Consider the option from Week 12 Assignment: an ATM call with 3 months to expiry. The current stock price is $20, the risk-free rate is 10% and volatility is 25% per annum.

A) Derive a closed-form expression for the sensitivity of the options delta with respect to implied volatility: d^2 C/ddS

B) Confirm and compare your analytical solution numerically

C) Is there an intuitive explaination for what this sensitivity means?

D) Plot (d/d)C vs S. Explain what happens, and why, as the call option goes in and out of the money.

E) Verify the interest rate tree that we created in class by pricing up a 5% two-year coupon bond (not callable) using discount factors and the tree.