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CHAPTER 6
Put-Call Parity and Synthetics
In order to understand more complex spread strategies involving two or more options, it is essential to understand the arbitrage relationship of the put-call pair. Puts and calls of the same month and strike on the same underlying have prices that are defined in a mathematical relationship. They also have distinctly related vegas, gammas, thetas, and deltas. This chapter will show how the metrics of these options are interrelated. It will also explore synthetics and the idea that by adding stock to a position, a trader may trade with indifference either a call or a put to the same effect.
Put-Call Parity Essentials
Before the creation of the Black-Scholes model, option pricing was hardly an exact science. Traders had only a few mathematical tools available to compare the relative prices of options. One such tool, put-call parity, stems from the fact that puts and calls on the same class sharing the same month and strike can have the same functionality when stock is introduced.
For example, traders wanting to own a stock with limited risk can buy a married put: long stock and a long put on a share-for-share basis. The traders have infinite profit potential, and the risk of the position is limited below the strike price of the option. Conceptually, long calls have the same risk/reward profile—unlimited profit potential and limited risk below the strike.
Exhibit 6.1
is an overview of the at-expiration diagrams of a married put and a long call.
EXHIBIT 6.1
Long call vs. long stock + long put (married put).
Married puts and long calls sharing the same month and strike on the same security have at-expiration diagrams with the same shape. They have the same volatility value and should trade around the same implied volatility (IV). Strategically, these two positions provide the same service to a trader, but depending on margin requirements, the married put may require more capital to establish, because the trader must buy not just the option but also the stock.
The stock component of the married put could be purchased on margin. Buying stock on margin is borrowing capital to finance a stock purchase. This means the trader has to pay interest on these borrowed funds. Even if the stock is purchased without borrowing, there is opportunity cost associated with the cash used to pay for the stock. The capital is tied up. If the trader wants to use funds to buy another asset, he will have to borrow money, which will incur an interest obligation. Furthermore, if the trader doesn’t invest capital in the stock, the capital will rest in an interest-bearing account. The trader forgoes that interest when he buys a stock. However the trader finances the purchase, there is an interest cost associated with the transaction.
Both of these positions, the long call and the married put, give a trader exposure to stock price advances above the strike price. The important difference between the two trades is the value of the stock below the strike price—the part of the trade that is not at risk in either the long call or the married put. On this portion of the invested capital, the trader pays interest with the married put (whether actually or in the form of opportunity cost). This interest component is a pricing consideration that adds cost to the married put and not the long call.
So if the married put is a more expensive endeavor than the long call because of the interest paid on the investment portion that is below the strike, why would anyone buy a married put? Wouldn’t traders instead buy the less expensive—less capital intensive—long call? Given the additional interest expense, they would rather buy the call. This relates to the concept of arbitrage. Given two effectively identical choices, rational traders will choose to buy the less expensive alternative. The market as a whole would buy the calls, creating demand which would cause upward price pressure on the call. The price of the call would rise until its interest advantage over the married put was gone. In a robust market with many savvy traders, arbitrage opportunities don’t exist for very long.
It is possible to mathematically state the equilibrium point toward which the market forces the prices of call and put options by use of the put-call parity. As shown in Chapter 2, the put-call parity states
where c is the call premium, PV(x) is the present value of the strike price, p is the put premium and s is the stock price.
Another, less academic and more trader-friendly way of stating this equation is
where Interest is calculated as
Interest = Strike × Interest Rate ×(Days to Expiration/365)
1
The two versions of the put-call parity stated here hold true for European options on non-dividend-paying stocks.
Dividends
Another difference between call and married-put values is dividends. A call option does not extend to its owner the right to receive a dividend payment. Traders, however, who are long a put and long stock are entitled to a dividend if it is the corporation’s policy to distribute dividends to its shareholders.
