Quanto Adjustments in the Presence of Stochastic Volatility

A Appendix
Lemma 1. Let ν be a mean-reversion Ornstein-Uhlenbeck process under the measure Q, i.e.:
Futhermore, let the function y be defined as
dν(t) = κ (θ − ν(t)) dt + δdW Q (t), ν(0) = ν0.
�
Q
y (t, T, ν(t)) = E e −s1
� Tt
ν(u) 2 � Tt
du−s2 ν(u)du+s3v(T ) �
for arbitrary complex numbers s1,s2, s3 and −s1ν(u) 2 − s2ν(u) is lower bounded. Then y has the following solution:
where the function D is given by
with
y (t, T, ν(t)) = D (t, T, ν(t), s1, s2, s3)
D (t, T, ν(t), s1, s2, s3) = e 1 2 A(t,T,s1,s2)ν(t) 2 +B(t,T,s1,s2,s3)ν(t)+C(t,T,s1,s2,s3)
A (t, T, s1, s2) = κ γ1
−
δ2 δ2 Ψ (γ1, γ2)
Φ (γ1, γ2)
B (t, T, s1, s2, s3) =
κθγ1 − γ2γ3 + γ3Ψ (γ1, γ2)
δ 2 γ1Φ (γ1, γ2)
− κθ
δ 2
C (t, T, s1, s2, s3) = − 1
κ
ln (Φ (γ1, γ2)) + (T − t)
�
2 2
2 2 2
κ θ γ1 − γ
+
2� 3
2δ2γ 3 � �
sinh (γ1 (T − t))
− γ1 (T − t)
1 Φ (γ1, γ2)
� �
cosh (γ1 (T − t)) − 1
Φ (γ1, γ2)
Proof. See the appendix of Schöbel and Zhu [9].
+ (κθγ1 − γ2γ3) γ3
δ2γ 3 1
Ψ (γ1, γ2) = sinh (γ1 (T − t)) + γ2 cosh (γ1 (T − t))
Φ (γ1, γ2) = cosh (γ1 (T − t)) + γ2 sinh (γ1 (T − t))
γ1 = � 2δ2s1+κ2 γ2 = 1 � 2 �
κ − 2δ s3
γ1
γ3 = κ 2 θ − s2δ 2 .
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