Let $r$ denote the riskless rate and let $\sigma>0$ be a fixed volatility parameter.

(a) Let $\left(S_{t}\right)_{t \geqslant 0}$ be a Black-Scholes asset with zero dividends:

$$S_{t}=S_{0} \exp \left(\sigma B_{t}+\left(r-\sigma^{2} / 2\right) t\right),$$

where $B$ is standard Brownian motion. Derive the Black-Scholes partial differential equation for the price of a European option on $S$ with bounded payoff $\varphi\left(S_{T}\right)$ at expiry $T$ :

$$\partial_{t} V+\frac{1}{2} \sigma^{2} S^{2} \partial_{S S} V+r S \partial_{S} V-r V=0, \quad V(T, \cdot)=\varphi(\cdot)$$

[You may use the fact that for $C^{2}$ functions $f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ satisfying exponential growth conditions, and standard Brownian motion $B$, the process

$$C_{t}^{f}=f\left(t, B_{t}\right)-\int_{0}^{t}\left(\partial_{t} f+\frac{1}{2} \partial_{B B} f\right)\left(s, B_{s}\right) d s$$

is a martingale.]

(b) Indicate the changes in your argument when the asset pays dividends continuously at rate $D>0$. Find the corresponding Black-Scholes partial differential equation.

(c) Assume $D=0$. Find a closed form solution for the time-0 price of a European power option with payoff $S_{T}^{n}$.