In the LIBOR market model, forward interest rates are log-normal under their respective forward measures. This note shows that their distributions under the other forward measures of the tenor structure have approximately log-normal tails.
The LIBOR market model [2] is one of the most popular models for pricing and hedging interest rate derivatives. Its state variables are forward interest rates F n (t) := F (t; T n-1 , T n ), spanning time periods [T n-1 , T n ], where
is a fixed tenor structure. Under the T M -forward measure Q M , which has as numeraire the zero coupon bond maturing at T M , the dynamics of the forward rates are
Here, σ n are some positive deterministic volatility functions, and W is a vector of standard Brownian motions with instantaneous correlations dW i (t)dW j (t) = ρ ij dt. Moreover, τ n = τ (T n-1 , T n ) denotes the year fraction between the tenor dates T n-1 and T n . Note that each rate F n is a geometric Brownian Motion under its own forward measure, while it has a non-zero drift under the other forward measures. A popular approximation of the above dynamics is obtained by “freezing the drift”:
Since the drifts are now deterministic, the new rates F fd n are just geometric Brownian motions, which allows for explicit pricing formulas for many interest-linked products. As a piece of evidence for the quality of this approximation, we show in the present note that, for fixed t > 0, the distribution of F fd n (t) has roughly the same tail heaviness as the distribution of F n (t).
If X is any log-normal random variable, so that log X ∼ N (µ, σ 2 ) for some real µ and positive σ, then (1) sup{v :
Our main result shows that F n (t) has approximately log-normal tails, in the sense that the left-hand side of ( 1) is finite and positive if X is replaced by F n (t). Furthermore, this “critical moment” is the same for F n (t) and the frozen drift approximation F fd n (t). Theorem 1. In the log-normal LIBOR market model, we have for all t > 0 and all
Proof. Note that the latter equality is obvious from (1), since F fd n (t) is log-normal with log-variance parameter σ 2 = t 0 σ n (s) 2 ds. We now show the first equality. Recall that the measure change from the T n -forward measure to the T n-1 -forward measure is effected by the likelihood process [1] dQ n dQ n-1
Therefore, putting φ(x) = exp(log 2 x), we obtain
Now let ε > 0 be arbitrary, and define q by 1 q + 1 1+ε = 1. Then Hölder’s inequality yields
By the finite moment assumption, we obtain the implication
Inductively, this leads to the implication
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