An adjustment must be made to the put-call parity to account for the possibility of a dividend payment. The equation must be adjusted to account for the absence of dividends paid to call holders. For a dividend-paying stock, the put-call parity states
The interest advantage and dividend disadvantage of owning a call is removed from the market by arbitrageurs. Ultimately, that is what is expressed in the put-call parity. It’s a way to measure the point at which the arbitrage opportunity ceases to exist. When interest and dividends are factored in, a long call is an equal position to a long put paired with long stock. In options nomenclature, a long put with long stock is a synthetic long call. Algebraically rearranging the above equation:
The interest and dividend variables in this equation are often referred to as the basis. From this equation, other synthetic relationships can be algebraically derived, like the synthetic long put.
A synthetic long put is created by buying a call and selling (short) stock. The at-expiration diagrams in
Exhibit 6.2
show identical payouts for these two trades.
EXHIBIT 6.2
Long put vs. long call + short stock.
The concept of synthetics can become more approachable when studied from the perspective of delta as well. Take the 50-strike put and call listed on a $50 stock. A general rule of thumb in the put-call pair is that the call delta plus the put delta equals 1.00 when the signs are ignored. If the 50 put in this example has a −0.45 delta, the 50 call will have a 0.55 delta. By combining the long call (0.55 delta) with short stock (–1.00 delta), we get a synthetic long put with a −0.45 delta, just like the actual put. The directional risk is the same for the synthetic put and the actual put.
A synthetic short put can be created by selling a call of the same month and strike and buying stock on a share-for-share basis (i.e., a covered call). This is indicated mathematically by multiplying both sides of the put-call parity equation by −1:
The at-expiration diagrams, shown in
Exhibit 6.3
, are again conceptually the same.
EXHIBIT 6.3
Short put vs. short call + long stock.
A short (negative) put is equal to a short (negative) call plus long stock, after the basis adjustment. Consider that if the put is sold instead of buying stock and selling a call, the interest that would otherwise be paid on the cost of the stock up to the strike price is a savings to the put seller. To balance the equation, the interest benefit of the short put must be added to the call side (or subtracted from the put side). It is the same with dividends. The dividend benefit of owning the stock must be subtracted from the call side to make it equal to the short put side (or added to the put side to make it equal the call side).
The same delta concept applies here. The short 50-strike put in our example would have a 0.45 delta. The short call would have a −0.55 delta. Buying one hundred shares along with selling the call gives the synthetic short put a net delta of 0.45 (–0.55 + 1.00).
Similarly, a synthetic short call can be created by selling a put and selling (short) one hundred shares of stock.
Exhibit 6.4
shows a conceptual overview of these two positions at expiration.
EXHIBIT 6.4
Short call vs. short put + short stock.
Put-call parity can be manipulated as shown here to illustrate the composition of the synthetic short call.
Most professional traders earn a short stock rebate on the proceeds they receive when they short stock—an advantage to the short-put–short-stock side of the equation. Additionally, short-stock sellers must pay dividends on the shares they are short—a liability to the married-put seller. To make all things equal, one subtracts interest and adds dividends to the put side of the equation.
Comparing Synthetic Calls and Puts
The common thread among the synthetic positions explained above is that, for a put-call pair, long options have synthetic equivalents involving long options, and short options have synthetic equivalents involving short options. After accounting for the basis, the four basic synthetic option positions are:
Because a call or put position is interchangeable with its synthetic position, an efficient market will ensure that the implied volatility is closely related for both. For example, if a long call has an IV of 25 percent, the corresponding put should have an IV of about 25 percent, because the long put can easily be converted to a synthetic long call and vice versa. The greeks will be similar for synthetically identical positions, too. The long options and their synthetic equivalents will have positive gamma and vega with negative theta. The short options and their synthetics will have negative gamma and vega with positive theta.
American-Exercise Options
Put-call parity was designed for European-style options. The early exercise possibility of American-style options gums up the works a bit. Because a call (put) and a synthetic call (put) are functionally the same, it is logical to assume that the implied volatility and the greeks for both will be exactly the same. This is not necessarily true with American-style options. However, put-call parity may still be useful with American options when the limitations of the equation are understood. With at-the-money American-exercise options, the differences in the greeks for a put-call pair are subtle.
