Vulcan Gravitational Fracking (VGF) stock sells for $77 a share. The annual interest rate is 8%, and the annual standard deviation on VGF stock is 20%. The time to expiration of all options is 9 months. You are short 1000 put options on VGF with a strike price of 75. To hedge you position, you can buy or sell VGF stock, or another put options on VGF at the money (i.e. with a strike price of 77).

Given all this:

A: Derive the position you will take if you want to delta hedge your position.

B: Derive the position you will take if you want to delta-gamma hedge your position.

C: Given your answers in A and B, graph (on the same graph) the net gain or loss in your positions if the price of VGF jumps to X, X a number between 1 and 100. (So you’ll want two lines on your graph, one with the return to delta hedging and one with the return to delta-gamma hedging.)