No strategy involving the term arbitrage is necessarily “simple.” The very concept of arbitrage relies on technicalities and being very precise, so detailed discussions of arbitrage strategies are usually pretty tough. However, many of you may have seen the sort of mail I received from one of my brokers last year regarding some deep ITM call options on General Electric (GE) I held (N.B. I ended up rolling these options to January 2018 about a month before the 2016 expiry).
Notice the bold text just above the table: “Long options for which early exercise is projected to be economically beneficial”
This is a reference to something that people in the options world call “dividend arbitrage” — a strategy to generate riskless profits by exercising an option on a dividend paying stock (usually, exercising an option is a bad thing to do — for reasons we discuss in our training classes). The strategy revolves around a relationship called “Put-Call Parity,” which is a simple thing and important for an intelligent option investor to understand.
This article is also included as an appendix of The Framework Investing, but I get enough questions related to early option exercise, that I thought I would post it here.
Put-Call Parity
Before the Black-Scholes-Merton model (BSM), there was no way to directly calculate the value of an option, but there was a way to triangulate put and call prices as long as one had three pieces of data:
- The stock’s price
- The risk-free rate
- The price of a call option to figure the fair price of the put, and vice versa
In other words, if you know the price of either the put or a call, as long as you know the stock price and the risk-free rate, you can work out the price of the other option. These four prices are all related by a specific rule termed put-call parity.
Put-call parity is only applicable to European options, so it is not terribly important to stock option investors most of the time. The one time it becomes useful is when thinking about whether to exercise early in order to receive a stock dividend—and that discussion is a bit more technical. I’ll delve into those technical details in a moment, but first, let’s look at the big picture. Using the intelligent option investor’s graphic format employed in this book, the big picture is laughably trivial.
Direct your attention to the following diagrams. What is the difference between the two?
If you say, “Nothing,” you are practically right but technically wrong. The image on the left is actually the risk-reward profile of a purchased call option struck at $50 paired with a sold put option struck at $50. The image on the right is the risk-reward profile of a stock trading at $50 per share.
This simple comparison is the essence of put-call parity. The parity part of put-call parity just means that accepting downside exposure by selling a put while gaining upside exposure by buying a call is basically the same thing as accepting downside exposure and gaining upside exposure by buying a stock.
What did I say? It is laughably trivial. Now let’s delve into the details of how the put-call parity relationship can be used to help decide whether to exercise a call option or not (or whether the call option you sold is likely to be exercised or not).
Dividend Arbitrage and Put-Call Parity
Any time you see the word arbitrage, the first thing that should jump to mind is “small differences.” Arbitrage is the science of observing small differences between two prices that should be the same (e.g., the price of IBM traded on the New York Stock Exchange and the price of IBM traded in Philadelphia) but are not. An arbitrageur, once he or she spots the small difference, sells the more expensive thing and buys the less expensive one and makes a profit without accepting any risk.
Because we are going to investigate dividend arbitrage, even a big-picture guy like me has to get down in the weeds because the differences we are going to try to spot are small ones. The weeds into which we are wading are mathematical ones, I’m afraid, but never fear—we’ll use nothing more than a little algebra. We’ll use these variables in our discussion:
K = strike price
CK = call option struck at K
PK = put option struck at K
Int = interest on a risk-free instrument
Div = dividend payment
S = stock price
Because we are talking about arbitrage, it makes sense that we are going to look at two things, the value of which should be the same. We are going to take a detailed look at the preceding image, which means that we are going to compare a position composed of options with a position composed of stock.
Let’s say that the stock at which we were looking to build a position is trading at $50 per share and that options on this stock expire in exactly one year. Further, let’s say that this stock is expected to yield $0.25 in dividends and that the company will pay these dividends the same day that the options expire.
Let’s compare the two positions in the same way as we did in the preceding big-picture image. As we saw in that image, a long call and a short put are the same as a stock. Mathematically, we would express this as follows:
CK – PK = SK
Although this is simple and we agreed that it’s about right, it is not technically so.
The preceding equation is not technically right because we know that a stock is an unlevered instrument and that options are levered ones. In the preceding equation, we can see that the left side of the equation is levered (because it contains only options, and options are levered instruments), and the right side is unlevered. Obviously, then, the two cannot be exactly the same.