Exhibit 6.5
is a comparison of the greeks for the 50-strike call and the 50-strike put with the underlying at $50 and 66 days until expiration.
EXHIBIT 6.5
Greeks for a 50-strike put-call pair on a $50 stock.
Call
Put
Delta
0.554
0.457
Gamma
0.075
0.078
Theta
0.020
0.013
Vega
0.084
0.084
The examples used earlier in this chapter in describing the deltas of synthetics were predicated on the rule of thumb that the absolute values of call and put deltas add up to 1.00. To be a bit more realistic, consider that because of American exercise, the absolute delta values of put-call pairs don’t always add up to 1.00. In fact,
Exhibit 6.5
shows that the call has closer to a 0.554 delta. The put struck at the same price then has a 0.457 delta. By selling 100 shares against the long call, we can create a combined-position delta (call delta plus stock delta) that is very close to the put’s delta. The delta of this synthetic put is −0.446 (0.554 − 1.00). The delta of a put will always be similar to the delta of its corresponding synthetic put. This is also true with call–synthetic-call deltas. This relationship mathematically is
This holds true whether the options are in-, at-, or out-of-the-money. For example, with a stock at $54, the 50-put would have a −0.205 delta and the call would have a 0.799 delta. Selling 100 shares against the call to create the synthetic put yields a net delta of −0.201.
If long or short stock is added to a call or put to create a synthetic, delta will be the only greek affected. With that in mind, note the other greeks displayed in
Exhibit 6.5
—especially theta. Proportionally, the biggest difference in the table is in theta. The disparity is due in part to interest. When the effects of the interest component outweigh the effects of the dividend, the time value of the call can be higher than the time value of the put. Because the call must lose more premium than the put by expiration, the theta of the call must be higher than the theta of the put.
American exercise can also cause the option prices in put-call parity to not add up. Deep in-the-money (ITM) puts can trade at parity while the corresponding call still has time value. The put-call equation can be unbalanced. The same applies to calls on dividend-paying stocks as the dividend date approaches. When the date is imminent, calls can trade close to parity while the puts still have time value. The role of dividends will be discussed further in Chapter 8.
Synthetic Stock
Not only can synthetic calls and puts be derived by manipulation of put-call parity, but synthetic positions for the other security in the equation—stock—can be derived, as well. By isolating stock on one side of the equation, the formula becomes
After accounting for interest and dividends, buying a call and selling a put of the same strike and time to expiration creates the equivalent of a long stock position. This is called a synthetic stock position, or a combo. After accounting for the basis, the equation looks conceptually like this:
This is easy to appreciate when put-call parity is written out as it is here. It begins to make even more sense when considering at-expiration diagrams and the greeks.
Exhibit 6.6
illustrates a long stock position compared with a long call combined with a short put position.
EXHIBIT 6.6
Long stock vs. long call + short put.
A quick glance at these two strategies demonstrates that they are the same, but think about why. Consider the synthetic stock position if both options are held until expiration. The long call gives the trader the right to buy the stock at the strike price. The short put gives the trader the obligation to buy the stock at the same strike price. It doesn’t matter what the strike price is. As long as the strike is the same for the call and the put, the trader will have a long position in the underlying at the shared strike at expiration when exercise or assignment occurs.
The options in this example are 50-strike options. At expiration, the trader can exercise the call to buy the underlying at $50 if the stock is above the strike. If the underlying is below the strike at expiration, he’ll get assigned on the put and buy the stock at $50. If the stock is bought, whether by exercise or assignment, the
effective price
of the potential stock purchase, however, is not necessarily $50.
