Parity & Profit

Put-Call Parity with Financing

Discounting the strike turns expiry parity into a cost-today identity.

highlighted = computed this step

Two matching portfolios

Portfolio A is a long call plus a bond that pays the strike at expiry. Portfolio B is a long put plus the stock. At expiry, both pay the larger of the stock price and the strike.

max⁡(ST−K,0)+K=max⁡(K−ST,0)+ST\max(S_T-K,0)+K=\max(K-S_T,0)+S_T
Put-call parity with financingCall plus bond and put plus stock have matching expiry payoffs.Payoff equality at expiryExpiry stockA: call + bondB: put + stockEqual payoff$90.00$100.00$100.00max(S_T,K) = $100.00$100.00$100.00$100.00max(S_T,K) = $100.00$120.00$120.00$120.00max(S_T,K) = $120.00

Checking the expiry payoffs

With strike $100.00, the table checks stock prices $90.00, $100.00, and $120.00. The two portfolios pay $100.00, $100.00, and $120.00 in the same rows.

payoffA=payoffB=max⁡(ST,K)\text{payoff}_A=\text{payoff}_B=\max(S_T,K)
Put-call parity with financingCall plus bond and put plus stock have matching expiry payoffs.Payoff equality at expiryExpiry stockA: call + bondB: put + stockEqual payoff$90.00$100.00$100.00max(S_T,K) = $100.00$100.00$100.00$100.00max(S_T,K) = $100.00$120.00$120.00$120.00max(S_T,K) = $120.00

Discounting the strike

Since the payoffs match, no-arbitrage says their costs today match. With rate 10%, the present value of the strike is $90.91, exact dollar fraction 1000/11. So C minus P equals spot minus PV(K): $100.00 minus $90.91 = $9.09, exact dollar fraction 100/11.

C−P=S0−PV(K)=$100.00−$90.91=$9.09\text{C}-\text{P}=S_{0}-\text{PV}(K)=\$100.00-\$90.91=\$9.09
Put-call parity with financingThe matching payoff portfolios must have the same cost today.Cost todayItemCost identityTodayPortfolio AC + PV(K)C + $90.91Portfolio BP + S0P + $100.00Implied C - PS0 - PV(K)$9.09

Scope and use

This is a no-arbitrage identity under frictionless assumptions: borrowing and lending at the one stated rate, with no transaction costs or fees. PV(K) discounts the strike at that rate. Parity pins the price difference C − P, not the individual call and put prices, and is descriptive — not a forecast or investment advice.

frictionless no-arbitrage identity\text{frictionless no-arbitrage identity}