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Consider the stanadard Black-Scholes model and a T-claim $\mathcal{X}$ of the form $\mathcal{X}=\Phi(S(T))$.

Denote the corresponding arbitrage free price process by $\Pi(t)$.

Show that, under the martingale measure $Q$,$\Pi(t)$ has a local return equal to the short rate of interest r. In other words show that $\Pi(t)$ has a differential of the form:

$$d\Pi(t)=r.\Pi(t)dt$+g(t)dW_t$$

Show that under the martingale measure $Q$, the process $Z(t)=\frac{\Pi(t)}{B(t)}$ is a martingale. More precisely, show that the stochastic differential for $Z$ has zero drift term,i.e, it is of the form:

$$dZ(t)=Z(t)\sigma_{Z}(t)dW_t$$

Determine also the diffusion process $\sigma_Z(t)$ (in terms of the pricing function F and its derivatives).

The first question is easily solved by recognizing f(t,S(t))=\frac{\pi(t)}{B(t)}, so that f(t,S(t)) satisfies the BS pde.

$df(t,S(t))=f_t dt+f_S S(t)dS(t)+\frac{1}{2} f_{SS} S^2(t) dS^2(t)\\=f_t dt+f_S S(t)r dt+f_S S(t)\sigma dW_t+\frac{1}{2}f_{SS}\sigma^2 dt=f_S S(t)\sigma dW_t +r f(t,S(t))$ as I wanted to prove.

My problem is with the first part of the second question. I do not want to use the Black-Scholes pde to prove that $\Pi(t)$ is a martingale, since I want to prove the risk-neutral valuation formula independently of the BS pde.

So the risk neutral valuation formula is given by $$\Pi(t)=B(t)E^Q_t\left(\frac{\Pi(T)}{B(T)}\right)\implies\frac{\Pi(t)}{B(t)}=E^Q_t\left(\frac{\Pi(T)}{B(T)}\right) $$, therefore by the tower property of the conditional expected values the right side of the expression is a martingale which implies $\Pi(t)$ to be a martingale.

Question:

Is this reasoning correct? If not, how would I proceed to prove $\Pi(t)$ is a martingale?

Thanks in advance!

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