For example, if the trader bought one 50-strike call at 3.50 and sold one 50-strike put at 1.50, he will effectively purchase the underlying at $52 upon exercise or assignment. Why? The trader paid a net of $2 to get a long position in the stock synthetically (3.50 of call premium debited minus 1.50 of put premium credited). Whether the call or the put is ITM, the effective purchase price of the stock will always be the strike price plus or minus the cost of establishing the synthetic, in this case, $52.
The question that begs to be asked is: would the trader rather buy the stock or pay $2 to have the same market exposure as long stock? Arbitrageurs in the market (with the help of the put-call parity) ensure that neither position—long stock or synthetic long stock—is better than the other.
For example, assume a stock is trading at $51.54. With 71 days until expiration, 26.35 IV, a 5 percent interest rate, and no dividends, the 50-strike call is theoretically worth 3.50, and the 50-strike put is theoretically worth 1.50.
Exhibit 6.7
charts the synthetic stock versus the actual stock when there are 71 days until expiration.
EXHIBIT 6.7
Long stock and synthetic long stock with 71 days to expiration.
Looking at this exhibit, it appears that being long the actual stock outperforms being long the stock synthetically. If the stock is purchased at $51.54, it need only rise a penny higher to profit (in the theoretical world where traders do not pay commissions on transactions). If the synthetic is purchased for $2, the stock needs to rise $0.46 to break even—an apparent disadvantage. This figure, however, does not include interest.
The synthetic stock offers the same risk/reward as actually being long the stock. There is a benefit, from the perspective of interest, to paying only $2 for this exposure rather than $51.54. The interest benefit here is about $0.486. We can find this number by calculating the interest as we did earlier in the chapter. Interest, again, is computed as the strike price times the interest rate times the number of days to expiration divided by the number of days in a year. The formula is as follows:
Inputting the numbers from this example:
The $0.486 of interest is about equal to the $0.46 disparity between the diagrams of the stock and the synthetic stock with 71 days until expiration. The difference is due mainly to rounding and the early-exercise potential of the American put. In mathematical terms
The synthetic long stock is approximately equal to the long stock position when considering the effect of interest. The two lines in
Exhibit 6.7
—representing stock and synthetic stock—would converge with each passing day as the calculated interest decreases.
This equation works as well for a synthetic short stock position; reversing the signs reveals the synthetic for short stock.
Or, in this case,
Shorting stock at $51.54 is about equal to selling the 50 call and buying the 50 put for a $2 credit based on the interest of 0.486 computed on the 50 strike. Again, the $0.016 disparity between the calculated interest and the actual difference between the synthetic value and the stock price is a function of rounding and early exercise. More on this in the “Conversions and Reversals” section.
Synthetic Stock Strategies
Ultimately, when we roll up our sleeves and get down to the nitty-gritty, options trading is less about having another alternative for trading the direction of the underlying than it is about trading the greeks. Different strategies allow traders to exploit different facets of option pricing. Some strategies allow traders to trade volatility. Some focus mainly on theta. Many of the strategies discussed in this section present ways for a trader to distill risk down mostly to interest rate exposure.
Conversions and Reversals
When calls and puts are combined to create synthetic stock, the main differences are the interest rate and dividends. This is important because the risks associated with interest and dividends can be isolated, and ultimately traded, when synthetic stock is combined with the underlying. There are two ways to combine synthetic stock with its underlying security: a conversion and a reversal.
Conversion
A conversion is a three-legged position in which a trader is long stock, short a call, and long a put. The options share the same month and strike price. By most metrics, this is a very flat position. A trader with a conversion is long the stock and, at the same time, synthetically short the same stock. Consider this from the perspective of delta. In a conversion, the trader is long 1.00 deltas (the long stock) and short very close to 1.00 deltas (the synthetic short stock). Conversions have net flat deltas.
The following is a simple example of a typical conversion and the corresponding deltas of each component.
Short one 35-strike call:
−0.63 delta
Long one 35-strike put:
−0.37 delta
Long 100 shares:
1.00 delta
0.00 delta
The short call contributes a negative delta to the position, in this case, −0.63. The long put also contributes a negative delta, −0.37. The combined delta of the synthetic stock is −1.00 in this example, which is like being short 100 shares of stock. When the third leg of the spread is added, the long 100 shares, it counterbalances the synthetic. The total delta for the conversion is zero.
