method for the Valuation of American Exchange Options

Primal-Dual Active-Set Method for the Valuation of American Exchange Options

Approximate pricing of American exchange options with jumps

AbstractThis paper presents a numerical method to price American exchange options based on jump‐diffusion processes. We first derive a closed‐form expression for the value of European exchange options, then decompose the value function of an American exchange option into a European counterpart, and an early exercise premium that is derived analytically. The early exercise boundary for an American exchange option approximately satisfies an algebraic equation that can be quickly numerically solved. Consequently, a formula is obtained for efficiently pricing American exchange options. The numerical results reveal that our pricing formula is robust and accurate.

A Neural Network Approach to Value R&D Compound American Exchange Option

In this paper we show as the neural network methodology, coupled with the Least Squares Monte Carlo approach, can be very helpful in valuing R&D investment opportunities. As it is well known, R&D projects are made in a phased manner, with the commencement of subsequent phase being dependent on the successful completion of the preceding phase. This is known as a sequential investment and therefore R&D projects can be considered as compound options. In addition, R&D investments often involve considerable cost uncertainty so that they can be viewed as an exchange option, i.e. a swap of an uncertain investment cost for an uncertain gross project value. Finally, the production investment can be realized at any time before the maturity date, after that the effects of R&D disappear. Consequently, an R&D project can be considered as a compound American exchange option. In this context, the Least Squares Monte Carlo method is a powerful and flexible tool for capital budgeting decisions and for valuing American-type options. But, using the simulated values as “targets”, the implementation of a neural network allows to extend the results for any R&D valuation and to abate the waiting time of Least Squares Monte Carlo simulation.

A numerical approach to pricing exchange options under stochastic volatility and jump-diffusion dynamics

We consider a method of lines (MOL) approach to determine prices of European and American exchange options when underlying asset prices are modeled with stochastic volatility and jump-diffusion dynamics. As with any other numerical scheme for partial differential equations (PDEs), the MOL becomes increasingly complex when higher dimensions are involved, so we first simplify the problem by transforming the exchange option into a call option written on the ratio of the yield processes of the two assets. This is achieved by taking the second asset yield process as the numéraire. Under the equivalent martingale measure induced by this change of numéraire, we derive the exchange option pricing integro-partial differential equations (IPDEs) and investigate the early exercise boundary of the American exchange option. We then discuss a numerical solution of the IPDEs using the MOL, its implementation using computing software and possible alternative boundary conditions at the far limits of the computational domain. Our analytical and numerical investigation shows that the near-maturity behavior of the early exercise boundary of the American exchange option is significantly influenced by the dividend yields and the presence of jumps in the underlying asset prices. Furthermore, with the numerical results generated by the MOL, we are able to show that key jump and stochastic volatility parameters significantly affect the early exercise boundary and exchange option prices. Our numerical analysis also verifies that the MOL performs more efficiently, than other finite difference methods or simulation approaches for American options, since the MOL integrates the computation of option prices, greeks and the early exercise boundary, and does so with the least error.

Representation of exchange option prices under stochastic volatility jump-diffusion dynamics

In this article, we provide representations of European and American exchange option prices under stochastic volatility jump-diffusion (SVJD) dynamics following models by Merton [Option pricing when underlying stock returns are discontinuous. J. Financ. Econ., 1976, 3(1-2), 125–144], Heston [A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud., 1993, 6(2), 327–343], and Bates [Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Rev. Financ. Stud., 1996, 9(1), 69–107]. A Radon–Nikodým derivative process is also introduced to facilitate the shift from the objective market measure to other equivalent probability measures, including the equivalent martingale measure. Under the equivalent martingale measure, we derive the integro-partial differential equation that characterizes the exchange option prices. We also derive representations of the European exchange option price using the change-of-numéraire technique proposed by Geman et al. [Changes of numéraire, changes of probability measure and option pricing. J. Appl. Probab., 1995, 32(2), 443–458] and the Fourier inversion formula derived by Caldana and Fusai [A general closed-form spread option pricing formula. J. Bank. Finance, 2013, 37, 4893–4906], and show that these two representations are comparable. Lastly, we show that the American exchange option price can be decomposed into the price of the European exchange option and an early exercise premium.