We can fix this problem by delevering the left side of the preceding equation. Any time we sell a put option, we have to place cash in a margin account with our broker. Recall that a short put that is fully margined is an unlevered instrument, so margining the short put should delever the entire option position. Let’s add a margin account to the left side and put $K in it:
CK – PK + K = S
This equation simply says that if you sell a put struck at K and put $K worth of margin behind it while buying a call option, you’ll have the same risk, return, and leverage profile as if you bought a stock—just as in our big-picture diagram.
But this is not quite right if one is dealing with small differences. First, let’s say that you talk your broker into funding the margin account using a risk-free bond fund that will pay some fixed amount of interest over the next year. To fund the margin account, you tell your broker you will buy enough of the bond account that one year from now, when the put expires, the margin account’s value will be exactly the same as the strike price. In this way, even by placing an amount less than the strike price in your margin account originally, you will be able to fulfill the commitment to buy the stock at the strike price if the put expires in the money (ITM). The amount that will be placed in margin originally will be the strike price less the amount of interest you will receive from the risk-free bond. In mathematical terms, the preceding equation becomes
CK – PK + (K – Int) = S
Now all is right with the world. For a non-dividend-paying stock, this fully expresses the technical definition of put-call parity.
However, because we are talking about dividend arbitrage, we have to think about how to adjust our equation to include dividends. We know that a call option on a dividend-paying stock is worth less because the dividend acts as a “negative drift” term in the BSM. When a dividend is paid, theory says that the stock price should drop by the amount of the dividend. Because a drop in price is bad for the holder of a call option, the price of a call option is cheaper by the amount of the expected dividend
Thus, for a dividend-paying stock, to establish an option-based position that has exactly the same characteristics as a stock portfolio, we have to keep the expected amount of the dividend in our margin account.[1] This money placed into the option position will make up for the dividend that will be paid to the stock holder. Here is how this would look in our equation:
CK – PK + (K – Int) + Div = S
With the dividend payment included, our equation is complete.
Now it is time for some algebra. Let’s rearrange the preceding equation to see what the call option should be worth:
CK = PK + Int – Div + (S – K)
Taking a look at this, do you notice that last term (S – K)? A stock’s price minus the strike price of a call is the intrinsic value. And we know that the value of a call option consists of intrinsic value and time value. This means that
So now let’s say that time passes and at the end of the year, the stock is trading at $70—deep ITM for our $50-strike call option. On the day before expiration, the time value will be very close to zero as long as the option is deep ITM. Building on the preceding equation, we can put the rule about the time value of a deep ITM option in the following mathematical equation:
PK + Int – Div ≈ 0
If the time value falls ever falls below 0, the value of the call would trade for less than the intrinsic value. Of course, no one would want to hold an option that has negative time value. In mathematical terms, that scenario would look like this:
PK + Int – Div < 0
From this equation, it follows that if
PK + Int < Div
your call option has a negative implied time value, and you should sell the option in order to collect the dividend.
This is what is meant by dividend arbitrage. But it is hard to get the flavor for this without seeing a real-life example of it. The following table shows the closing prices for Oracle’s stock and options on January 9, 2014, when they closed at $37.72. The options had an expiration of 373 days in the future—as close as I could find to one year—the one-year risk-free rate was 0.14 percent, and the company was expected to pay $0.24 worth of dividends before the options expired.
In the theoretical option portfolio, we are short a put, so its value to us is the amount we would have to pay if we tried to flatten the position by buying it back—the ask price. Conversely, we are long a call, so its value to us is the price we could sell it for—the bid price.
Let’s use these data to figure out which calls we might want to exercise early if a dividend payment was coming up.
There are only two strikes that might be arbitraged for the dividends—the two furthest ITM call options. In order to realize the arbitrage opportunity, you would wait until the day before the ex-dividend date, exercise the stock option, receive the dividend, and, if you didn’t want to keep holding the stock, sell it and realize the profit.
Notes
[1] A penny saved is a penny earned. We can think of the option being cheaper by the amount of the dividend, so we will place the amount that we save on the call option in savings.
[2] This is calculated using the following equation:
Interest = strike × r × percent of 1 year
In the case of the $18 strike, interest = 18 × 0.14% × (373 days/365 days per year) = $0.03.
Want to Learn More about Options? Contact Us!