For the past couple of weeks I’ve been writing about creating synthetic positions, which means making a position that looks like a call out of stock and puts, or vice versa. Today I’ll wrap up that topic for now, with an extension of last week’s discussion about conversions and reversals (which you can read here).
Last week, I mentioned that it would be possible to take advantage of mis-pricing of options. Here are the details.
I noted that an arbitrage opportunity would exist whenever puts or calls were overpriced in relation to each other. As a quick review, below is the equation that describes the relationship between stock and option prices. It is one of the expressions of the concept of put-call parity:
For a put and a call at the same strike price,
Call Price = Put price + Stock price – Strike Price
If the two sides of this equation are out of balance, a guaranteed profit can be made by betting that the inequality will go away. This can be done by creating a conversion (if the call price is too high); or a reverse conversion, which is also called a reversal, if the call price is too low. By the time the options expire, the inequality must go away. Here is why.
At expiration, no time value remains, and every option is worth exactly its intrinsic value. All options that are in the money (ITM), by any amount, are worth exactly that in-the-money amount. All options that are out of the money (OTM), are worth zero.
Let’s look at a stock, its $100 strike put, and its $100 strike call.
On expiration day, here are some example stock prices, and the resulting option prices.
|Strike Price||Stock Price||Call Price||Put Price|
|K||S||C||P||Put Price +Stock-Strike|
As you can see, all of these sets of prices satisfy the put-call parity equation above, which is C = P+S-K. Put-call parity is simply what must be true as a consequence of the fact that all options are worth their intrinsic value at expiration.
Before expiration, there are a couple of small additions to be made to the put-call parity equation to account for interest and dividends. The idea is that the buyer of a call could be earning interest on the cash he’s saving by not buying the stock; so we add the interest he could make to the call price. On the other hand, the call buyer is also not getting the dividends that the stock owner would get. So we have to mark down the call by the amount of the dividends to be received during the call’s lifetime. The revised formula is:
Call Price = Put Price + Stock Price – Strike Price + Interest on Stock price – Dividends
Using I for Interest and D for Dividends, we can write
C = P + S – K + I – D
This can be rearranged to solve for any of the variables when the others are known. If we know the price of the stock and the put, we can compute the price of the call. If we know the call price, we can compute the put price.
If at any time before expiration this equation doesn’t compute, then the options are mispriced. The amount of the mispricing can be captured by selling short the overpriced option and buying the underpriced one. As we do this, to keep the sold option from being naked, we also take a position in the underlying. We buy the underlying if we’re shorting the call (creating a covered call). We sell the underlying short if we’re shorting the put (creating a covered put write).
OK, so just how would we realize our profit? By unwinding the trade whenever the option prices change so that the equation balances again. If this doesn’t happen before expiration, it must certainly happen then. At expiration time, we will exercise or be assigned on whichever option is in the money; the stock we receive or give as a result of that exercise will automatically liquidate our stock position. What we’re left with will be profit, in the amount of the original mis-pricing.
Using the example above, assume that the stock is at $101. Of the options at the $100 strike, the Put is priced at $2 and the call is priced at $2.90. Interest rate (the risk-free US Treasury rate) is 0% and a $.25 dividend is expected before expiration.
So the call price should equal P + S – K + I – D. In this case that’s $2 + $101 – $100 + (0 * $100) – $.25 = $2.75. But the call is actually priced higher than that, at $2.90. The call is overpriced by $.15, which could be captured using a conversion.
The conversion would work like this: We could buy the stock at $101.00; sell the call at $2.90; buy the put at $2.00. Net cash out of pocket is $100.10.
A little later, we’d collect the dividend of $.25, since we own the stock. Subtracting that from our $100.10 cost makes the net cash out of pocket $99.85.
Expiration arrives. If the stock is still at 101, the put expires worthless. The call will be assigned, so we’d lose the stock and be paid $100 for it. Subtracting our adjusted cost of $99.85, we have a profit of $.15. This is exactly the amount of the call’s overpricing.
What if the stock is not unchanged – say it’s dropped from 101 to $99. In that case, the short call expires worthless. We exercise our put and sell the stock for $100. Subtracting our $99.85 cost, we once again have a profit of $.15. And this will be true no matter what the price of the stock is. The profit was locked in at the moment we bought the conversion.
Since that’s true, any trader who sees that mis-pricing opportunity will get busy doing conversions, which means selling calls and buying puts. Before long the buying pressure on the puts and the selling pressure on the calls will force them back into their correct relationship. And because these relationships reliably hold, the existence of synthetics is made possible.
Comprehending these relationships will help you to understand the workings of options, including all the possibilities that exist in synthetic positions.
For questions or comments on this article, contact me at firstname.lastname@example.org.