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Price sensitivities are important in any model for pricing derivatives because the sensitivities determine the trading strategy for hedging the derivative. The sensitivities are more commonly known as Greeks and in the LIBOR market model can be computed using Monte Carlo methods or likelihood ratio methods (LRMs). The difference between the two methods is in the dependence of the parameter under consideration (i.e., the initial value of a price or interest rate), for sensitivity estimation. In Monte Carlo methods, the dependence of the parameter is made in the underlying stochastic process via sample paths, which lead to estimators that differentiate the paths of the process, known as pathwise derivatives.29 The LRM puts the dependence in the probability measure and leads to estimators based on differentiating probability densities. The following discussion is based on the work of Glasserman and Zhao (1999). Assume a given tenor structure that is a nite set of dates 0 = T0 < T1 < . . . < TN < TN+1, representing maturities spaced three months for six months apart. We also assume that the day-count fractions i = Ti+1 Ti, i = 0, . . . , N, are all equal to a xed (i.e., = 0.25 years). However, in practice, day-count conventions lead to intervals of different sizes. We assume the index of the next tenor date at time t satis-

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Suppose = 0, where 0 is an n n matrix of 0 s. Then we get: dx(t) = dt (9.20)

10:18

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Working Group 4 (WG 4) and the Technical Support Group (TSG)

where, for example, C( + ) is the value computed with , a small fraction of (i.e., = 0.001 ). Thus, we can compute vega, rho, and theta as: v 1.664 1.661 = 10 2 (0.00015) 1.662 1.663 = 9.09 2 (0.000055) 1.663 1.662 = 0.667 2 (0.00075)

246 RPR 103611 (* S) 257 IC 9564 (* R)

5.19 4.53 4.12 3.84 3.63

s.isspace()

158 166 167 167 168 171

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While the LSM model is convenient for pricing swaptions since it is consistent with Black s formula, in general there are no analytical formulas for pricing interest rate derivatives involving swap rates. Consequently, forward LIBOR rates, rather than forward swap rates, are used to model the yield curve. However, we may be interested in computing the dynamics of forward swap rates under the numeraire used for forward rates, namely P(t, Tn). Thus, we are interested in expressing the dynamics of forward swap rates under the measure QP( ,Tn). Following Brigo and Mercurio, by applying the change-of-drift formula given in equation (1.68), we obtain the percentage drift mn,N (t) for Sn,N under QP( ,Tn) as C (t ) mn = 0dt d ln(Sn, N (t ))d ln n, N P(t , Tn ) The covariation term is computed as follows: N i P(t , Ti ) i N 1 i = ln i = n +1 = ln = (Fn +1 (t ), . . . , FN (t )) P(t , Tn ) i = n +1 j = n +1 1 + j Fj (t ) (13.46)

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FIGURE 5.14 d)

591 593

The initial condition of the central limit theorem states roughly that higher-order moments of the distributions, such as how it is skewed, become less and less important relative to the standard deviation as n .

where x, y, fx , fy are the sampling intervals, and n1 , m1 , as well as n2 , m2 are integers satisfying M1 n1 ; m1 M2 n2 ; m2 M1 M2 4:4-2

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