Stochastic Volatility Modeling

Local volatility 35

2.3.1.1 An exact solution

The exact solution is taken from [

58

] and [

19

]. It relies on the mapping of

S

to

an asset X that does not jump on dividend dates.

Let us assume that dividends consist of two portions: a xed cash amount and a

proportional part. The dividend d

i

falling at time t

i

is given by:

d

i

= y

i

S

t

−

i

+ c

i

When looking for a security that does not experience dividend jumps the forward

naturally comes to mind. However, we would have to pick an arbitrary maturity

T

for the forward – the local volatility function would change whenever an option

with maturity longer than T was priced.

Let us instead use a driftless process

X

which starts ...