A note on the never-early-exercise region of American power exchange options

This note discusses how the never-early-exercise region of American power exchange options is influenced by the nonlinearity from its power coefficients. We consider a class of models which satisfy the power invariant property and show that early exercise depends crucially on the quantities termed effective dividend yields. Our mathematical analysis extends an existing model-free result and indicates how early exercise should depend on parameters. A numerical analysis is conducted to complement the analytical results and provide further observations.

Perpetual Exchange Options under Jump-Diffusion Dynamics

This paper provides a pricing formula for perpetual exchange options, where the dynamics of the underlying assets are driven by jump-diffusion processes. It is an extension of Gerber and Shiu, and also Wong, who have priced perpetual exchange options under the pure-diffusion setting, and that of Gerber and Shiu, who have also considered perpetual options on single assets under jump-diffusion dynamics. It complements the results of Cheang and Chiarella, who derive a probabilistic representation of the American exchange option price under jump-diffusion dynamics.

On the valuation of American exchange options: an analytical approximation

We adopt a quadratic approach to the valuation of the option to exchange one asset for another, when the option owner has the right to exercise before option expiration. Accurate pricing results are obtained and tested against competitive models in the literature, building on the hypothesis that option value is the product of two functions, one being a function of time and the other one being a function of the stock prices.

SOLVING A TWO VARIABLES FREE BOUNDARY PROBLEM ARISING IN A PERPETUAL AMERICAN EXCHANGE OPTION PRICING MODEL

We investigate an American exchange option (AEO) pricing problem. Under the perfect market assumption, an AEO pricing problem can be modeled as a free boundary problem (FBP). The FBP is converted into an integral equation by using the Green’s function. When the expiration date tends to infinity, we obtain a time-invariant constant of the exercise boundary. Moreover, we provide a pricing formula for valuating the early exercise premium of the perpetual AEO.

An Analytic Formula for the Price of American Exchange Options

In this article we demononstrate how to find an analytic solution to the pricing of American exchange options using homotogy methods. As an aside we derive formula for American call and put options when the underlying pays a continuous dividend. These solutions are given in terms of power series.

The generalized perpetual American exchange-option problem

This paper revisits a general optimal stopping problem that often appears as a special case in some finance applications. The problem is essentially of the same form as the investment-timing problem of McDonald and Siegel (1986) in which the underlying processes are two correlated geometric Brownian motions (GBMs) with drifts less than the discount rate. By contrast, we attempt to analyze the underlying optimal stopping problem to its full generality without imposing any restriction on the drifts of the GBMs. By extending the first passage time approach of Xia and Zhou (2007) to the current context, we manage to obtain a complete and explicit characterization of the solution to the problem on all possible drift domains. Our analysis leads to a new and interesting observation that the underlying optimal stopping problem admits a two-sided optimal continuation region on some certain parameter domains.

Generalized Analytical Upper Bounds for American Option Prices

AbstractThis paper generalizes and tightens Chen and Yeh's (2002) analytical upper bounds for American options under stochastic interest rates, stochastic volatility, and jumps, where American option prices are difficult to compute with accuracy. We first generalize Theorem 1 of Chen and Yeh (2002) and apply it to derive a tighter upper bound for American calls when the interest rate is greater than the dividend yield. Our upper bounds are not only tight, but also converge to accurate American call option prices when the dividend yield or strike price is small or when volatility is large. We then propose a general theorem that can be applied to derive upper bounds for American options whose payoffs depend on several risky assets. As a demonstration, we utilize our general theorem to derive upper bounds for American exchange options and American maximum options on two risky assets.

Pricing American exchange options in a jump‐diffusion model

AbstractA way to estimate the value of an American exchange option when the underlying assets follow jump‐diffusion processes is presented. The estimate is based on combining a European exchange option and a Bermudan exchange option with two exercise dates by using Richardson extrapolation as proposed by R. Geske and H. Johnson (1984). Closed‐form solutions for the values of European and Bermudan exchange options are derived. Several numerical examples are presented, illustrating that the early exercise feature may have a significant economic value. The results presented should have potential for pricing over‐the‐counter options and in particular for pricing real options. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:257–273, 2007

On the Valuation of American Exchange Options: An Analytical Approximation

We adopt a quadratic approach to the valuation of the option to exchange one asset for another, when the option owner has the right to exercise prior to option expiration. Accurate pricing results are obtained and tested against competitive models in the literature, building on the hypothesis that option value is the product of two functions, one being a function of time, and the other one being a function of the stock prices.

