Put-⁠Call Parity

What is put-call parity?

Put-call parity is the no-arbitrage relationship that links the prices of a European call and put with the same strike and expiration to the underlying price, dividends, and the present value of the strike. A long call plus a short put pays the same at expiration as buying the stock at the strike, so, absent arbitrage, the call price minus the put price equals the stock price minus the present values of the dividends and the strike.

Two packages that pay the same at expiration must cost the same today, once the rate and dividends are priced in. The clean equality applies to European options; American equity options add the value of early exercise.

What is the put-call parity formula?

For a stock with known discrete dividends, C − P = S − PV(dividends) − K × exp(−rT). C is the call price, P the put price, S the stock price, K the strike, r the continuously compounded rate, and T time in years.

How put-call parity creates synthetic stock

Long call plus short put pays S_T − K at expiration. Above K, the call contributes the difference. Below K, the short put contributes the same signed difference. Adding a cash investment that grows to K gives S_T.

That final payoff explains synthetic long exposure. The option package and stock also have different initial funding and dividend cash flows, which the parity equation prices explicitly.

Put-call parity example: six-month options

With stock at $100, a $100 strike, a 4% annual continuously compounded rate, and half a year remaining, the strike’s present value is $98.02. The call-minus-put value is $1.98.

A $6.20 call therefore pairs with a $4.22 put, and the long-call/short-put package costs $1.98. Its expiration P/L after subtracting that entry debit is S_T − $101.98. Book the financing as its own leg: the $98.02 invested at 4% is what grows to the $100 strike.

Conversion vs reversal: trading a parity gap

When synthetic long exposure is expensive relative to stock after carry, sell the synthetic and buy stock. That conversion combines long stock, long put, and short call.

When synthetic long exposure is cheap, buy the synthetic and sell stock. That reversal combines short stock, long call, and short put. The relative price determines the direction.

At zero carry, suppose stock and strike are both $100, the call is $7, and the put is $4. The synthetic costs $3 more than parity. A conversion costs $97 and pays $100 at expiration. Taking the reversal instead receives $97 initially against a $100 final obligation, locking in the wrong side of the difference before costs.

Implied forward prices and American options

With discount factor D, the option-implied forward is F = K + (C − P) / D. Compare matching European pairs across strikes to examine the forward encoded in the chain.

American options add the value of early exercise. Value their exercise choices and cash-flow timing when comparing the legs. Use the exact contract terms and executable prices to build the complete package.

Parity connects familiar strategies through their economic exposure. Compare protective puts, covered calls, synthetic stock, and financed option positions by their full payoff; the options profit calculator makes the equivalence visible.

Limitations of put-call parity

• Use bounds for American equity options. The clean equality holds only for European options, and American equity options can be exercised early, especially around dividends or with deep in-the-money puts, so the relationship becomes a set of bounds.
• Check the financing rate and the dividend estimate before trading a gap: an error in either one produces an apparent mispricing that disappears at the correct inputs.
• Price the complete package before trading a gap. The conversion or reversal locks in a profit only after it clears bid-ask prices, financing, borrow, fees, and assignment risk.
• Parity fixes the call and put relative to each other and the stock; whether both are cheap or rich in volatility terms comes from implied volatility.

Bottom line

Put-call parity ties a European call and put to the stock, dividends, and the discounted strike, so the two options are priced together. Use it to read implied forwards, recognize synthetic equivalents, and check quotes, and price the complete conversion or reversal at executable prices before treating any gap as an arbitrage.