How do you create synthetic positions using options, and what is put-call parity?

Synthetic positions are one of the most elegant concepts in derivatives. By combining options and/or the underlying asset, you can replicate the payoff of another instrument. Put-call parity is the equation that makes this possible.

Put-Call Parity (European Options)

c + PV(X) = p + S

Where:

• c = European call premium
• p = European put premium
• S = Current stock price
• PV(X) = Present value of the strike price (discounted at the risk-free rate)

This equation says: owning a call + lending money = owning a put + owning the stock. Both sides produce identical payoffs at expiration.

Rearranging for Synthetic Positions:

Numerical Example — Evergreen Tech Stock

Evergreen Tech trades at $80. European options with 6-month expiry, K = $80:

• Call price: $6.50
• Put price: $4.20
• Risk-free rate: 5% annually
• PV(X) = $80 / (1.05)^0.5 = $78.06

Verify parity: $6.50 + $78.06 = $84.56 vs $4.20 + $80 = $84.20

The small difference ($0.36) reflects transaction costs and bid-ask spreads. In theory, they should be equal.

Synthetic Long Stock:

Buy the call ($6.50) + Sell the put ($4.20) + Invest PV(X) = $78.06

Net cost: $6.50 - $4.20 + $78.06 = $80.36 (approximately $80, the stock price)

At expiration:

• If S_T > 80: Exercise call, receive stock worth S_T. Put expires. Bond pays $80, used to pay strike.
• If S_T < 80: Call expires. Put is exercised against you, you buy stock at $80 from bond proceeds. You own stock worth S_T.

In both cases, you end up with the stock — identical to just buying it directly.

Why Synthetic Positions Matter:

1. If the call is mispriced relative to the put, you can create an arbitrage using put-call parity
2. Sometimes it's cheaper to replicate a position synthetically than to buy it directly
3. Short-selling restrictions can be bypassed with a synthetic short (sell call + buy put)

Exam Tip: CFA Level I commonly gives you three of the four components and asks you to determine the fourth using put-call parity. Memorize the rearrangements.

Practice synthetic positions in our CFA Derivatives question bank.