Most of the conversion’s other greeks are pretty flat as well. Gamma, theta, and vega are similar for the call and the put in the conversion, because they have the same expiration month and strike price. Because the trader is selling one option and buying another—a call and a put, respectively—with the same month and strike, the greeks come very close to offsetting each other. For all intents and purposes, the trader is out of the primary risks of the position as measured by greeks when a position is converted. Let’s look at a more detailed example.
A trader executes the following trade (for the purposes of this example, we assume the stock pays no dividend and the trade is executed at fair value):
Sell one 71-day 50 call at 3.50
Buy one 71-day 50 put at 1.50
Buy 100 shares at $51.54
The trader buys the stock at $51.54 and synthetically sells the stock at $52. The synthetic price is computed as −3.50 + 1.50 − 50. Therefore, the stock is sold synthetically at $0.46 over the actual stock price.
Exhibit 6.8
shows the analytics for the conversion.
EXHIBIT 6.8
Conversion greeks.
This position has very subtle sensitivity to the greeks. The net delta for the spread has a very slightly negative bias. The bias is so small it is negligible to most traders, except professionals trading very large positions.
Why does this negative delta bias exist? Mathematically, the synthetic’s delta can be higher with American options than with their European counterparts because of the possibility of early exercise of the put. This anomaly becomes more tangible when we consider the unique directional risk associated with this trade.
In this example, the stock is synthetically sold at $0.46 over the price at which the stock is bought. If the stock declines significantly in value before expiration, the put will, at some point, trade at parity while the call loses all its time value. In this scenario, the value of the synthetic stock will be short at effectively the same price as the actual stock price. For example, if the stock declines to $35 per share then the numbers are as follows:
or
With American options, a put this far in-the-money with less than 71 days until expiry will be all intrinsic value. Interest, in this case, will not factor into the put’s value, because the put can be exercised. By exercising the put, both the long stock leg and the long put leg can be closed for even money, leaving only the theoretically worthless call. The stock-synthetic spread is sold at 0.46 and essentially bought at zero when the put is exercised. If the put is exercised before expiration, the profit potential is 0.46 minus the interest calculated between the trade date and the day the put is exercised. If, however, the conversion is held until expiration, the $0.46 is negated by the $0.486 of interest incurred from holding long stock over the entire 71-day period, hence the trader’s desire to see the stock decline before expiration, and thus the negative bias toward delta.
This is, incidentally, why the synthetic price (0.46 over the stock price) does not exactly equal the calculated value of the interest (0.486). The trader can exercise the put early if the stock declines and capitalize on the disparity between the interest calculated when the conversion was traded and the actual interest calculation given the shorter time frame. The model values the synthetic at a little less than the interest value would indicate—in this case $0.46 instead of $0.486.
The gamma of this trade is fairly negligible. The theta is slightly positive. Rho is the figure that deserves the most attention. Rho is the change in an option’s price given a change in the interest rate.
The −0.090 rho of the conversion indicates that if the interest rate rises one percentage point, the position as a whole loses $0.09. Why? The financing of the position gets more expensive as the interest rate rises. The trader would have to pay more in interest to carry the long stock. In this example, if interest rises by one percentage point, the synthetic stock, which had an effective short price of $0.46 over the price of the long stock before the interest rate increase, will be $0.55 over the price of the long stock afterward. If, however, the interest rate declines by one percentage point, the trader profits $0.09, as the synthetic is repriced by the market to $0.37 over the stock price. The lower the interest rate, the less expensive it is to finance the long stock. This is proven mathematically by put-call parity. Negative rho indicates a bearish position on the interest rate; the trader wants it to go lower. Positive rho is a bullish interest rate position.
