Suppose L_i(t)satisfies the LLM model as in Problem 8.30. Here, we would like to find the stochastic differential equation of L_i(t) under the spot LIBOR measure QM̃ whose numeraire is the discrete money market account process M̃. The discrete dynamics of M̃ is defined by

Problem 8.30

Consider the Lognormal LIBOR Market (LLM) model for the LIBOR L_i(t), i = 0, 1, ··· ,n − 1, defined on the tenor structure {T₀,T₁, ··· ,T_n} where 0 0 1 n. Let v_i(t) denote the scalar volatility function of L_i(t), and write L_i(0) = L_i,₀,i = 0, 1, ··· ,n − 1. Now, L_i(t) satisfies

where Z_i(t) is Q_Ti-Brownian. We write ρ_ij (t) as the correlation coefficient between Z_i(t) and Z_j (t) such that ρ_ij (t)dt = dZ_i(t)dZ_j (t). Define

Under the terminal measure Q_Tn , show that X_i(t) satisfies the following stochastic differential equation

with initial condition: X_i(0) = 0, and Z̃_iⁿ⁺¹ (t) is Q_Tn -Brownian under the Girsanov transformation: Q_Ti → Q_Tn.

Note that Z̃ⁿ₁ , ··· ,Z̃ⁿ_n are Q_Tn -Brownian and

since an equivalent change of a probability measure preserves the correlation between the Brownian processes. We write formally 

and subsequently μ̂_i(t) is determined. Recall

Show that the dynamics of Li(t) under the spot LIBOR measure Q_M̃ is given by

where Z̃_i(t) is Q_M̃ -Brownian. 

Transcribed Image Text:

j-1 M(T;) = [I k=0 1 B(Tk, Tk+1) with M(To) = 1, j = 1, 2,. 9 n.