Conversion Arbitrage: Locking a Parity Mispricing

Key Takeaways

• Conversion arbitrage holds long stock, a long put, and a short call at one strike and date.
• It captures a put-call parity violation when the call is rich relative to the put.
• The locked profit is fixed and direction-neutral once the trade is executed.
• Edges are tiny and vanish fast, so commissions and assignment risk usually decide profitability.

What It Is

A conversion combines a long stock position with a synthetic short stock built from options: a long put and a short call at the same strike and expiration. The synthetic short cancels the price risk of the real shares.

What is left is a fixed payoff determined by the strikes, the option prices, and carry. Conversion arbitrage uses this structure to exploit a mispricing in put-call parity, the law that links the prices of a call, a put, the stock, and a bond. When the relationship breaks, the conversion captures the gap.

The Intuition

Put-call parity says a long call and a short put at the same strike equal a financed position in the stock. If the call is too expensive relative to the put, you want to sell the rich call and buy the cheap put, then offset with stock so direction does not matter.

That is exactly a conversion. You sell the overpriced call, buy the underpriced put, and own the shares. The combined position cannot lose to price moves because the long stock and the synthetic short net out. You simply collect the mispricing.

How Conversion Arbitrage Works

The legs and the parity condition:

Conversion = long stock + long put + short call   (same K, same expiration)

Put-call parity (no dividends):
  C - P = S - K * e^(-rT)

Conversion profits when the call is rich:
  C - P  >  S - K * e^(-rT)

At expiration the stock is either above or below the strike. If above, the short call is assigned and you deliver your shares at the strike. If below, you exercise your long put and sell your shares at the strike. Either way you receive the strike, so your outcome was fixed the moment you opened the trade. The profit is the parity gap minus carry, commissions, and any dividends.

Worked Example

A stock trades at 50. The 50-strike call is trading at 2.20 and the 50-strike put at 1.90, both with the same near-term expiration. Assume negligible carry and no dividend for simplicity.

Parity says the call minus the put should roughly equal the stock minus the discounted strike, which is near zero here. Instead the call is 0.30 richer than the put. You build the conversion: buy 100 shares at 50, buy the 50 put for 1.90, sell the 50 call for 2.20. The net option credit is 0.30.

If the stock finishes at 55, the short call is assigned and you sell shares at 50; the put expires worthless. If it finishes at 45, you exercise the put and sell at 50; the call expires worthless. In both cases you end with the strike plus the 0.30 option credit, locking roughly 30 dollars per contract before costs. The profit did not depend on direction.

Common Mistakes

1. Underpricing the costs. The edge is often a few cents per share. Commissions, the bid-ask spread on three legs, and financing on the stock can swallow the entire profit.

2. Ignoring dividends. A dividend before expiration changes parity and the early-exercise calculus on the short call. Miss it and the locked profit can become a loss.

3. Forgetting early assignment. An American short call deep in the money, especially near an ex-dividend date, can be assigned early, unwinding the hedge before you intended.

4. Assuming the edge persists. Market makers correct parity violations in seconds. Retail traders rarely see the mispricing before it closes.

5. Confusing conversion with reversal. A conversion holds long stock when the call is rich. A reversal shorts stock when the put is rich. Mixing them reverses your exposure.

Frequently Asked Questions

What is conversion arbitrage in simple terms? Conversion arbitrage holds shares plus a long put and a short call at one strike to pocket a pricing error. When the call is too expensive next to the put, you lock a small fixed profit no matter where the stock moves.

How does conversion arbitrage affect investment decisions? It is a market-neutral way to capture a put-call parity mispricing rather than a directional bet. In the worked example, a 0.30 parity gap produced about 30 dollars of locked profit per contract before costs.

What is a real-world example of conversion arbitrage? A market maker sees a 50-strike call trading richer than parity allows, sells the call, buys the matching put, and buys the stock, banking the gap with no exposure to the stock's direction.

How can investors avoid losing the edge in conversion arbitrage? Account for commissions, spreads, dividends, and financing before trading, prefer liquid names with tight markets, and watch ex-dividend dates that can trigger early assignment on the short call.

How is conversion arbitrage different from reversal arbitrage? A conversion uses long stock with a long put and short call when the call is overpriced. A reversal uses short stock with a long call and short put when the put is overpriced. They are mirror images.

Sources

1. OIC (The Options Industry Council). "Put/Call Parity." https://www.optionseducation.org/advancedconcepts/put-call-parity
2. Damodaran, A. (NYU Stern). "Options Arbitrage." https://pages.stern.nyu.edu/~adamodar/New_Home_Page/invfables/optionarb.htm
3. Corporate Finance Institute. "Put-Call Parity." https://corporatefinanceinstitute.com/resources/derivatives/put-call-parity/
4. Cboe Options Institute. "Options Education and Strategy Resources." https://www.cboe.com/optionsinstitute/

Disclaimer

This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.