Put-call parity formula with a worked example

For a European option on a non-dividend-paying asset, the standard relationship is:

C + PV(K) = P + S

Where:

Rearranging the identity gives a synthetic position. A long call plus a short put with the same strike and expiration resembles long exposure to the underlying, but the financing, exercise, assignment, dividend, and settlement terms still have to line up.

• C is the call value
• P is the put value
• S is the spot price of the underlying
• K is the strike price
• PV(K) is the present value of paying the strike at expiration

Assume the following values for matching European options:

Calculate the two sides:

The values match. This means the two portfolios have the same modeled expiration cash flows under the stated assumptions. It does not mean the stock will rise, the options will be profitable, or a live order can be filled at those marks.

• Spot price S = $100
• Strike K = $100
• Present value of the strike PV(K) = $97
• Call value C = $8
• Put value P = $5
• Left side: C + PV(K) = $8 + $97 = $105
• Right side: P + S = $5 + $100 = $105

The strike is paid at expiration, not today. Discounting it with the relevant financing rate makes the timing comparable with the spot and option prices. If rates change, PV(K) changes even when the strike and spot stay the same.

For a continuously compounded rate r and time to expiration T, a simplified expression is:

PV(K) = K × e^(-rT)

The exact discounting convention, day count, collateral terms, and funding rate depend on the product and market. Do not substitute a headline interest rate for the financing assumption without checking the contract and quote timestamp.

If the underlying pays known dividends before expiration, the simplified no-dividend formula needs an adjustment. One common discrete-dividend form is:

C + PV(K) + PV(dividends) = P + S

With a continuous dividend yield q, a common form is:

C + PV(K) = P + S × e^(-qT)

The sign and cash-flow placement are easier to verify by building both portfolios to the same expiration payoff. Ex-dividend dates, uncertain dividends, stock borrow, and contract adjustments can make a small apparent mismatch reasonable rather than exploitable.

Before using the formula, confirm:

1. Call and put reference the same underlying and deliverable 2. Strike, expiration, multiplier, and currency match 3. Exercise style and settlement method are compatible 4. Spot, option quotes, rates, dividends, and borrow use aligned timestamps 5. The modeled cash flows occur at the same dates

The clean textbook relationship is most direct for European-style options. American-style equity options can be exercised early, especially around dividends, so early-exercise value and assignment risk may prevent a simple equality from being tradable.

Rearrange the no-dividend identity:

These are payoff relationships, not interchangeable account instructions. A synthetic position may have different margin, borrow, liquidity, exercise, assignment, tax, and settlement behavior than the asset or option it resembles.

• Long call plus short put plus financed strike can replicate long stock exposure
• Long put plus long stock can replicate a call plus the present value of the strike
• Short call plus long put can represent a short-stock-like payoff after the cash flows are aligned

Use the ask for legs you must buy and the bid for legs you plan to sell. Then include:

• commissions and exchange fees
• stock borrow availability and borrow cost
• dividends and financing
• exercise and assignment procedures
• settlement dates and currency conversion
• order size, quote age, and leg timing

A midpoint gap can disappear when the package is priced at executable quotes. A parity difference is not a trade until every leg can be entered, maintained, and settled under the same assumptions.

Record one row for each matched contract:

1. Underlying, deliverable, strike, expiration, multiplier, and exercise style 2. Spot price, call bid and ask, and put bid and ask 3. Rate, dividend schedule, borrow assumption, and quote timestamps 4. Present-value calculation and the side of the formula being tested 5. All transaction costs and the order size that could actually be executed 6. The remaining gap after costs and an explanation for any difference

This article explains put-call parity for education. It does not identify a guaranteed arbitrage, predict prices, or replace the current contract specification and broker agreement. Put-call parity explained provides the broader relationship and practical context.