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

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Abstract

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.

Introduction

An exchange option gives its holder the right to exchange one asset for another. Earlier studies can be dated back to Fisher  [3] and Margrabe  [9] who derived the closed-form pricing formula for a European exchange option under the classical Black–Scholes model. Lindset  [7] extended the pricing formula to Merton’s jump–diffusion model  [10] and proposed to use the Geske–Johnson method  [4] to price its American version. One extension from the plain-vanilla exchange option is the power exchange option where nonlinear dependence is introduced by its power coefficients. As seen in Johnson and Tian  [6] and Blenman and Clark  [1], power exchange options provide more flexibility in the design of indexed executive stock options. Under the Black–Scholes model, the price of its European version was derived in closed form in  [1]. By using the martingale property of the underlying stock prices,  [1] also gave a sufficient condition under which its American version should never be exercised early.

The merit of this sufficient condition for the never-early-exercise (NEE) property is that it is model-free. However, the fact that it applies to a wide range of models also makes it a conservative condition. In fact, there are plenty of option parameters which actually lead to NEE but cannot be identified by this condition. In this note we consider a specific class of stock price models and show that the sufficient condition for NEE can be considerably weakened such that much more option parameters can be identified as never-early-exercise. In the model class of interest, we assume the power invariant property holds, meaning that the powered process of stock price remains in the same family as the original stock price process. This property enables us to introduce the power martingale condition and define the effective dividend yields which play important roles in the analysis of early exercise. A number of popular stock price models belong to this model class, including the Black–Scholes models, jump–diffusion models  [10], and variance gamma models  [8]. In fact, it contains all the exponential Levy models. A commonly used model not satisfying the power invariant property is the Heston stochastic volatility model  [5].

When the sufficient condition for NEE is met, the American power exchange option price must be equal to the price of its European version. Contrarily, if the condition is not satisfied, early exercise may be possible and it is of interest to discuss the value contributed by early exercise. Taking the perspective of effective dividend yields, the pricing problem can be reduced to its plain-vanilla version except that the dividend yields are no longer nonnegative. This gives rise to some new results which cannot be seen in the plain-vanilla setting. We derive upper bounds on the American power exchange option price and show that they take different forms according to the signs of effective dividend yields. These results enable us to see in what way the real-valued effective dividend yields affect the value and likelihood of early exercise. Moreover, they provide some computational implications for the pricing of American power exchange options. When an option’s parameters are in the NEE region (condition for NEE is met), it can be valued by standard European option pricing methods. When the parameters are outside the NEE region, the upper bounds (which may be evaluated in the same way as European options) provide an indication of whether it is worth incurring the computational cost of accurately pricing the American version.

To investigate the contribution of early exercise in a more accurate way, we apply the Geske–Johnson method  [4], [7] to conduct a numerical analysis which provides further observations. We find that the numerical NEE region generally covers an even wider range of option parameters than the theoretical condition would suggest. Through our numerical examples, we provide in-depth discussions on how early exercise is influenced by the two power coefficients.

Section snippets

Mathematical analysis

Consider the power exchange option which gives the payoff (S1tn1S2tn2)+ where S1t,S2t are stock price processes and the power coefficients n1,n2 are positive real numbers. The vanilla exchange option corresponds to the special case n1=n2=1. Denote the option maturity time as T and let r,q1,q20 respectively stand for the nonnegative interest rate and the dividend yields of the two stock price processes. For the market to be free of arbitrage, the stock prices must satisfy the martingale

Numerical analysis

The main insight of the preceding mathematical analysis is that early exercise is more (less) likely to happen if (Q1,Q2) moves away from (closer to) quadrant (II) (the NEE region) on the Q1Q2 plane. To further quantify the value of early exercise and its dependence on (Q1,Q2), this section provides a numerical analysis based on the Geske–Johnson (GJ) method  [4]. Let BPEk denote the price of a Bermudan power exchange option which is exercisable at k equally spaced time points over option

Acknowledgment

The authors acknowledge the support from the Ministry of Science and Technology of Taiwan under the grant number 101-2410-H-011-003.

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