CT8 October 2010 Question 5iii)

[Mbmr99cc]

In this question, the risk-neutral pricing model gives us the following equation to solve:

V_0 = e^(-r(T-t)) * E_Q[S_1^2 | S_0]​

I understand we need to derive S_t to resolve S_1^2 any further...

In the next step

dS_t = r * S_t * dt + sigma * S_t * dZ_t ?​

What is the reasoning for substituting r in for mu here?

Thanks in advance!

 

[John Potter]

When we price, we must do so under the risk-neutral probability measure Q - otherwise we will allow an arbitrage opportunity. Under Q, all risky assets are expected to increase at the same rate as cash (i.e. earn risk-free rate r). So, a quick way to sort this out is just to sub in r for mu. The exact maths behind all of this is a corollary to the CMG theorem but the most important point to understand is that under Q, E[St| Fw] = Sw exp(r(t-w)) we expect the share to go up like cash

John