Stochastic Volatility Models - Computational Methods for Quantitative Finance - page 106

(S, Y 1 ,...,Y n v ) ∈ R

n v + 1

can be described by the system of SDEs (compare

with (8.1))

d Z t = b(Z t ) d t + Σ(Z t ) d W t ,Z 0 = z, (9.3)

n v + 1 -valued Brownian motion, and the coefficients b : R n v + 1

where W is R →

1 ) satisfy the Lipschitz-continuity (8.2) and the
linear growth condition (8.3). We again denote by

n v + 1 , Σ : R n v + 1 (n v + 1 ) × (n v +

R → R

Q = ΣΣ .

9.1.1 Heston Model

= √ y wit h n v =

The Heston model is given by ξ(y) 1in( 9.2 ) and Y following

β √ Y t d W t . The Brownian motion W
might be correlated with the Brownian motion W 1

a CIR process, i.e. d Y t = α(m − Y t ) d t +

in ( 9.1 ) that drives the asset

price S . We th us intro duce a Brownian motion W 2 independent of W 1

and write

1

W t =

ρW t ρ 2 W t being the instantaneous correlation co-

efficient. Under a non-unique EMM, the coefficients b , Σ in ( 9.3 ) for the Heston
model are therefore given by

+ − , with ρ ∈[− 1 , 1 ]

,

rs

b(z) = (9.4)

α(m − y) − λ(s, y)

s √ y

0

βρ √ yβ 1

Σ(z) = , (9.5)

− ρ 2 √ y

where z (s, y) and the function λ appearing in ( 9.4 ) represents the price of volati l-

ity risk. Different choices of λ are given in the literature, for example, λ(s, y)

=

c √ y

=

(together with ρ = 0) in [6] and λ(s, y) = cy in [79]. For simplicity, we will choose

λ ≡ 0 in the following.

9.1.2 Multi-scale Model

(Y 1 ,...,Y n v ) evolves according to We assume that each component of Y =

d Y t c k (Y t ) d t g k (Y t ) d W t ,k = + = 1 ,...,n v , (9.6)

with coefficients c k ,g k : R → R smooth and at most linearly growing.

As in the Heston model, the Brownian motions W 1 and W t

1 ,...,n v ,are
not independent to each other, but are allowed to have a correlation structure L ∈
R

, k =

(n v + 1 ) × (n v + 1 )

of the form
(W t , W t ,...,W n v ) = LW t

t

(W 1 ,...,W n v + 1 )

with W = a standard (n v + 1 ) -dimensional Brownian motion

and