Put-Call Parity¶

Put-call parity is a fundamental relationship between European call and put option prices. It is a no-arbitrage condition that must hold in frictionless markets, and provides a powerful tool for pricing, hedging, and detecting arbitrage opportunities.

This section derives put-call parity, verifies it for the Black-Scholes formula, and explores its applications.

Statement of Put-Call Parity¶

Section goal: the identity \(C - P = S - Ke^{-rT}\) as a forward-replication relation.

1. The Parity Relationship¶

For European options on a non-dividend-paying stock with the same strike \(K\) and maturity \(T\):

or equivalently at time \(t=0\):

In words: The difference between call and put prices equals the difference between the current stock price and the present value of the strike.

Forward-contract reading. The right-hand side \(S - Ke^{-rT}\) is the time-\(0\) value of a forward contract to buy the stock at strike \(K\) on date \(T\). So put-call parity reads, fundamentally,

A long call combined with a short put at the same strike replicates this forward; the no-arbitrage derivation below makes the replication rigorous.

2. Alternative Forms¶

Rearranging gives equivalent expressions:

Each form is useful for different applications.

No-Arbitrage Derivation¶

Recall (see § Put-Call Parity: The Bridge): comparing Portfolio A (long call + zero-coupon bond paying \(K\)) with Portfolio B (long put + stock) yields identical terminal payoffs \(\max(S_T, K)\) in every state, so by no-arbitrage their initial values agree:

The derivation is model-free: it requires only a risk-free bond and the absence of arbitrage, not the Black-Scholes dynamics. Equivalently, "long call \(+\) short put" replicates a forward struck at \(K\) (slope-\(1\) line through \((K, 0)\)). The remainder of this section verifies parity algebraically against the BS formula and explores its applications.

Verification with Black-Scholes¶

Section goal: direct algebraic confirmation that the BS formulas satisfy parity.

1. Black-Scholes Formulas¶

Recall:

2. Compute C - P¶

3. Use Symmetry Property¶

For the standard normal CDF:

Therefore:

Verified: Put-call parity holds exactly for the Black-Scholes formula. ✓

Arbitrage Opportunities¶

Section goal: how to exploit a parity violation.

If put-call parity is violated, arbitrage opportunities exist.

1. Case 1: C - P > S - Ke^-rT (Call overpriced relative to put)¶

Arbitrage strategy:

1. Sell call (receive \(C\))
2. Buy put (pay \(P\))
3. Buy stock (pay \(S\))
4. Borrow \(Ke^{-rT}\) at rate \(r\)

Net cash flow at \(t=0\):

(positive cash inflow = free money)

At maturity \(T\):

• If \(S_T > K\): Exercise call against you (deliver stock for \(K\)), put expires, repay loan \(K\). Net: \(0\)
• If \(S_T \leq K\): Call expires, exercise put (sell stock for \(K\)), repay loan \(K\). Net: \(0\)

Result: Guaranteed profit at \(t=0\), zero cash flow at \(T\) → Arbitrage

2. Case 2: C - P < S - Ke^-rT (Put overpriced relative to call)¶

Reverse all four legs of Case 1: buy call, sell put, short stock, lend \(Ke^{-rT}\). The initial cash flow \(-C + P + S - Ke^{-rT} > 0\) is again pocketed risk-free, and the same case analysis at \(T\) shows all positions close at zero net.

Applications¶

Section goal: Greek symmetries, synthetic instruments, and replication uses of parity.

1. Synthetic Positions¶

Put-call parity allows creation of synthetic positions:

These are useful when an option is illiquid or mispriced, or when only one option trades actively (compute the other via \(P = C - S + Ke^{-rT}\)).

2. Early Exercise of American Options¶

The remainder of this subsection extends parity to American options and may be skipped on first reading.

For American options on non-dividend-paying stocks, equality breaks and parity becomes a two-sided inequality:

The upper bound uses \(C_{\text{Am}} = C_{\text{Eu}}\) (early exercise of an American call on a non-dividend stock is never optimal) together with \(P_{\text{Am}} \geq P_{\text{Eu}}\) and European parity. The lower bound captures the worst case from the call-minus-put holder's perspective — that the put is exercised against them immediately when \(S < K\), leaving net cash \(S - K\). The width of the band, \(K(1 - e^{-rT})\), is the early-exercise premium of the American put over the European put: it is exactly the interest the put holder forgoes by waiting, and it shrinks to zero as \(r \to 0\) or \(t \to T\). So early exercise is rationally driven by the time value of the strike payment, and the inequality \(C_{\text{Am}} - P_{\text{Am}} < S - Ke^{-rT}\) measures by how much. (Exercise 7 derives both bounds explicitly.)

