91Ó°ÊÓ

Verify that \(A S^{a} e^{\gamma t}\) satisfies the Black-Scholes PDE for $$a=\left(\frac{1}{2}-\frac{r-\delta}{\sigma^{2}}\right) \pm \sqrt{\left(\frac{r-\delta}{\sigma^{2}}-\frac{1}{2}\right)^{2}+\frac{2(r-\gamma)}{\sigma^{2}}}$$

Assuming that the stock price satisfies equation \((20.27),\) verify that \(K e^{-r(T-r)}+\) \(S(t) e^{-\delta(T-t)}\) satisfies the Black-Scholes equation, where \(K\) is a constant. What is the boundary condition for which this is a solution?

Verify that \(S(t) e^{-3(T-t)} N\left(d_{1}\right)\) satisfies the Black-Scholes equation.

Verify that \(e^{-r(T-r)} N\left(d_{2}\right)\) satisfies the Black-Scholes equation.

Suppose that a derivative claim makes continuous payments at the rate \(\Gamma .\) Show that the Black-Scholes equation becomes $$V_{t}+\frac{1}{2} \sigma^{2} S^{2} V_{s s}+(r-\delta) S V_{s}+\Gamma-r V=0$$

For the following four problems assume that \(S\) follows equation (21.5) and \(Q\) follows equation \((21.35) .\) Suppose \(S_{0}=\$ 50, Q_{0}=\$ 90, T=2, r=0.06, \delta=0.02\) \(\delta_{Q}=0.01, \sigma=0.3, \sigma_{Q}=0.5,\) and \(\rho=-0.2 .\) Use Proposition 21.1 to find solutions to the problems. Optional: For each problem, verify the solution using Monte Carlo. What is the value of a claim paying \(Q(T)^{-1} S(T) ?\) Check your answer using Proposition 20.4.

For the following four problems assume that \(S\) follows equation (21.5) and \(Q\) follows equation \((21.35) .\) Suppose \(S_{0}=\$ 50, Q_{0}=\$ 90, T=2, r=0.06, \delta=0.02\) \(\delta_{Q}=0.01, \sigma=0.3, \sigma_{Q}=0.5,\) and \(\rho=-0.2 .\) Use Proposition 21.1 to find solutions to the problems. Optional: For each problem, verify the solution using Monte Carlo. An agricultural producer wishes to insure the value of a crop. Let \(Q\) represent the quantity of production in bushels and \(S\) the price of a bushel. The insurance payoff is therefore \(Q(T) \times V[S(T), T],\) where \(V\) is the price of a put with \(K=\$ 50\) What is the cost of insurance?