This paper values American options on foreign assets in a stochastic interest rate economy using a two-point Geske and Johnson (1984) technique. The method requires the valuation of just two options: a European option and a twice-exercisable option. I first derive the risk-neutral distributions of asset prices under two forward risk-adjusted measures. Closed form solutions for European options on foreign assets are then obtained by applying these risk-neutral distributions. This article also provides analytic solutions for pricing twice exercisable options that are at most two-dimensional even though the valuation problem involves four risk factors at two exercise dates. I report the results of numerical evaluations of American option values using my method and show how they vary with the interest rate parameters. I also verify the accuracy of the proposed method by comparing with the benchmark values obtained from the least-square method of Longstaff and Schwartz (2001).

This 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.

This paper extends the seminal Cox-Ross-Rubinstein ((1979), CRR hereafter) binomial model to multiple assets. It differs from previous models in that it is derived under the complete market environment specified by Duffie and Huang (1985) and He (1990). The complete market assumption requires the number of states to grow linearly with the number of assets. However, the number of correlations grows at a faster rate, causing the CRR model to be indirectly extendable. We solve such a problem by recognizing that the fast growing correlation number is matched by the number of the angles of the edges of a hypercube spanned by the risky assets. As a result, we derive a solution that allows the number of equations to equal the number of risky assets and the riskless bond. The resulting tree structure hence provides the same intuition of pricing and hedging contingent claims as that provided by the CRR model. Finally, the proposed model is not only as easy to implement as the one-dimensional CRR model but also it is more memory efficient than the existing multi-factor lattice models.