1. Black-Scholes PDE:

Show that the Black-Scholes price for a European Put with strike K and expiration T

P (t, S_t) = K e^-r(T-t) Φ (-d₂(T - t, S_t)) - S_t Φ (-d₁(T - t, S_t)) ,

where

d₁(T - t, S_t) = {log{S_t/K} + (r + 1/2 . σ²)(T - t)}/(σ √T - t)

and

d₂(T - t.S_t) = d₁(T - t, S_t) - σ √(T - t)

and Φ(.) is the cumulative distribution function of the standard normal distribution function, satisfies the Black- Scholes PDE with the terminal condition for the European Put. What is the Delta?

2. More Black-Scholes Formulas:

Assume the Black-Scholes model with interest rate r > 0 and volatility σ > 0. Calculate the Black-Scholes price at time 0 of the following binary options:

(a) Cash-or-nothing call with payoff

CC(S) = C . 1_{S≥K}, C > 0.

(b) Cash-or-nothing put with payoff

CP (S) = C . 1_{{S, C > 0.}

(c) Asset-or-nothing call with payoff

AC(S) = S . 1_{S≥K}.

(d) Asset-or-nothing put with payoff

AP (S) = S . 1_{S.

3. Delta-Hedging:

Assume today is 01/31/2007. The S&P 500 index is at $1,428.65, its volatility is 17.17%, and the risk-free interest rate is 5%. You work in an investment bank that sells a European Put option at the money (i.e. the strike is K = $1, 428.65) expiring in one year (at t = T = 1). For simplicity, assume that the year has 256 (=2⁸) (trading-)days.

In the attached file "HW5Series.xls" you find the actual evolution of the S&P 500 over the next year, i.e. over the next 256 days, as well as 3 fictional evolutions (simulated paths). You are in charge of hedging the option, you have to trade stock and money market account to hedge your bank's position. Proceed as follows:

(a) Determine the option price at time zero by the Black-Scholes formula. That's what your customers pay.

(b) Using the option premium which your bank obtains, set up a hedging portfolio for each of the 4 given paths and rebalance your position every 16/4/1 days, respectively.

(c) At time T = 1, report your profit or loss (p&l) for each of the 12 = 4 × 3 cases (4 different paths × three different rebalancing periods).1

(d) Interpret your results.

Don't send me Excel Files! Please write up a report (max. 1 page) with your profits/losses and your interpretation.