3. Arbitrage Detection and Implied Rates¶

Compare observed market prices to put-call parity. Define the deviation:

If \(|\Delta|\) exceeds transaction costs, an arbitrage opportunity exists. Conversely, if call, put, and stock prices are known, one can extract the implied risk-free rate:

Generalizations¶

Recall (see § Put-Call Parity: The Bridge): parity extends by replacing \(S\) with the carry-adjusted spot — continuous dividends give \(C - P = Se^{-qT} - Ke^{-rT}\), discrete dividends give \(C - P = (S - De^{-rt_d}) - Ke^{-rT}\), FX (Garman-Kohlhagen) gives \(C - P = Xe^{-r_fT} - Ke^{-r_dT}\), and futures give \(C - P = e^{-rT}(F - K)\).

Numerical Example¶

Section goal: a worked check of parity at typical option parameters.

Market data:

• Stock price: \(S_0 = 50\)
• Strike: \(K = 50\)
• Time to maturity: \(T = 0.5\) years
• Risk-free rate: \(r = 4\%\)
• Call price: \(C_0 = 4.50\)

Question: What should the put price be?

Solution:

From put-call parity:

Answer: The put should be priced at $3.51.

Check: \(C - P = 4.50 - 3.51 = 0.99\) and \(S - Ke^{-rT} = 50 - 49.01 = 0.99\) ✓

Option Strategies from Parity¶

Section goal: synthetic forwards, conversions, and reverse conversions.

Put-call parity underpins several standard strategies. A conversion (long stock + long put + short call) locks in a risk-free payoff of \(K\) at maturity, equivalent to a bond. Its reverse, a reversal, creates a synthetic short bond. Combining conversions and reversals at two different strikes produces a box spread with guaranteed payoff \(K_2 - K_1\), which must cost \((K_2 - K_1)e^{-rT}\) by no-arbitrage.

Put-call parity was first rigorously derived by Hans Stoll (1969), before Black-Scholes. Unlike the pricing formula itself, parity is model-independent: it requires only the absence of arbitrage, not any specific distributional assumptions.

Summary¶

Put-call parity \(C - P = S - Ke^{-rT}\) is a model-independent no-arbitrage relationship. It enables synthetic position construction, arbitrage detection, and one-from-the-other pricing. It generalizes to dividends, foreign exchange, and futures. As a pure consequence of the law of one price, it is more fundamental than any specific pricing formula.

Exercises¶

Exercise 1. A stock trades at \(S_0 = 75\). A European call with \(K = 80\), \(T = 0.25\), and \(r = 5\%\) is priced at \(C_0 = 2.80\). Using put-call parity, determine the price of the corresponding European put.

Exercise 2. Verify put-call parity algebraically for the Black-Scholes formulas. Starting from \(C - P = [S\mathcal{N}(d_1) - Ke^{-rT}\mathcal{N}(d_2)] - [Ke^{-rT}\mathcal{N}(-d_2) - S\mathcal{N}(-d_1)]\), use the identity \(\mathcal{N}(x) + \mathcal{N}(-x) = 1\) to show that \(C - P = S - Ke^{-rT}\).

Exercise 3. The following prices are observed: \(S_0 = 100\), \(C_0 = 12.50\), \(P_0 = 8.00\), \(K = 100\), \(T = 1\), \(r = 3\%\). Check whether put-call parity holds. If it does not, describe the arbitrage strategy and compute the risk-free profit.

Exercise 4. Derive put-call parity for a stock paying continuous dividends at rate \(q\):

Construct the two replicating portfolios and show their terminal values are identical.

Exercise 5. A box spread consists of buying a bull call spread (\(K_1\), \(K_2\)) and buying a bear put spread (\(K_1\), \(K_2\)). Show that its payoff at maturity is always \(K_2 - K_1\), regardless of \(S_T\), by using put-call parity applied at both strikes. What should the box spread cost at time \(0\)?

Exercise 6. Using put-call parity, derive the implied risk-free rate from the following market data: \(S_0 = 50\), \(C_0 = 6.20\), \(P_0 = 4.10\), \(K = 50\), \(T = 0.5\). Solve for \(r\) and compare with prevailing treasury rates.

Exercise 7. Explain why put-call parity does not hold exactly for American options. Show that for American options on a non-dividend-paying stock, the following inequality holds:

Hint: use the fact that the American call equals its European counterpart, while the American put is worth at least as much as the European put.