Options Pricing: Put-Call Parity for Exam FM

Options Basics: Calls and Puts

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). European options can only be exercised at expiration, while American options can be exercised at any time up to expiration.

Option values depend on six factors: 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 option pricing models like Black-Scholes. Understanding the payoff structure of options is essential for analyzing hedging strategies and combined positions.

Key terminology: an option is "in the money" when exercising it would be profitable (S > K for a call, S < K for a put), "at the money" when S = K, and "out of the money" when exercise would not be profitable. The intrinsic value is max(S - K, 0) for a call and max(K - S, 0) for a put. The time value is the option premium minus the intrinsic value.

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 equation says that a call plus a zero-coupon bond equals a put plus the stock.

The intuition behind put-call parity is that both sides replicate the same payoff at expiration. The left side (call + bond) pays max(S_T, K): if S_T > K, exercise the call and get S_T; if S_T <= K, the call expires worthless and the bond pays K. The right side (put + stock) also pays max(S_T, K): if S_T < K, exercise the put and get K; if S_T >= K, the put expires worthless and the stock is worth S_T.

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 because the option holder does not receive dividends.

Applying Put-Call Parity

Put-call parity is used to find one option price from the other. Example: A stock trades at $45 with no dividends. A 6-month European call with strike $40 is priced at $7. The risk-free rate is 5% per year. Find the put price. Using C - P = S_0 - K*v^T: 7 - P = 45 - 40*(1.05)^(-0.5) = 45 - 39.04 = 5.96. So P = 7 - 5.96 = $1.04.

Put-call parity also identifies arbitrage opportunities. If the market prices of C, P, S_0, and K*v^T do not satisfy the parity equation, an arbitrageur can construct a riskless profit by buying the cheap side and selling the expensive side of the equation.

Payoff and Profit Diagrams

The payoff diagram shows the option value at expiration as a function of S_T. For a long call, the payoff is zero when S_T <= K and rises linearly when S_T > K. For a long put, the payoff is zero when S_T >= K and rises linearly when S_T < K. Short positions are the mirror images of their long counterparts.

The profit diagram accounts for the option premium. Profit from buying a call is max(S_T - K, 0) - C*(1+i)^T (the future value of the premium is subtracted because the premium is paid at time 0 but the payoff is at time T). The profit from buying a put is max(K - S_T, 0) - P*(1+i)^T. For short positions, the profit is the negative of the long position's profit.

The break-even point is where profit equals zero. For a long call, the break-even is S_T = K + C*(1+i)^T. For a long put, the break-even is S_T = K - P*(1+i)^T.

Option Strategies and Combinations

Exam FM tests several common option strategies. A straddle is a long call and a long put with the same strike and expiration; it profits from large price movements in either direction. A bull call spread is a long call at strike K_1 and a short call at higher strike K_2; it profits from moderate price increases with limited risk. A bear put spread is a long put at K_2 and a short put at K_1; it profits from moderate price decreases.

A collar combines a long put at K_1 with a short call at K_2 (K_2 > K_1), protecting against downside while capping the upside. If the put and call premiums are equal, the collar is zero-cost. The payoff of a collar on a long stock position is: S_T if K_1 <= S_T <= K_2, K_1 if S_T < K_1, and K_2 if S_T > K_2. This creates a bounded range of outcomes.

Synthetic Positions

Put-call parity allows the construction of synthetic positions. A synthetic long forward is a long call and a short put with the same strike: payoff = (S_T - K) - 0 + 0 - (K - S_T) = S_T - K for all S_T. A synthetic stock is a long call, a short put, and a long bond: C - P + K*v^T = S_0. These synthetic positions can be used when direct positions are unavailable or more expensive.

On Exam FM, you should be able to identify equivalent positions using put-call parity. For instance, owning a stock and buying a put is equivalent to buying a call and a bond. This equivalence is useful for checking your work and for understanding why different strategies produce the same economic outcome.

Study Tips and Common Pitfalls

The most common error is forgetting to account for the future value of the premium in profit calculations. The premium is paid at time 0, but the payoff occurs at time T, so the cost should be accumulated to time T for a proper profit comparison. Another frequent mistake is confusing payoff diagrams (which start at zero for out-of-the-money options) with profit diagrams (which start at a negative value equal to the accumulated premium).

When applying put-call parity, always check whether dividends are present. The formula changes from C - P = S_0 - K*v^T (no dividends) to C - P = S_0*e^(-delta*T) - K*e^(-rT) (continuous dividends). Practice at least 5 put-call parity problems and 5 profit diagram problems, including combined strategies like spreads and collars.