Question: Consider a Black-Scholes model with r = 6%, = 0.25, S(0) = 50 and = 0. Suppose you sold ten (10) 90-day Puts with strike

Consider a Black-Scholes model with r = 6%, = 0.25, S(0) = 50 and = 0. Suppose you sold ten (10) 90-day Puts with strike K = 50 (at price P₁(0) determined by the Black-Scholes formula). You wish to hedge your risks and in particular are worried about the stock moving up and down.

1. Using the underlying stock as well as 180-day Puts with strike K = 50 (priced at P₂(0) again using the B-S formula) construct a Delta-Gamma neutral portfolio.

2. Suppose that you also invest in bonds such that the initial value of your portfolio is exactly zero. Thus your overall portfolio is x shares of stock, short 10 contracts of 90-day Puts, y contracts of the 180-day Puts and z zero-coupon bonds. You should write out explicitly the numeric values of x, y, z above.

Produce a table listing the value of your portfolio tomorrow V(1/365) in the cases that the stock goes to S(1/365) = 48, 50, 52.

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