ABSTRACT
Moreover, with the aim of enhancing the model’s ability to fit both caplet and swaption smiles, we let the stochastic volatility process have non-zero correlation with the forward rates. In our case, because of the presence of stochastic volatility, we prefer to work in the spot measure Ps. In this measure, the dynamics of the forward rates in the SABR-LMM model reads :
dLk = a2Bk(t, F )dt+ σk(t)aCk(Lk)dZk da = νadZn+1; , dZidZj = ρij(t)dt i, j = 1, · · · , n+ 1
with
Ck(Lk) = φkL βk k
Bk(t, F ) = k∑
τjρkjσk(t)σj(t)Ck(Lk)Cj(Lj) 1 + τjLj
We introduce the constants φk for normalization purposes, so that σk(t = 0) = 1, for all k. The forward rate dynamics under the forward measure Pk is much simpler
dLk(t) = σk(t)aCk(Lk)dWk
da = −νa2 k∑
τjρjaσj(t)Cj(Lj) 1 + τjLj
dt+νadWn+1, dWkdWn+1 = ρka(t)dt
with initial conditions a(t = 0) = a0 and Lk(t = 0) = L0k. Note that as shown in chapter 6, the log-normal SABR model defines a martingale as long as 0 ≤ βk < 1 or ρka ≤ 0 for βk = 1. The possibility of moment explosions due to volatility being log-normally distributed in the case of the SABR-LMM model is an open question.
