Now for this week: we're kicking off the post with an image of pack mules as a play on one of today's topics, cost-to-carry. We'll be getting into some math that option owners use to decide when the time might be right to exercise -- and I figured pictures of gorgeous canyons might grab your attention faster than pictures of math symbols. Anyway, one of the key factors in those calculations is cost-to-carry.

The other big factor is synthetic relationships, or how one pair of options positions can have very similar risks and rewards to an entirely different pair, making them essentially synthetic versions of each other. Let's actually start by explaining this concept.

(This post also includes a little bonus: we're going to explore a very important relationship, that between the price of a put and the price of a call when both have same strike price and expiration date. In order to do the early exercise topic justice, it's useful to explain this relationship first.)

Can X actually equal Y?

Sometimes, yes. Consider the risk, rewards and market outlook of the following pair: long stock plus a long put. In many ways, this resembles holding just a long call: you benefit if the market rises, and you have limited and known risk if the market falls.

The profit and loss graphs below make this similarity a little clearer. The X-axis refers to the stock price at expiration, and the y-axis refers to the profit or loss for each position. Let's assume the stock is at 80, the interest rate is 5.3613%, volatility is 27.95% and expiration is 61 days away, with no dividend upcoming. Let's also assume we're using the ATM put and call in this example, with a strike price for each of 80.

The first graph shows the P&L of a long stock position (gray), the P&L for a long put (green), and the P&L for the combined position (red). Compare this red line with the second graph, which depicts the simple P&L of the long call (blue). It's easy to see that buying the protective put and buying the call results in a very similar P&L at expiration.

Before we run away and say they're not just similar but identical, let's graph both on the same graph and see if they are any subtle differences.

When we graph each position on top of the other (protective put in red and long call in blue), the point where each strategy crosses the X-axis is called the breakeven point at expiration. Notice the small difference in the breakeven number of these two positions (83.35 vs 84). That's because the put is trading for a little less than the call. Why? It has to or there would be an arbitrage situation. Let me explain: we have two strategies that are mostly the same as far as risk and reward are concerned, but not with regard to carry costs.

Imagine you had no money and wanted to place this trade, and your broker was willing to lend you the money required. If this all came together, and the put and call were trading for the same amount, which trade would you use your funds on: a protective put or a long call? I hope you said long call, because if the put and call are exactly the same price, it makes sense to choose the call. With the call, you won't have to borrow additional money to buy the stock (remember, a protective put is a long put PLUS long stock), and your breakeven would be same. Since both trades have basically the same risk and reward, to make them truly comparable we have to even things out. How? If we take the carry-cost of the stock out of the price of the put, then everything comes into equal balance. Here is the quick and dirty math involved:

Cost of stock = 100 x 80 = 8000

Carry-cost of stock = 8000 x .053613 = 428.90

(If we could put this 8000 dollars into an interest-bearing cash account, this is how much money we'd earn annually in interest. By holding long stock, we're forgoing that interest income: that's what "cost-to-carry" actually means.)

Translate to days = 428.90/365 = 1.17 of interest income lost per day

Carry-costs over life of position =1.17 x 61 days to expiration = 71.37/100 = 0.71

Compare this number, 0.71 to the different between breakevens in our graph above, which is .75 (84 - 83.35 = .75). That's pretty much identical, but not exactly. Even though 75 cents might seem like a small discrepancy, it's an important one and not a rounding error. In fact, this relationship has to hold true or institutional traders would buy the cheap strategy and sell the expensive strategy and collect money above and beyond the carry-cost with no additional risk. That's the arbitrage opportunity I referred to before.

Time to play market-maker

To understand why this matters, put on your mesh jacket and imagine you're a market-maker for a second. An apparently tiny difference like this matters more to market makers than retail customers because of their role to provide liquidity to the marketplace. That role often boxes them into positions they didn't plan on holding, so they're always considering alternative "ways out" of various trades.

Market makers are also looking for the smallest edge they can possibly get. Given their huge size and very low transactional costs, it can make financial sense for them to find a trade that makes them a penny and try to make that penny 10,000 times in a single day.

From our graph discussion above, you know being long that ITM call has pretty much the same risks and rewards as being long stock and long a put. The big question a market marker is asking himself or herself is this: when does it make more sense for me to convert my long ITM call into its synthetic equivalent (long stock + long put)? The decisions they make on that point impacts early assignment risk for individual investors like you and me, so it pays to stay aware of them.

Cost-to-carry

That question brings us back to the pack mules and cost-to-carry, because the decision hangs on whether it's cheaper to hold the long call or the long-stock-plus-long-put, "cheaper" in terms of the interest costs.

Let's evaluate each choice and weigh them against each other. Say you exercise that long call: you'll capture the ITM value, but you'll also lose any time-value the calls had left. You'll also lose the limited risk of holding a call.