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.

A modified finite-lived American exchange option methodology applied to real options valuation

Investment opportunities that are deferrable over a finite period of time are finite-lived American exchange options. Although such investment opportunities can be valued using the Carr [Carr, P., 1988, The Valuation of Sequential Exchange Opportunities, Journal of Finance 43:5, 1235–1256.] model, the derivation of this model is not very accurate when correctly formulated. Furthermore, past applications (e.g., [Taudes, A. 1998, Software Growth Options, Journal of Management Information Systems 15: 1, 165–185.]) have implemented the model without correcting for an important typo in the Carr paper. While such investment opportunities are more accurately valued using the Carr [Carr, P. 1995, The Valuation of American Exchange Options with Application to Real Options, in: L.Trigeorgis, ed, Real Options in Capital Investment: Models, Strategies and Applications (Praeger, Westport, Connecticut, London).] model, this model suffers from the problem known as “non-uniform convergence”. This paper proposes a modified approach for estimating the Carr [Carr, P., 1988, The Valuation of Sequential Exchange Opportunities, Journal of Finance 43:5, 1235–1256.] model that emits more accurate output values with a minimal addition of mathematical and computational cost. The paper then demonstrates the superiority of this modified model for three real investment opportunities.

Valuation of expansion flexibility in flexible manufacturing system investments using sequential exchange options

In this paper, a valuation approach that modifies traditional discounted cash flow (DCF) methodology is presented to incorporate the option premium of expansion flexibility while evaluating a flexible manufacturing system investment. Expansion flexibility allows for changing the production capacity in response to deviations in demand, while disregarding the assumption that management makes an irrevocable decision based on its future market expectations. Expansion flexibility provides a key strategic advantage by avoiding the large financial commitment at the initial investment stage and enabling investment in a phased manner according to the changes in market conditions. The aim of this paper is to present a thorough valuation methodology that accounts for both the benefits of keeping the option to expand alive and the loss of market share to competing firms if the expansion investment were delayed. In this paper, the value of expansion flexibility is computed using sequential exchange options. The proposed method employs an analytic approximation scheme for valuing American exchange options on dividend-paying assets. A numerical example demonstrates the application of the valuation framework. The options approach that incorporates the expansion flexibility option into the analysis results in a higher value than the standard DCF approach, which ignores the value of the option to expand. We also perform sensitivity analyses to see whether the expansion option increases in value in cases of high uncertainty where management can respond flexibly to new market information, and where the investment analysis ignoring flexibility yields a marginally positive net present value. The results of the proposed valuation framework are presented in comparison with a previous model that ignores the opportunity cost of delaying expansion investment.

On the option valuation and decomposition of exchange option

In this paper, we shall find the unique rational price associated with the exchange option. Also, we find the decomposition of Snell envelope and value function of the American exchange option.

An Options Approach to Valuing Expansion Flexibility in Flexible Manufacturing System Investments

ABSTRACT Flexibility in manufacturing processes provides an ability to change or even reverse the decisions made in earlier periods. The traditional economic evaluation methods of investments in flexible manufacturing systems ignore the value of flexibility, which should be one of the key issues in the justification process. Options approach appears as a means of overcoming the limitations of conventional discounted cash flow methods. In this work, a methodology for valuing expansion flexibility of flexible manufacturing systems is presented. Expansion flexibility in a phased manufacturing investment can be valued by viewing an initial investment as being analogous to purchasing an option to exchange one risky asset for another risky asset within a time period from the initial investment. While keeping the option to expand is of value, a thorough analysis requires that the opportunity cost of delaying expansion be taken into account. In this paper, an analytic approximation methodology for valuing sequential American exchange options on dividend paying stocks is employed for valuing expansion flexibility. A comprehensive numerical example is presented to illustrate the approach, and sensitivity analyses are performed.