### What is a Reverse Conversion?

A reverse conversion is a form of arbitrage that enables options traders to profit from an overpriced put option no matter what the underlying does. The trade consists of selling a put and buying a call to create a synthetic long position while shorting the underlying stock. As long as the put and call have the same underlying strike price and expiration date, the synthetic long position will have the same risk/return profile as ownership of an equivalent amount of underlying stock. This means the trade (short stock and synthetically long the stock with options) is hedged, with the profit coming only from the mispricing of the option premiums.

### Key Takeaways

- A reverse conversion is an arbitrage situation in the options market where a put is overpriced or a call underpriced (relative to the put), resulting in a profit to the trader no matter what the underlying does.
- The reverse conversion is created by shorting the underlying, buying a call, and selling a put. The call and put have the same strike price and expiry.
- A put option is typically cheaper, not more expensive, than the equivalant call option. Therefore, finding a reverse conversion trade is rare.

### Understanding Reverse Conversion

A reverse conversion is a form of arbitrage that enables traders to capitalize on situations where the put is overpriced by selling a put, buying a call, and shorting the underlying stock. The call covers the short stock if it rises above the strike price and the put is covered by the shorted stock if the stock price is less than the strike price. The profit when initiating these positions is the theoretical arbitrage gain from the position that comes from the overpriced put option.

Reverse conversion arbitrage is a type of put-call parity, which says that put and call option positions should be roughly equal when paired with the underlying stock and T-bills, respectively.

To illustrate this concept, consider two scenarios:

- A trader that purchases one call option with a strike price of $100 and a T-bill with a face value of $10,000 that matures on the call's expiration date will always have a minimum portfolio value equal to $10,000.
- A trader that purchases one put option with a strike price of $100 and $10,000 worth of stock will always have a minimum value equal to the strike price of the put option.

Since traders are unlikely to have an impact on interest rates, arbitrageurs take advantage of option price dynamics to ensure that the put and call values equal out for a given asset. In fact, the put-call parity theorem is central to determining the theoretical price of a put option using the Black-Scholes model.

### Example of a Reverse Conversion

In a typical reverse conversion transaction or strategy, a trader short sells stock and hedges this position by buying its call and selling its put. Whether the trader makes money depends on the borrowing cost of the shorted stock and the put and call premiums. All of this may render a return better than a low-risk money market trade, which is where their capital would likely be parked if they can't find worthwhile trades.

Assume it is June and a trader sees an overpriced October put in Apple Inc. (AAPL).

The stock is currently trading at $190, and so they short 100 shares. They buy a $190 call and short a $190 put.

The call costs $13.10, while the put is trading at $15. This is a rare situation since puts typically cost less than equivalent calls.

No matter what happens the trader makes a profit of $190 ($15 - $13.10 = $1.90 x 100 shares), less commissions.

Here's how the scenarios could play out. The trader initially receives $19,000 from the stock short sale, as well as $1,500 from the sale of the put. They pay $1,310 for the call. Therefore, their net inflow is $19,190.

Assume Apple drops to $170 over the life of the option. The call expires worthless, the short stock is assigned because the put buyer is in the money. The trader provides the shares at $190 costing $19,000. Another way to look at it is that the put is losing $2,000 while the short stock is making $2,000. The two items net out at the strike price of $190 which is equivalent to $19,000 on a one contract (100 shares) position.

If the price of Apple rises to $200, the put option expires worthless, the short position is covered at a cost of $20,000 ($200 x 100 shares) but the call option is worth $1,000 ($200 - $190 x 100 shares), netting the trader $19,000 in costs.

In both cases, the trader pays $19,000 to exit the position, but their initial inflow was $19,190. Their profit, whether Apple rises or falls, is $190. This is why this is an arbitrage trade.

Borrowing costs and commissions have been excluded in the example above for simplicity but are an important factor in the real world. Consider, for example, that if it costs $10 in commissions per trade, it will cost $30 to initiate the three required positions. The cost to exit will depend on whether the options are exercised or if the positions are exited prior to expiration. In either case, there will be additional costs to exit the trade, plus the borrowing costs on the shorted stock.