In our contribution the goal is to find the analytic solution of the characteristic function (ch.f.)ofxT, given the initial data under the hybrid Heston–Hull–White model. That is, we want to find aclosed form expression for
Φ(ω;x0,v0,r0) :=E(exp(iωxT)|x0,v0,r0).
A first observation on the model is the following: If xt satisfies
dxt=(rt−12vt)dt+√vtd ̃W1,t,
then St=exp(xt) satisfies the Heston–Hull–White model, as can be seen by applying Itô’s lemma.This paper has a twofold aim:
- Solve the problem under the assumptionρ13=ρ23=0.
- Solve the problem under the assumptionρ23=0, and under the additional assumption thatκη=λ2/4, in which case√vtis governed by an Ornstein–Uhlenbeck process.
It is organized as follows: in section 1, we decompose the three correlated Wiener processes intothree independent ones, and establish some notation. In section 2, we eliminate two noises byexploiting the Gaussianity of thert-distribution, as well as the fact thatxtdoes not occur on the r.h.s.of the equations. In section 3, we obtain the ch.f. ofxTin the aforementioned two cases.