But a one-percentage-point change in the interest rate in one day is a big and uncommon change. The question is: is rho relevant? That depends on the type of position and the type of trader. A 0.090 rho would lead to a 0.0225 profit-and-loss (P&(L)) change per one lot conversion on a 25-basis-point, or quarter percent, change. That’s just $2.25 per spread. This incremental profit or loss, however, can be relevant to professional traders like market makers. They trade very large positions with the aspiration of making small incremental profits on each trade. A market maker with a 5,000-lot conversion would stand to make or lose $11,250, given a quarter-percentage-point change in interest rate and a 0.090 rho.
The Mind of a Market Maker
Market makers are among the only traders who can trade conversions and reversals profitably, because of the size of their trades and the fact that they can buy the bid and sell the offer. Market makers often attempt to leg into and out of conversions (and reversals). Given the conversion in this example, a market maker may set out to sell calls and in turn buy stock to hedge the call’s delta risk (this will be covered in Chapters 12 and 17), then buy puts and the rest of the stock to create a balanced conversion: one call to one put to one hundred shares. The trader may try to put on the conversion in the previous example for a total of $0.50 over the price of the long stock instead of the $0.46 it’s worth. He would then try to leg out of the trade for less, say $0.45 over the stock, with the goal of locking in a $0.05 profit per spread on the whole trade.
Reversal
A reversal, or reverse conversion, is simply the opposite of the conversion: buy call, sell put, and sell (short) stock. A reversal can be executed to close a conversion, or it can be an opening transaction. Using the same stock and options as in the previous example, a trader could establish a reversal as follows:
Buy one 71-day 50 call at 3.50
Sell one 71-day 50 put at 1.50
Sell 100 shares at 51.54
The trader establishes a short position in the stock at $51.54 and a long synthetic stock position effectively at $52.00. He buys the stock synthetically at $0.46 over the stock price, again assuming the trade can be executed at fair value. With the reversal, the trader has a bullish position on interest rates, which is indicated by a positive rho.
In this example, the rho for this position is 0.090. If interest rates rise one percentage point, the synthetic stock (which the trader is long) gains nine cents in value relative to the stock. The short stock rebate on the short stock leg earns more interest at a higher interest rate. If rates fall one percentage point, the synthetic long stock loses $0.09. The trader earns less interest being short stock given a lower interest rate.
With the reversal, the fact that the put can be exercised early is a risk. Since the trader is short the put and short stock, he hopes not to get assigned. If he does, he misses out on the interest he planned on collecting when he put on the reversal for $0.46 over.
Pin Risk
Conversions and reversals are relatively low-risk trades. Rho and early exercise are relevant to market makers and other arbitrageurs, but they are among the lowest-risk positions they are likely to trade. There is one indirect risk of conversions and reversals that can be of great concern to market makers around expiration: pin risk. Pin risk is the risk of not knowing for certain whether an option will be assigned. To understand this concept, let’s revisit the mind of a market maker.
Recall that market makers have two primary functions:
1. Buy the bid or sell the offer.
2. Manage risk.
When institutional or retail traders send option orders to an exchange (through a broker), market makers are usually the ones with whom they trade. Customers sell the bid; the market makers buy the bid. Customers buy the offer; the market makers sell the offer. The first and arguably easier function of market makers is accomplished whenever a marketable order is sent to the exchange.
Managing risk can get a bit hairy. For example, once the market makers buy April 40 calls, their first instinct is to hedge by selling stock to become delta neutral. Market makers are almost always delta neutral, which mitigates the direction risk. The next step is to mitigate theta, gamma, and vega risk by selling options. The ideal options to sell are the same calls that were bought—that is, get out of the trade. The next best thing is to sell the April 40 puts and sell more stock. In this case, the market makers have established a reversal and thereby have very little risk. If they can lock in the reversal for a small profit, they have done their job.
