This paper describes a practical algorithm based on Monte Carlo simulation for the pricing of multi-dimensional American (i.e., continuously exercisable) and Bermudan (i.e., discretelyexercisable) options. The method generates both lower and upper bounds for the Bermudan option price and hence gives valid confidence intervals for the true value. Lower bounds can be generated using any number of primal algorithms. Upper bounds are generated using a new Monte Carlo algorithm based on the duality representation of the Bermudan value function suggested independently in Haugh and Kogan (2001) and Rogers (2001). Our proposed algorithm can handle virtually any type of process dynamics, factor structure, and payout specification. Computational results for a variety of multi-factor equity and interest rate options demonstrate the simplicity and efficiency of the proposed algorithm. In particular, we use the proposed method to examine and verify the tightness of frequently used exercise rules in Bermudan swaption markets.

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Abstract: Bermudan swaptions have until recently been valued using only one-factor models such as the Black-Derman-Toy (BDT) or Black-Karasinski (BK) models. The LIBOR Market (LM) model which is a more general multi-factor model is becoming increasingly popular as a benchmark model. Whereas the BDT and BK models can be approximated using a lattice facilitating easy valuation of Bermudan swaption, the LM model doesn’t conform to the lattice framework and as such the valuation seems very difficult. Monte-Carlo simulation is a popular alternative to the lattice framework for derivatives valution. In order to facilitate valuation of Bermudan swaptions the Monte-Carlo simulation technique must be extended. A few methods doing this are presently available, eg [And98]. A common feature of these methods is that the estimated option premia are only lower bounds on the true premia. The Stochastic Mesh method proposed by [BG97b] for valuation of Bermudan (equity) options with applications to equity options provides a lower and an upper bound. We have applied this method to the LM model and use this to verify the premia found by Andersen. We will also apply the approach suggested in [LS98] to the LM model and verify the premia found using that approach. As it turns out this approach is a special case of the [And98] approach. Furthermore we also examine the impact on the Bermudan swaption premia when moving from a LM model with only one factor to a LM model with multiple factors and do indeed find a significant—but not dramatic—impact. We find the [And98] and [LS98] approaches to be mutually consistent and in line with results obtained from low-biased Stochastic Mesh estimates.

The value of American options depends on the exercise policy followed by option holders. Market frictions, risk aversion, or model risk, for example, can result in sub-optimal behavior. We study the sensitivity of American options to suboptimal exercise strategies. We show that a measure of this sensitivity is the Gamma of the American option at the optimal exercise price. Precisely, if B is the optimal exercise price, but exercise is either brought forward when or delayed until a price eB has been reached, the cost of suboptimal exercise is given by 12 (B) (B eB)2,where (B) denotes the American option Gamma. This result holds in a multi-factor setting and does not depend on the exercise policy followed by the option holder. We interpret this Gamma as a Greek of American-style securities, which is related to the early-exercise property. Hence, American options with small Gamma are less sensitive to suboptimal exercise, if B and eB are not far away, which has a quadratic cost. We are grateful to Peter Carr, Francis Longsta¤, and Richard Stanton for helpful comments. This paper has been accepted for presentation at the 2008 Global Derivative Securities Conference (Paris) and the 2008 WFA meetings (Hawaii). We welcome additional comments and references that we can have inadvertently missed. Any remaining errors are our own.

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We propose an implementation of the LIBOR market model, adapting the recombining node methodology of Ho, Stapleton and Subrahmanyam (1995). Initial tests suggest that the method provides a fast and accurate approach for the valuation of path dependent interest rate derivatives such as Bermudan-style swaptions. The lattice based approach illustrated here provides an efficient alternative to Monte-Carlo simulation implementation of the LMM.

for a careful reading of this manuscript. The paper also profited by comments from seminar

In Boyle et al. (2000) we propose a simulation method for pricing high-dimensional American style derivatives. The method exploits the uniformity property of the low discrepancy sequences so that the resulting biased high estimator can achieve higher rate of convergence of quasi-Monte Carlo method. In this paper, we extend this work by proposing another estimator that is biased low. It has the computational advantage that it can be obtained concurrently with the high-biased estimator using a recursive valuation approach. Some numerical examples are conducted to demonstrate its efficiency. We also show that further enhancement to the proposed estimator is possible by incorporating the standard variance reduction technique such as the use of control variates.

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profited by comments from seminar participants at Stanford University, the INFORMS Annual