Question

As a hedge fund manager, you own a portfolio composed of (1) a long position in a European call and (2) short position in a put written on the same stock, with the same strike and the same maturity. Answer the following questions.

i) You are interested in generating arbitrage profits. Is there a standard traded derivative instrument you should trade if you observe a differential between the value of your portfolio and the price of that derivative instrument?
ii) What is the value of your option portfolio at time t as a function of the option contract’s primitives (stock price, drift parameter, diffusion parameter, strike price, maturity date, risk-free rate).

Answer 1

As a hedge fund manager, you can trade standard traded derivative instruments to generate arbitrage profits. The value of your option portfolio can be calculated using the Black-Scholes option pricing model.

Step-by-step explanation:

As a hedge fund manager, if you observe a differential between the value of your portfolio and the price of a derivative instrument, you can trade standard traded derivative instruments to generate arbitrage profits. One such instrument that you can trade is the futures contract on the underlying stock.

The value of your option portfolio at time t can be calculated using the Black-Scholes option pricing model. This model calculates the value of an option based on the underlying stock price, the strike price, the time to expiration, the risk-free rate, and the volatility of the stock.

The formula for the value of a European call option is: C = S * N(d1) - X * e^(-r * T) * N(d2), and for a European put option is: P = X * e^(-r * T) * N(-d2) - S * N(-d1), where S is the stock price, X is the strike price, T is the time to expiration, r is the risk-free interest rate, N() represents the cumulative standard normal distribution, d1 = (ln(S/X) + (r + (sigma^2)/2) * T) / (sigma * sqrt(T)), and d2 = d1 - sigma * sqrt(T).