his chapter follows
[Piterbarg2003a]
.
The SDEs (
Heston
equations
)
were covered in the section
Heston
equation
. We will be considering the quantity
in its martingale measure
.
We also perform a slight modification of the first
equation:


(Displaced Heston equations)

where the
is some number
.
Such transformation preserves analytical tractability and allows for modelling
of skew in volatility smile for short maturities. The term structure of
volatility smile prompts introduction of time dependency of the parameter
.
Such dependency is incompatible with the analytical tractability that was
established in the section (
Heston
equation
).
