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which is a continuous nonvanishing function for s I . Notice that the coef cient of 3 in the Taylor expansion of Cov( 1 , 2 ) is equal to (b + b )/2. This follows either by direct computation or by taking into account the fact that det[Var( )] is a symmetric function of the pair s, t. Set (s, t) = det[Var( )]. The behavior of (s, t) as s, t t can be obtained from (s, t) = det Var(X(s), X(t), X (s), X (t))T det[Var(X(s), X(t))T ]

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exercise features To price Bermudan options, many banks use one- or two-factor short rate models such as the BDT (1990), Black and Karasinski (1991), Hull and White (1990), and Ritchken and Sanksarasubramanian (1995) A characteristic common to each of these models is that they can be implemented numerically in low-dimension lattices (such as nite differences or binomial trees), which are well suited for handling the free boundary problem that arises for options with early exercise features1 The ease of implementation that these models provide comes at the expense of realism For instance, these models usually have only one driving Brownian motion, implying perfect correlation of all forward rates Since the models are based on the short rate as the underlying quantity, there usually are not enough degrees of freedom to precisely calibrate these models to quoted market instruments such as caps and swaptions.

Jacobs: Not necessarily. The amount of leverage of a given strategy is within the investor s control. Although Federal Reserve Board Regulation T permits leverage of up to two-to-one for equity strategies, for example, the investor can choose not to lever. Thus, given an initial $100 in capital, the investor could invest $50 long and sell short $50; the amount at risk is then identical to that of a $100 long-only investment.

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Trying to improve the t by using time-dependent parameters in these short rate models can lead to a nonstationary and largely unpredictable term structure evolution2 LIBOR market models (LMM) provide a more realistic framework since they can incorporate volatility skews, price liquid instruments in closed form, and have enough degrees of freedom to allow for a good t while maintaining a largely stationary term structure3 But, due to the high number of state variables in the LMM, recombining lattices are not computationally feasible and valuation of all contingent claims is almost always done by Monte Carlo simulation Although exible and easy to implement, Monte Carlo has slow convergence and dif culties dealing with early-exercise features of American- and Bermudan-style options The rst problem is overcome with variance reduction techniques such as antithetics and control variates.

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The second problem was long thought to be a problem that Monte Carlo could not overcome However, the work of Longstaff and Schwartz (2001), Tilley (1993), and Broadie and Glasserman (1997a) has proven this belief to be incorrect although practical implementation obstacles still exist There are several methods for pricing American-style derivatives in the LMM or closely related HJM (1992) model that have been proposed in the literature The rst is the use of nonrecombining trees, known as bushy trees, which is set up to approximate the continuous-time dynamics of interest rates (see Heath et al (1992) and Brace, Gatarek, and Musiela (1997)) Backward induction algorithms can be applied and early exercise features easily incorporated Unfortunately, as the number of time steps increases, the number of nodes (states) increases exponentially making its use computationally expensive.

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