Put-Call Parity Calculator

Put-call parity is a fundamental principle in options pricing that establishes a fixed relationship between the price of a European call option and a European put option with the identical underlying asset, strike price, and expiration date. The core equation states that the sum of the call option price and the present value of the strike price equals the sum of the underlying spot price and the put option price: C + PV(K) = S + P. Here, C is the call price, PV(K) is the present value of the strike price discounted at the risk-free rate, S is the spot price, and P is the put price. The present value of the strike price is calculated using the formula PV(K) = K / (1 + r)^t, where K is the strike price, r is the annual risk-free rate, and t is the time to expiry in years.

Imagine you are analyzing a European option on a stock with a current spot price (S) of $100 and a strike price (K) of $105 expiring in exactly 1 year (t = 1). The annualized risk-free interest rate (r) is 5% (0.05). Suppose the market trading price for the put option (P) is $8.00, and you want to find the theoretical fair value of the call option (C). First, compute the present value of the strike price: PV(K) = 105 / (1 + 0.05)^1 = 105 / 1.05 = $100.00. Next, apply the put-call parity relationship: C + PV(K) = S + P, which translates to C + 100 = 100 + 8. Solving for the call price yields C = 8.00. Therefore, a fair call option price under strict parity conditions should be $8.00.

When working with options models, always ensure that your input prices match the correct exercise style, as standard put-call parity strictly applies to European-style options rather than American-style contracts. Additionally, pay close attention to dividend yields if the underlying asset distributes cash flows during the option's lifespan, as unadjusted models can lead to calculation errors. Finally, remember that transaction costs and borrowing constraints in real-world markets can occasionally maintain a temporary price discrepancy without offering a true riskless profit.