Consider the payoff $g(S_T)$ shown in the figure below. Consider Black-Scholes model for the price of a risky asset with $T = 1$, $r = .04$, and $\sigma = .02$ and dividends are paid quarterly with dividend yield $10\%$. Take $S_0 = 10$, $K_1 = 9$, and $K_2 = 11$. Find the Black-Scholes price, $\Delta$, $\Gamma$, $\rho$, and $\mathcal{V}$ of this option at time $t = 0$. Find $\Theta$ at time $t = 0$ without taking derivatives with respect to $S$.
Solution: The payoff is, $$g(S_t) = (S_t - K_1)_{+} - 2(S_t - \frac{(K_1 + K_2)}{2})_{+} + (S_t - K_2)_{+}$$ The Black-Scholes formula with dividend gives \begin{align*} V(t = 0,S) &= e^{-r\tau}\hat{\mathbb{E}}[g(\tilde{d}S_T)]\\ &= \tilde{d}\left(BS_{call}(\frac{K_1}{\tilde{d}}) - 2BS_{call}(\frac{K_1+K_2}{2\tilde{d}}) + BS_{call}(\frac{K_2}{\tilde{d}})\right) \end{align*} where $$\tilde{d} = \left(1 - \frac{d}{4} \right)^{4} = .9037$$ So, $$V(t = 0,S) = e^{-r\tau}\hat{\mathbb{E}}[g(\tilde{d}S_T)] = (.9037)((0) - 2(0) + (0)) \approx 0 $$ For the Greeks we have $$\Delta = \partial_S V(t = 0,S) = \tilde{d}\left[\Phi(d_1(\frac{K_1}{\tilde{d}})) + \Phi(d_1(\frac{K_1+K_2}{2\tilde{d}})) + N(d_1(\frac{K_2}{\tilde{d}})) \right] \approx 0$$ $$\Gamma = \partial_{SS}V(t = 0, S) = 0$$ $$\rho = \partial_r V(t = 0,S) = \left( e^{-rt}(\frac{K_1}{\tilde{d}})(t)\Phi(d_2) + e^{-rt}(\frac{K_1+K_2}{2\tilde{d}})(t)\Phi(d_2) + e^{-rt}(\frac{K_2}{\tilde{d}})(t)\Phi(d_2)\right) \approx 0$$ $$\mathcal{V} = (S\sqrt{t}\Phi(d_1) + S\sqrt{t}\Phi(d_1) + S\sqrt{t}\Phi(d_1)) \approx 0$$