Suppose that each Libor rate is driven by a separate Wiener process dL_i (t) = σ_i (L_i(t),t) dWⁱ_t (under the measure with numeraire being the discount bond maturing at the pay date of the Libor rate) with correlation structure given by dWⁱ_t  dWⁱ_t  = ρ_ij (t)dt. Using the same approach as in Section 9.1.1, find the drift of each Libor rate under the terminal measure.

Section 9.1.1

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In the analysis below, we shall consider a single Wiener process Wt driving all Libor rates. The ideas can be extended to the general case of multiple Wiener drivers above, but the notation becomes more involved and it does not add any explanatory value, so we will not pursue this here. Consider the Libor rate Li(t) under the Ti-forward measure (i.e. a martingale) with SDE dL;(t) = ơi(Li(t),t)dW. Consider a change of measure from the T-forward measure to the Ti+1-forward measure with Radon Nikodym derivative given by Notice that R(t) dR(t) = dWi = dQTi+1 dQTi -(t) D(t,Ti+1)D(0,T) D(t, Ti) D(0, Ti+1) D(0, T;) (1 + Li+1(t) Ti+1)D(0, Ti+1)* Thus, Girsanov's Theorem gives =... D(0, Ti) Ti+1dLi+1(t) (1 + Li+1(t) Ti+1)²D(0, Ti+1) _Ti+1®i+1(L;+1(t),t) -R(t)dWi+¹ 1 + Li+1(t) Ti+1 so that dL;(t) = 0;(Li(t),t) |- _Ti+1i+1(Li+1(t),t), 1 + Li+1(t) Ti+1 Ti+10i+1(Li+1(t),t) 1+ Li+1(t) Ti+1 _Ti+1i+1(Li+1(t),t) :(Li(t), t) 1+ Li+1(t) Ti+1 Working inductively, 2 we see that dLi(t) + oi(Li(t), t) |- dt + dwi+¹, Ti+1i+1(Li+1(t),t)ơi(Li(t),t) 1+ Li+1(t) Ti+1 N = -σi(Li(t), t) Σ j=i+1 -dt + oi(Li(t), t)dWi+¹. -dt + dWi+¹ Ti+20i+2(Li+2(t), t) 1+ Li+2(t) Ti+2 Tjøj(Lj(t), t) 1 + Lj(t) Tj N dLi(t) ≈ −oi(Li(t), t) Σ j=i+1 - dt -dt + dWi+² Wi+2] -dt + oi(Li(t), t)dWN. Thus, we have the drift of each Libor rate Li(t) under the measure where the discount bond with the longest maturity Ty is numeraire (i.e. the terminal measure). From the above equation, it is clear that under the terminal measure, the drift of Li(t) depends on {Lj(t)}+1. Further, since D(t, T;) = I=1 1+L;7;, to get discount factors for any expiry T, we require L₁(t). Hence, even a one-factor LMM is Markovian in only N state variables (L;(t)}1, if it is described by N Libor rates. N This is a major drawback of the LMM. First, a fairly large number of state variables is involved. For example, if we are interested in a 30-year swap in US dollars, since Libor rates are quarterly, we have a total of 120 state variables. This means computation time would be about 120 times that if we only needed one state variable. Furthermore, there is no chance to put the LMM on a PDE (which can realistically handle only up to three state variables). (There are some approaches that involve freezing the drift and hence making this a one-factor model, e.g. the simplest involves using TjOj(Lj(0), t) dt + oi (Li(t), t)dWN, (1 + Lj (0) Tj) and other approaches involving predictor-corrector or Brownian bridge techniques exist. But these approaches tend not to be able to give sufficiently accurate forward swap rates for long expiries, e.g. of 10 years or more. So, they are not much utilised in practice.) This means we need to somehow figure out how to deal with early exercise on a Monte Carlo, where we do not know the conditional expectation of the continuation value at a future point in time. We will discuss this further in Section 9.4.