After you've exercised, you're long stock. To limit the risk on that position, you'd add a long put at the same strike price. Whew! Now you're protected on the downside again.

But there's a new problem: buying stock costs money. To pay for long stock, you had to either borrow money or use your cash, and either way, it's costing you money in interest. You can figure out exactly how much interest costs you with the following formula:

Strike price x interest rate x days to expiration
_______________________________________ 
                                    365

Let's apply this to a concrete example. You bought December 80 calls on stock XYZ, and you had 44 days left until expiration. Let's say the current interest rate is 5.3%.  That's:

80 x 0.053 x 44     = 0.51
____________    
          365  

In other words, you're paying about 51 cents in interest for every share of XYZ you're holding, from now until expiration. If you're long 100 shares, that's 51 dollars in interest costs. (If you're a monstro market-maker, you might be multiplying that figure by a lot more than 100 shares, so this cost can add up.)

Dividends' effect on cost-to-carry

Let's throw another tempting factor into the mix: maybe you're motivated to capture an upcoming dividend. Certainly receiving a dividend could offset the cost of buying the put and your interest costs.

To get the dividend, you have to own the stock before the ex-dividend date, which explains the spike in early exercise you'll usually see for calls on stocks with upcoming large dividends.

If you were asking yourself above: "How do I know if 51 dollars is a lot or a little in terms of cost-to-carry?", here comes the answer. As the owner of the 100 strike call, you're weighing the dividend versus buying the 100 strike put plus the lost interest.

Dividend VS. buying put + lost interest income

So let's do the math for this example. Let's say the December 80 put costs 0.50, and the upcoming dividend is 0.20. Scoring that 0.20 dividend doesn't seem as attractive as saving 1.01 (0.50 cost of buying put + 0.51 in lost interest = 1.01), right? When you compare the 0.20 dividend to the combined cost of buying the put plus interest, as the option owner you'd probably decide not to exercise.

That math might change if we were closer to expiration when the ex-dividend date arrives. If you were only 10 days from expiration, and still anticipating a dividend, the above calculation for interest costs comes out differently:

80 x 0.053 x 10 = 0.12
_____________
          365

Now you're comparing the same 0.20 dividend to interest costs (0.12) plus the cost of buying the put (0.05) -- 0.20 versus 0.17. (Here's how I got that 0.05 estimated price for the put. If expiration is really close and the call is way ITM, the put will be way OTM, making the December 80 put pretty cheap - only 0.05.) That 0.20 dividend, in other words, starts to look a lot more attractive when it costs you only 0.17 to get it. Again, it could be MUCH more attractive if you're a market maker dealer who can multiply that 3-cent gain over tens of thousands of shares.

A few caveats

If you're a put owner considering early exercise on a dividend-paying stock, the situation is reversed. If you were to exercise, you'd get short stock, which generates cash (plus the interest on that cash) in your account.

As a short stock holder, though, you'd be obligated to pay the dividend to the actual owner of the shares you shorted. That's why most put owners don't exercise a put around an ex-dividend date, because the interest received from selling the shares usually isn't enough to offset the cost of paying out the dividend. (These facts notwithstanding, keep in mind that puts generally seem to get exercised more often the calls, as outlined in last week's post.)

Another caveat is the interest rate you use in your calculation. Market makers can get very different interest rates from what you or I can get. If you were only paying a 4% borrowing rate in the above example, the decision to exercise at 44 days until expiration might look a lot more attractive than it does at 5.3%. Your best bet is to use the current broker call rate to make your calculations.

Now let's leave the fantasy world of market makers behind, with their ultra-low commissions and unlimited margin, and return to us regular investors. As a long call owner and individual investor, does it usually make since to exercise a call to capture a dividend? Usually, it doesn't, so don't try to get too fancy in your strategy. Bottom line is this: if you're long a call and you want out, for individual investors it usually makes most to just sell it. What we've covered today is useful information for the call seller trying to assess his or her risk of early assignment. Many times a market marker will be on the other side of that trade, and as the seller you are not in control of what happens next.

That's it for today, folks. Next week we'll get into some more complex scenarios: how to recover if your spread gets hit with a surprise early assignment. See you next week!

Regards,
Brian (OG)

[image: GC07 SouthKaibab 3815 by erthsister on flickr]

Options involve risk and are not suitable for all investors. Please read Characteristics and Risks of Standardized Options.

While implied volatility represents the consensus of the marketplace as to the future level of stock price volatility or probability of reaching a specific price point there is no guarantee that this forecast will be correct.

Any strategies discussed or securities mentioned, are strictly for illustrative and educational purposes only and are not to be construed as an endorsement, recommendation, or solicitation to buy or sell securities.