How do you use put-call parity to create synthetic positions and spot arbitrage opportunities?

Put-call parity is one of the most powerful relationships in derivatives, and the CFA Level II exam tests both the mechanics and the applications. Let's break it down.

The Fundamental Relationship:

C + PV(K) = P + S

Or equivalently: C - P = S - PV(K)

Where C = European call price, P = European put price, S = current stock price, K = strike price, and PV(K) = present value of the strike discounted at the risk-free rate.

Creating Synthetic Positions:

You can rearrange the equation to replicate any component:

Arbitrage Example:

Suppose Oakridge Semiconductor stock trades at $85. A 6-month European call (K = $80) is priced at $9.50, and the corresponding put is priced at $2.80. The risk-free rate is 4%.

First, compute PV(K): $80 / (1.04)^0.5 = $80 / 1.0198 = $78.45

Check parity: C - P = $9.50 - $2.80 = $6.70

S - PV(K) = $85.00 - $78.45 = $6.55

The left side ($6.70) exceeds the right side ($6.55) by $0.15. The call is relatively overpriced (or the put is relatively underpriced).

Arbitrage Strategy:

• Sell the call at $9.50
• Buy the put at $2.80
• Buy the stock at $85.00
• Borrow PV(K) = $78.45 at the risk-free rate

Net initial cost: -$9.50 + $2.80 + $85.00 - $78.45 = -$0.15 (you receive $0.15)

At expiration, regardless of the stock price, your position nets to zero (the protective put + short call creates a floor and ceiling at K). You keep the $0.15 risk-free profit.

Key exam tip: Put-call parity only holds for European options on the same underlying, same strike, and same expiration. American options can deviate because of the early exercise premium. The exam often includes distractors that confuse European and American option scenarios.

Practice more derivatives problems in our CFA Level II question bank on AcadiFi.