How do you use put-call parity to create synthetic positions and spot arbitrage opportunities?
I understand the basic put-call parity formula C - P = S - PV(K), but I'm struggling to apply it. How do I actually use this to construct synthetic positions like a synthetic call or synthetic stock? And how would I identify an arbitrage opportunity if parity is violated?
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:
| Synthetic Position | Formula | Components |
|---|---|---|
| Synthetic call | C = P + S - PV(K) | Buy put + buy stock + borrow PV(K) |
| Synthetic put | P = C - S + PV(K) | Buy call + short stock + lend PV(K) |
| Synthetic stock | S = C - P + PV(K) | Buy call + sell put + lend PV(K) |
| Synthetic bond | PV(K) = S - C + P | Buy stock + sell call + buy put |
Arbitrage Example: Suppose Oakridge Semiconductor stock trades at 80) is priced at 2.80. The risk-free rate is 4%.
First, compute PV(K): 80 / 1.0198 = $78.45
Check parity: C - P = 2.80 = 85.00 - 6.55
The left 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: -2.80 + 78.45 = -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.
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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.
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