What happens if the market makers still have the reversal in inventory at expiration? If the stock is above the strike price—40, in this case—the puts expire, the market makers exercise the calls, and the short stock is consequently eliminated. The market makers are left with no position, which is good. They’re delta neutral. If the stock is below 40, the calls expire, the puts get assigned, and the short stock is consequently eliminated. Again, no position. But what if the stock is exactly at $40? Should the calls be exercised? Will the puts get assigned? If the puts are assigned, the traders are left with no short stock and should let the calls expire without exercising so as not to have a long delta position after expiration. If the puts are not assigned, they should exercise the calls to get delta flat. It’s also possible that only some of the puts will be assigned.
Because they don’t know how many, if any, of the puts will be assigned, the market makers have pin risk. To avoid pin risk, market makers try to eliminate their position if they have conversions or reversals close to expiration.
Boxes and Jelly Rolls
There are two other uses of synthetic stock positions that form conventional strategies: boxes and rolls.
Boxes
When long synthetic stock is combined with short synthetic stock on the same underlying within the same expiration cycle but with a different strike price, the resulting position is known as a box. With a box, a trader is synthetically both long and short the stock. The two positions, for all intents and purposes, offset each other directionally. The risk of stock-price movement is almost entirely avoided. A study of the greeks shows that the delta is close to zero. Gamma, theta, vega, and rho are also negligible. Here’s an example of a 60–70 box for April options:
Short 1 April 60 call
Long 1 April 60 put
Long 1 April 70 call
Short 1 April 70 put
In this example, the trader is synthetically short the 60-strike and, at the same time, synthetically long the 70-strike.
Exhibit 6.9
shows the greeks.
EXHIBIT 6.9
Box greeks.
Aside from the risks associated with early exercise implications, this position is just about totally flat. The near-1.00 delta on the long synthetic stock struck at 60 is offset by the near-negative-1.00 delta of the short synthetic struck at 70. The tiny gammas and thetas of both combos are brought closer to zero when they are spread against each another. Vega is zero. And the bullish interest rate sensitivity of the long combo is nearly all offset by the bearish interest sensitivity of the short combo. The stock can move, time can pass, volatility and interest can change, and there will be very little effect on the trader’s P&(L). The question is: Why would someone trade a box?
Market makers accumulate positions in the process of buying bids and selling offers. But they want to eliminate risk. Ideally, they try to be
flat the strike
—meaning have an equal number of calls and puts at each strike price, whether through a conversion or a reversal. Often, they have a conversion at one strike and a reversal at another. The stock positions for these cancel each other out and the trader is left with only the four option legs—that is, a box. They can eliminate pin risk on both strikes by trading the box as a single trade to close all four legs. Another reason for trading a box has to do with capital.
Borrowing and Lending Money
The first thing to consider is how this spread is priced. Let’s look at another example of a box, the October 50–60 box.
Long 1 October 60 call
Short 1 October 60 put
Short 1 October 70 call
Long 1 October 70 put
A trader with this position is synthetically long the stock at $60 and short the stock at $70. That sounds like $10 in the bank. The question is: How much would a trader be willing to pay for the right to $10? And for how much would someone be willing to sell it? At face value, the obvious answer is that the equilibrium point is at $10, but there is one variable that must be factored in: time.
In this example, assume that the October call has 90 days until expiration and the interest rate is 6 percent. A rational trader would not pay $10 today for the right to have $10 90 days from now. That would effectively be like loaning the $10 for 90 days and not receiving interest—A losing proposition! The trader on the other side of this box would be happy to enter into the spread for $10. He would have interest-free use of $10 for 90 days. That’s free money! Certainly, there is interest associated with the cost of carrying the $10. In this case, the interest would be $0.15.
This $0.15 is discounted from the price of the $10 box. In fact, the combined net value of the options composing the box should be about 9.85—with differences due mainly to rounding and the early exercise possibility for American options.
A trader buying this box—that is, buying the more ITM call and more ITM put—would expect to pay $0.15 below the difference between the strike prices. Fair value for this trade is $9.85. The seller of this box—the trader selling the meatier options and buying the cheaper ones—would concede up to $0.15 on the credit.
