Put-Call Parity

Put-call parity is one of the most fundamental relationships in options pricing. It states that the price of a European call and a European put — on the same underlying, with the same strike and expiration — are linked by a simple equation:

C - P = S - K × e^-rT

Or equivalently: C + K × e^-rT = P + S

This says that owning a call and lending money (left side) is equivalent to owning a put and the underlying stock (right side). Both positions produce the same payoff at expiration regardless of where the stock price ends up. Since they have the same payoff, they must have the same price — otherwise, you could buy the cheap one and sell the expensive one for a risk-free profit (arbitrage).

The beauty of put-call parity is that it is model-independent. It doesn't depend on the Black-Scholes model, the normal distribution, or any other pricing assumptions. It follows purely from the no-arbitrage principle, making it one of the strongest results in financial theory.

Let's prove put-call parity using a payoff comparison at expiration. Consider two portfolios:

Portfolio A: Long one European call (strike K, expiry T) + Cash of K × e^-rT (invested at the risk-free rate).

Portfolio B: Long one European put (strike K, expiry T) + Long one share of the underlying stock.

At expiration (time T), the cash has grown to K. Now compare payoffs:

If S_T > K (stock above strike):

• Portfolio A: Call pays (S_T - K) + cash K = S_T
• Portfolio B: Put expires worthless (0) + stock S_T = S_T

If S_T ≤ K (stock at or below strike):

• Portfolio A: Call expires worthless (0) + cash K = K
• Portfolio B: Put pays (K - S_T) + stock S_T = K

Both portfolios produce max(S_T, K) in every scenario. Since they have identical payoffs, they must have equal prices today:

C + Ke^-rT = P + S ⇒ C - P = S - Ke^-rT

Consider a stock trading at S = $100. A 3-month European call with strike K = $100 is priced at C = $6.50. The risk-free rate is r = 4%. What should the put be worth?

Apply put-call parity:

C - P = S - Ke^-rT

6.50 - P = 100 - 100 × e^{-0.04 × 0.25}

6.50 - P = 100 - 100 × 0.9900 = 100 - 99.00 = 1.00

P = 6.50 - 1.00 = $5.50

Arbitrage opportunity: If the market put price is $4.80 (less than $5.50), put-call parity is violated. You could:

1. Buy the put at $4.80
2. Buy the stock at $100
3. Sell the call at $6.50
4. Borrow $99.00 at the risk-free rate

Net cost today: $4.80 + $100 - $6.50 - $99.00 = -$0.70 (you receive $0.70 upfront)

At expiration, the long put + long stock position offsets the short call in every scenario, and the loan repayment is exactly covered. You pocket $0.70 risk-free. In practice, market makers and arbitrageurs monitor put-call parity continuously, so violations are quickly corrected.

Options traders use put-call parity every day for several purposes:

• Synthetic positions: Put-call parity allows you to create any one of the four components (call, put, stock, bond) from the other three. A synthetic long stock = long call + short put. A synthetic call = long stock + long put. These synthetic positions are used when the direct instrument is hard to trade or when they offer a price advantage.
• Arbitrage monitoring: Market makers continuously monitor put-call parity to detect and trade mispricings. Algorithms at firms like Citadel Securities flag violations in real time.
• Dividend estimation: For stocks that pay dividends, the put-call parity relationship changes to C - P = S - PV(dividends) - Ke^-rT. Traders use this to back out the market's implied dividend forecast from option prices.
• Interview questions: Put-call parity is a staple of options trading interviews. You should be able to derive it, apply it to find missing prices, and identify arbitrage opportunities when it's violated.