Enter Values
Put-Call Parity Formula
S + p = c + PV(X)
S = Spot price |
p = Put price |
c = Call price |
PV(X) = Present value of strike
Calculation Result
Theoretical Call Price
$9.76
PV of Strike
$95.24
Left Side (S + p)
$105.00
Right Side (c + PV(X))
$105.00
Parity Difference
+$0.00
Formula Breakdown
Put-Call Parity: S + p = c + X/(1+r)T
Rearranged to solve for your selected variable
Arbitrage Interpretation
| Scenario | Condition | Strategy |
|---|---|---|
| Call Underpriced | S + p > c + PV(X) | Buy call, sell protective put |
| Put Underpriced | S + p < c + PV(X) | Buy protective put, sell call |
| Fair Value | S + p = c + PV(X) | No arbitrage |
Understanding Put-Call Parity
Video Explanation
Video: Put-Call Parity Explained
What is Put-Call Parity?
Put-call parity is a fundamental relationship in options pricing that connects the prices of European call and put options with the same strike price and expiration date. It states that a protective put (holding stock + buying put) has the same payoff as a fiduciary call (buying call + investing in risk-free bonds).
Put-Call Parity Equation
Discrete: S + p = c + X/(1+r)T
Continuous: S + p = c + Xe-rT
Protective Put = Fiduciary Call
Continuous: S + p = c + Xe-rT
Protective Put = Fiduciary Call
Equivalent Portfolios
Protective Put
Own stock + Buy put
Provides downside protection while retaining unlimited upside potential. Total cost: S + p
Fiduciary Call
Buy call + Invest PV(X) in bonds
Limited loss to call premium while capturing upside. Total cost: c + PV(X)
Arbitrage Opportunities
When market prices deviate from put-call parity, arbitrage opportunities arise:
- If S + p > c + PV(X): The protective put is overpriced. Sell stock + sell put + buy call + buy bonds for risk-free profit.
- If S + p < c + PV(X): The fiduciary call is overpriced. Buy stock + buy put + sell call + borrow at risk-free rate.
Important: Put-call parity holds exactly only for European options (no early exercise). For American options, early exercise possibility means the relationship is an inequality rather than equality.
Key Assumptions
- European options only (no early exercise)
- Same strike price and expiration date for both options
- No dividends during the option's life (or adjust formula)
- No transaction costs or taxes
- Ability to borrow and lend at the risk-free rate
Dividend Adjustment: When the underlying pays dividends, adjust the formula: (S - PV(Dividends)) + p = c + PV(X)
Frequently Asked Questions
Put-call parity is a fundamental relationship between European call and put options with the same strike price and expiration date. The formula states that S + p = c + PV(X), where S is the spot price, p is the put price, c is the call price, and PV(X) is the present value of the strike price. This relationship shows that a protective put (stock + put) has the same payoff as a fiduciary call (call + bond).
Put-call parity only holds exactly for European options because American options can be exercised early. The possibility of early exercise means American options may have additional value (the early exercise premium) not captured by the put-call parity relationship. For American options, the relationship becomes an inequality rather than equality.
When the underlying asset pays dividends, the put-call parity formula must be adjusted by subtracting the present value of expected dividends from the spot price. The modified formula is: (S - PV(Dividends)) + p = c + PV(X). This adjustment accounts for the fact that the stock price will drop by the dividend amount on the ex-dividend date.
Arbitrage opportunities arise when market prices deviate from the put-call parity relationship. If the left side (S + p) does not equal the right side (c + PV(X)), traders can construct risk-free profits by buying the underpriced side and selling the overpriced side. In practice, these opportunities are rare and brief due to sophisticated traders quickly exploiting any mispricing.
This calculator provides results for European options. For American options, it gives an approximation, but the actual relationship is more complex due to the early exercise premium that American options carry. The results can still be useful as a reference point, but keep in mind that American option prices may deviate from put-call parity.
A synthetic position replicates the payoff of one instrument using a combination of others. Using put-call parity, you can create: a synthetic call (buy stock + buy put), a synthetic put (sell stock + buy call + invest PV(X)), or synthetic stock (buy call + sell put + invest PV(X)). These synthetic positions are useful when the actual instrument is unavailable or mispriced.
Disclaimer
This calculator is for educational purposes only and assumes European options with no dividends. Actual options pricing involves additional factors like volatility, bid-ask spreads, and transaction costs. For precise pricing, use the Black-Scholes model or consult professional tools. This tool should not be used for trading decisions.
