Risk Neutral Pricing Exercise

Question

I have the following exercise: A financial security pays off a dollar amount of $S_T^2$. Using Ito`s Lemma, what is the price today $V_t$ of this security? (S follows a Geometric Brownian Motion $dS = \mu Sdt + \sigma S dB$ )

I know that $V_t = e^{-r(T-t)}E^Q[V_T]$. So I tried to calculate $E^Q[V_T]$ with Ito`s Lemma:

1. $dV/dt = 0$
2. $dV/dS = 2S$
3. $d^2V/dS^2 = 2$

So Itos Lemma yields $dV = (2\mu + \sigma^2)S^2 dt + 2\sigma S^2dB $. Now, I have to solve this: $\int_{t}^{T} S^2 = (2\mu + \sigma^2) \int_{t}^{T} S^2 ds + 2\sigma \int_{t}^{T} S^2dB $, but I don`t know how to get the result $V_t = S^2e^{(2r + \sigma^2)(T-t)}$.

Can anyone help me with this exercise?

Answer 1

A few comments:

The question says "using Ito's lemma". We can also do it without using Ito's Lemma by simply calculating $E^Q(S_T^2)$ -- this should give you something that agrees with your form of $V_t$ that you were not sure how to get to.

To use Ito's Lemma in your calculation, I'm not entirely sure what approach the exercise is looking for, but if you recall solving the SDE of GBM, you can do something similar here to get an expression of the form $$\log(S^2_T) - \log(S^2_t) = (2r-\sigma^2)\int_t^Tdu + 2\sigma \int_t^Td\tilde{W_u} $$

which is a few short steps from the solution you're seeking.