Put-Call Parity

Easy2–3 hrsPut-Call ParityNo-ArbitrageSynthetic

Overview

Put-call parity is the bedrock of options pricing — derived purely from no-arbitrage, not from any model. It says: a portfolio of (long call, short put, same strike/expiry) must equal a forward contract. If it doesn't, you have free money. In interviews, always check PCP before doing any more complex options math — if the market violates it, the question might be asking you to find the arbitrage.

How to Recognize

→Given put price, asked for call price (or vice versa)
→Asked to construct a synthetic position
→'Is this options price arbitrage-free?'
→Forward price relationship to options

Step-by-Step Approach

1.PCP: C − P = S − K·e^(−rT) — always start here
2.If violated: buy the cheap side, sell the expensive side → riskless profit
3.Synthetic long call = long put + long stock + short bond
4.For dividends: replace S with S − PV(Div) in the formula
5.For futures: C − P = (F − K)·e^(−rT)

Key Formulas

C - P = S - Ke^{-rT}

(European, no dividends)

C - P = F e^{-rT} - Ke^{-rT} = (F-K)e^{-rT}

(futures)

03Synthetic long forward: long call + short put (same strike) = S − Ke^{−rT}

P = C - S + Ke^{-rT}

→ put from call

Worked Examples

0Quick Example

100, K=

100, r=5%, T=1yr, C=

10. PCP: P = 10 − 100 + 100·e^(−0.05) = 10 − 100 + 95.12 =

5.12.

Problem

S= $50, K=$ 50, r=0%, T=1yr. A call trades at $6, a put trades at$ 8. Is there an arbitrage?

Solution

PCP: C − P should equal S − K·e^(−rT) = 50 − 50 = 0 (since r=0%).

Market: C − P = 6 − 8 = −2. But PCP says it should be 0.

Put is 'overpriced' relative to the call (or call is underpriced).

Arbitrage: buy the underpriced side, sell the overpriced side.

Buy call ( $6), sell put ($ 8), sell stock ( $50), invest$ 50 at 0% (hold cash).

Cash inflow now: 8 − 6 + 50 − 50 = $2. At expiry, all positions cancel regardless of S. Lock in$ 2 profit.

Answer

Arbitrage profit: buy call + sell put + sell stock = net + $2 today. At expiry: if S>50, exercise call (+S−50), put expires (0), buy back stock (−S) → 0 net. If S<50, call expires (0), put exercised (pay 50−S), buy stock (−S) → 0 net. Always pocket the$ 2.

Common Mistakes

!
Forgetting to discount the strike: it's K·e^(−rT), not K, when interest rates are nonzero.
!
Applying PCP to American options as an equality — it's only a bound.
!
Confusing forward price F with spot S. For futures: use F instead of S·e^(rT).
!
Signs: C − P = S − Ke^(−rT), not the other way.

Practice Problems

Click "Show Answer" to reveal

C= $5, S=$ 100, K=$95, r=10%, T=0.5yr. What is the put price?

Hint: P = C − S + K·e^(−rT). Compute e^(−0.05) ≈ 0.9512.

Stock pays a $3 dividend in 6 months. S=$ 50, K= $50, r=0%, T=1yr. European call =$ 6. What is the put price?

Hint: Adjust S: use S − PV(Div) = 50 − 3 = 47.

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