1. Use the Put-Call parity and the Black-Scholes formula for the European call from Proposition 1 to derive a similar formula for the European put option.

2. Hedging a long position:

Consider a bank that has a long position in a European Put written on the stock price in a one period binomial model with u = 2, v = ½, S = S₀ = 4, and r = 1/16. The Put expires at time T = 1 and has strike price K = 3. At time zero, the bank owns this option, which ties up capital V₀. The bank wants to earn the interest rate 6.25% on this capital until time one (i.e. "no matter what the markets do", the bank wants to have 1.0625 • V₀ at time one). Specify how the bank's trader should invest in the stock and money markets to accomplish this.

3. Lookback option:

Suppose as in the Excel-File a three-period binomial model with u = 2, v = ½, S₀ = 4, and r = 1/16. Consider a lookback option that pays off

V₃ = max_0≤n≤3 S_n - S₃

at time three. Compute

(a) V₀ and ?₀;

(b) V₁ and ?₁.

4. American Puts, Calls, and Straddles:

Suppose as in the Excel-File a three-period binomial model with u = 2, v = ½ , S₀  = 4, and r =  1/16.

(a) Determine the price at time zero, denoted V₀^P, of the American put that expires at time three and has intrinsic value g_P(s) = (4 - s)⁺.

(b) Determine the price at time zero, denoted V₀^C, of the American call that expires at time three and has intrinsic value g_C(s) = (s - 4)⁺.

(c) Determine the price at time zero, denoted V₀^S , of the American straddle that expires at time three and has intrinsic value g_S(s)  =  g_P(s)  + g_C(s).

(d) Explain why it is possible that V₀^S < V₀^P + V₀^C.