Can someone explain put-call parity with a real arbitrage example? I don't get why it must hold.

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AcadiFi · Accepted Answer

Put-call parity is one of the most elegant relationships in finance, and once you see the arbitrage logic, it clicks permanently. Let me walk you through it from the ground up.

**The Formula:**

**c + PV(X) = p + S**

Where:
- c = price of a European call option
- PV(X) = present value of the strike price (i.e., X / (1 + r)^T)
- p = price of a European put option
- S = current stock price

Both options have the **same strike price (X)** and **same expiration date (T)**.

**The Intuition:**

The left side (call + bond) and the right side (put + stock) produce **identical payoffs at expiration** in every possible scenario:

| At Expiration | S_T > X | S_T < X | S_T = X |
|---------------|---------|---------|----------|
| Call + Bond | (S_T - X) + X = S_T | 0 + X = X | 0 + X = X |
| Put + Stock | 0 + S_T = S_T | (X - S_T) + S_T = X | 0 + X = X |

Both portfolios always pay S_T when the stock finishes above X, and X when below. Since the payoffs are identical, the prices **must** be equal. Otherwise, free money exists.

**Arbitrage Example:**

Cedarline Corp stock trades at **S = $75**. European options with a 1-year expiration and strike of $80 are available. The risk-free rate is 5%.

- Call price: c = $6.00
- Put price: p = $8.50

**Check if parity holds:**

- Left side: c + PV(X) = $6.00 + $80 / 1.05 = $6.00 + $76.19 = **$82.19**
- Right side: p + S = $8.50 + $75.00 = **$83.50**

The right side is **overpriced** by $1.31. Arbitrage strategy: **sell the expensive side, buy the cheap side.**

**The Trade:**
1. **Sell** the put for $8.50
2. **Short sell** the stock for $75.00
3. **Buy** the call for $6.00
4. **Invest** $76.19 in a risk-free bond (matures to $80.00)

**Net cash flow at initiation:**
+$8.50 + $75.00 - $6.00 - $76.19 = **+$1.31** (immediate profit)

**At expiration — two scenarios:**

**If S_T = $90 (stock rises above strike):**
- Call payoff: $90 - $80 = +$10
- Put expires worthless: $0
- Bond matures: +$80
- Buy stock to cover short: -$90
- Net: +$10 + $0 + $80 - $90 = **$0**

**If S_T = $65 (stock falls below strike):**
- Call expires worthless: $0
- Put exercised against you: -($80 - $65) = -$15
- Bond matures: +$80
- Buy stock to cover short: -$65
- Net: $0 - $15 + $80 - $65 = **$0**

In both cases, the expiration cash flow is **zero**, and you already pocketed **$1.31 at initiation**. That's risk-free profit.

**Key Exam Points:**
- Put-call parity applies only to **European** options
- If the equation doesn't balance, sell the overpriced side and buy the underpriced side
- The relationship can be rearranged: p = c + PV(X) - S (to find any one variable)

For more derivatives practice with arbitrage scenarios and option strategies, check out our CFA Level I question bank on AcadiFi.

Can someone explain put-call parity with a real arbitrage example? I don't get why it must hold.

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