SCFII-3 Pricing in Markovian Settings

Stochastic differential equation

Markov property:

Four-step procedure to get a PDE for g

(1) Find a martingale

(2) Take its differential

(3) Set term to 0

(4) Replace random variables by real variables

Step (1)

for , we have

Step (2)

Step (3)

Set the term (drift term) to zero, i.e.

Step (4)

Replace with the real variables, then

This is the PDE for .

The model is inspired by the volatility smile in reality. It assumes the stock price has

under risk-neutral .

Goal: Price a call option on .

Since is a martingale under , we have

Besides, we assume

we have

Set term to zero we have

then the PDE is

Note that if is much larger than , the option is always almost in the money, then

Assume the stock price has below SDE

(the same setting as the Black-Scholes).

The payoff function of Asian option is

The price of the Asian option will depend on and .

Suppose there is a portfolio having initial values and delta of s.t.

then

Using the Itǒ formula method and match the term and the term we will have

then the PDE is

Note that

The Markov property thus goes

Boundary Conditions

We have three parameters for the price of Asian option, i.e. .