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Put-Call Parity

Put-call parity is a fundamental relationship between call and put prices. It states that a call and put with the same strike and expiry must be priced consistently. If they are not, there is free money on the table.

Interactive Parity Calculator​

Adjust the inputs and watch how the parity relationship holds across different market conditions. Click on "PUT-CALL PARITY" to see the formula with your current values.

Spot Price$100.0k
$70k$130k
Days to Expiry30d
1d180d
Implied Volatility50%
10%150%
Interest Rate5%
0%20%
PUT-CALL PARITY
click to see formula
Call - Put
$5909
C
-
$5499
P
= $410
=
Spot - PV(Strike)
$100.0k
S
-
$99.6k
PV(K)
= $410
Call Price (C)
$5909
Put Price (P)
$5499
Spot (S)
$100.0k
PV(Strike)
$99.6k
EQUIVALENT PORTFOLIOS
Portfolio A
Long Call
+
Cash = PV(K)
$105499
=
Portfolio B
Long Put
+
Long Stock
$105499
Both portfolios have identical payoffs at expiry, so they must have the same value today.
Parity holds. No arbitrage opportunity. The call and put are correctly priced relative to each other.

Why It Works​

The insight is that two portfolios with identical payoffs at expiry must have the same value today.

Portfolio APortfolio B
Long callLong put
+ Cash worth PV(K)+ Long underlying

At expiry, both portfolios are worth max(S, K):

  • If S > K: Portfolio A exercises the call, Portfolio B holds the stock
  • If S < K: Portfolio A keeps the cash (now worth K), Portfolio B exercises the put

Same payoff = same price today. Any difference is an arbitrage opportunity.

No-Arbitrage Implications​

Put-call parity is a no-arbitrage condition. If violated:

If...Then...Arbitrage
C - P > S - PV(K)Calls overpriced vs putsSell call, buy put, buy stock, borrow PV(K)
C - P < S - PV(K)Puts overpriced vs callsBuy call, sell put, sell stock, lend PV(K)

In practice, small deviations exist due to:

  • Bid-ask spreads
  • Transaction costs
  • Borrowing costs
  • Execution risk

But large deviations get arbitraged away quickly.

Practical Uses​

1. Sanity Check Pricing​

If you see a call and put with the same strike, you can verify they are priced consistently:

C−P≈S−PV(K)C - P \approx S - PV(K)

If they are not, something is off: maybe stale quotes or a data error.

2. Synthetic Positions​

Put-call parity lets you create synthetic positions:

Want ThisBuild It With
Long callLong put + Long stock - Borrow PV(K)
Long putLong call + Short stock + Lend PV(K)
Long stockLong call + Short put + Lend PV(K)

This is useful when one leg is cheaper or more liquid than the direct position.

3. Understanding Skew​

The relationship also implies a connection to the forward price:

C−P=Forward−K⋅e−rTC - P = \text{Forward} - K \cdot e^{-rT}

This connects option prices to the forward, which is key for understanding volatility skew.

Put-call parity in its simple form only holds for European options.

For American options, the relationship becomes an inequality:

S−K≤C−P≤S−K⋅e−rTS - K \leq C - P \leq S - K \cdot e^{-rT}

Early exercise rights create additional value that breaks the strict equality. See Exercise Styles for more details.

On Hypercall: All options are European-style, so put-call parity holds exactly (minus transaction costs).

Connection to Other Concepts​

Put-call parity connects several important ideas:

  • Forward pricing: The relationship implies a forward price
  • No-arbitrage: Violations create risk-free profit opportunities
  • Synthetic replication: Any position can be built from others
  • Delta hedging: A delta-neutral portfolio earns the risk-free rate

Understanding parity helps you see that calls and puts are not independent. They are two views of the same underlying uncertainty.

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