Options Pricing: Put-Call Parity for Exam FM

Options Basics

A call option gives the holder the right (not obligation) to buy an asset at the strike price K. A put option gives the right to sell at K. The payoff of a European call at expiration is max(S_T - K, 0) and of a European put is max(K - S_T, 0). American options can be exercised at any time up to expiration.

Option values depend on: the underlying price S, strike K, time to expiration T, risk-free rate r, volatility, and dividends. For Exam FM, the focus is on payoff diagrams, profit diagrams, and the put-call parity relationship rather than the Black-Scholes formula.

Put-Call Parity

For European options on a non-dividend-paying stock: C - P = S_0 - K*v^T, where C is the call premium, P is the put premium, S_0 is the current stock price, K is the strike, and v^T is the present value factor. Equivalently, C + K*v^T = P + S_0. This says a call plus a zero-coupon bond equals a put plus the stock (a "synthetic forward").

With continuous dividends at rate delta: C - P = S_0 * e^(-delta*T) - K * e^(-r*T). The prepaid forward price of the stock replaces S_0.

Profit Diagrams

The profit from buying a call is max(S_T - K, 0) - C*(1+i)^T (accounting for the future value of the premium). The profit from buying a put is max(K - S_T, 0) - P*(1+i)^T. Exam FM problems ask you to draw profit diagrams for various positions (long call, short call, long put, short put) and combinations (straddles, spreads, collars). The break-even point is where profit equals zero. For a long call, the break-even is S_T = K + C*(1+i)^T.