Jelly Rolls
A jelly roll, or simply a roll, is also a spread with four legs and a combination of two synthetic stock trades. In a box, the difference between the synthetics is the strike price; in a roll, it’s the contract month. Here’s an example:
Long 1 April 50 call
Short 1 April 50 put
Short 1 May 50 call
Long 1 May 50 put
The options in this spread all share the same strike price, but they involve two different months—April and May. In this example, the trader is long synthetic stock in April and short synthetic stock in May. Like the conversion, reversal, and box, this is a mostly flat position. Delta, gamma, theta, vega, and even rho have only small effects on a jelly roll, but like the others, this spread serves a purpose.
A trader with a conversion or reversal can roll the option legs of the position into a month with a later expiration. For example, a trader with an April 50 conversion in his inventory (short the 50 call, long the 50 put, long stock) can avoid pin risk as April expiration approaches by trading the roll from the above example. The long April 50 call and short April 50 put cancel out the current option portion of the conversion leaving only the stock. Selling the May 50 calls and buying the May 50 puts reestablishes the conversion a month farther out.
Another reason for trading a roll has to do with interest. The roll in this example has positive exposure to rho in April and negative exposure to rho in May. Based on a trader’s expectations of future changes in interest rates, a position can be constructed to exploit opportunities in interest.
Theoretical Value and the Interest Rate
The main focus of the positions discussed in this chapter is fluctuations in the interest rate. But which interest rate? That of 30-year bonds? That of 10- or 5-year notes? Overnight rates? The federal funds rate? In the theoretical world, the answer to this question is not really that important. Professors simply point to the riskless rate and continue with their lessons. But when putting strategies like these into practice, choosing the right rate makes a big difference. To answer the question of which interest rate, we must consider exactly what the rates represent from the standpoint of an economist. Therefore, we must understand how an economist makes arguments—by making assumptions.
Take the story of the priest, the physicist, and the economist stranded on a desert island with nothing to eat except a can of beans. The problem is, the can is sealed. In order to survive, they must figure out how to open the can. The priest decides he will pray for the can to be opened by means of a miracle. He prays for hours, but, alas, the can remains sealed tight. The physicist devises a complex system of wheels and pulleys to pop the top off the can. This crude machine unfortunately fails as well. After watching the lack of success of his fellow strandees, the economist announces that he has the solution: “Assume we have a can opener.”
In the spirit of economists’ logic, let’s imagine for a moment a theoretical economic microcosm in which a trader has two trading accounts at the same firm. The assumptions here are that a trader can borrow 100 percent of a stock’s value to finance the purchase of the security and that there are no legal, moral, or other limitations on trading. In one account the trader is long 100 shares, fully leveraged. In the other, the trader is short 100 shares of the same stock, in which case the trader earns a short-stock rebate.
In the long run, what is the net result of this trade? Most likely, this trade is a losing proposition for the trader, because the interest rate at which the trader borrows capital is likely to be higher than the interest rate earned on the short-stock proceeds. In this example, interest is the main consideration.
But interest matters in the real world, too. Professional traders earn interest on proceeds from short stock and pay interest on funds borrowed. Interest rates may vary slightly from firm to firm and trader to trader. Interest rates are personal. The interest rate a trader should use when pricing options is specific to his or her situation.
A trader with no position in a particular stock who is interested in trading a conversion should consider that he will be buying the stock. This implies borrowing funds to open the long stock position. The trader should price his options according to the rate he will pay to borrow funds. Conversely, a trader trading a reversal should consider the fact that he is shorting the stock and will receive interest at the rate of the short-stock rebate. This trader should price his options at the short-stock rate.
A Call Is a Put
The idea that “a put is a call, a call is a put” is an important one, indeed. It lays the foundation for more advanced spreading strategies. The concepts in this chapter in one way or another enter into every spread strategy that will be discussed in this book from here on out.
Note
1
. Note, for simplicity, simple interest is used in the computation.