Option pricing with dividends (Exercise 6.3 continued). Consider an underlying asset price process \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)paying dividends at the continuous-time rate \(\delta>0\), and modeled as

\[
d S_{t}=(\mu-\delta) S_{t} d t+\sigma S_{t} d B_{t}
\]

where \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion.

a) Show that as in Lemma 5.14, if \(\left(\eta_{t}, \xi_{t}ight)_{t \in \mathbb{R}_{+}}\)is a portfolio strategy with value

\[
V_{t}=\eta_{t} A_{t}+\xi_{t} S_{t}, \quad t \geqslant 0
\]

where the dividend yield \(\delta S_{t}\) per share is continuously reinvested in the portfolio, then the discounted portfolio value \(\widetilde{V}_{t}\) can be written as the stochastic integral

\[
\widetilde{V}_{t}=\widetilde{V}_{0}+\int_{0}^{t} \xi_{u} d \widetilde{S}_{u}, \quad t \geqslant 0,
\]

b) Show that, as in Theorem 7.4, if \(\left(\xi_{t}, \eta_{t}ight)_{t \in[0, T]}\) hedges the claim payoff \(C\), i.e. if \(V_{T}=C\), then the arbitrage-free price of the claim payoff \(C\) is given by

\[
\pi_{t}(C)=V_{t}=\mathrm{e}^{-(T-t) r} \mathbb{E}^{*}\left[C \mid \mathcal{F}_{t}ight], \quad 0 \leqslant t \leqslant T,
\]

where \(\mathbb{E}^{*}\) denotes expectation under a risk-neutral probability measure \(\mathbb{P}^{*}\).

c) Compute the price at time \(t \in[0, T]\) of a European call option in a market with dividend rate \(\delta\) by the martingale method.

d) Compute the Delta of the option.