the
trade becomes a loser. In a way, Stacie is still selling volatility.
Covered Call
The problem with selling a naked call is that it has unlimited exposure to
upside risk. Because of this, many traders simply avoid trading naked calls.
A more common, and some would argue safer, method of selling calls is to
sell them covered.
A covered call is when calls are sold and stock is purchased on a share-
for-share basis to cover the unlimited upside risk of the call. For each call
that is sold, 100 shares of the underlying security are bought. Because of the
addition of stock to this strategy, covered calls are traded with a different
motivation than naked calls.
There are clearly many similarities between these two strategies. The
main goal for both is to harvest the premium of the call. The theta for the
call is the same with or without the stock component. The gamma and vega
for the two strategies are the same as well. The only difference is the stock.
When stock is added to an option position, the net delta of the position is
the only thing affected. Stock has a delta of one, and all its other greeks are
zero.
The pivotal point for both positions is the strike price. That’s the point the
trader wants the stock to be above or below at expiration. With the naked
call, the maximum payout is reaped if the stock is below the strike at
expiration, and there is unlimited risk above the strike. With the covered
call, the maximum payout is reaped if the stock is above the strike at
expiration. If the stock is below the strike at expiration, the risk is
substantial—the stock can potentially go to zero.
Putting It on
There are a few important considerations with the covered call, both when
putting on, or entering, the position and when taking off, or exiting, the
trade. The risk/reward implications of implied volatility are important in the
trade-planning process. Do I want to get paid more to assume more
potential risk? More speculative traders like the higher premiums. More
conservative (investment-oriented) covered-call sellers like the low implied
risk of low-IV calls. Ultimately, a main focus of a covered call is the option
premium. How fast can it go to zero without the movement hurting me? To
determine this, the trader must study both theta and delta.
The first step in the process is determining which month and strike call to
sell. In this example, Harley-Davidson Motor Company (HOG) is trading at
about $69 per share. A trader, Bill, is neutral to slightly bullish on Harley-
Davidson over the next three months. Exhibit 5.7 shows a selection of
available call options for Harley-Davidson with corresponding deltas and
thetas.
EXHIBIT 5.7 Harley-Davidson calls.
In this example, the May 70 calls have 85 days until expiration and are
2.80 bid. If Harley-Davidson remained at $69 until May expiration, the 2.80
premium would represent a 4 percent profit over this 85-day period (2.80 ÷
69). That’s an annualized return of about 17 percent ([0.04 / 85)] × 365).
Bill considers his alternatives. He can sell the April (57-day) 70 calls at
2.20 or the March (22-day) 70 calls at 0.85. Since there is a different
number of days until expiration, Bill needs to compare the trades on an
apples-to-apples basis. For this, he will look at theta and implied volatility.
Presumably, the March call has a theta advantage over the longer-term
choices. The March 70 has a theta of 0.032, while the April 70’s theta is
0.026 and the May 70’s is 0.022. Based on his assessment of theta, Bill
would have the inclination to sell the March. If he wants exposure for 90
days, when the March 70 call expires, he can roll into the April 70 call and
then the May 70 call (more on this in subsequent chapters). This way Bill
can continue to capitalize on the nonlinear rate of decay through May.
Next, Bill studies the IV term structure for the Harley-Davidson ATMs
and finds the March has about a 19.2 percent IV, the April has a 23.3
percent IV, and the May has a 23 percent IV. March is the cheapest option
by IV standards. This is not necessarily a favorable quality for a short
candidate. Bill must weigh his assessment of all relevant information and
then decide which trade is best. With this type of a strategy, the benefits of
the higher theta can outweigh the disadvantages of selling the lower IV. In
this case, Bill may actually like selling the lower IV. He may infer that the
market believes Harley-Davidson will be less volatile during this period.
So far, Bill has been focusing his efforts on the 70 strike calls. If he trades
the March 70 covered call, he will have a net delta of 0.588 per contract.
That’s the negative 0.412 delta from shorting the call plus the 1.00 delta of
the stock. His indifference point if the trade is held until expiration is
$70.85. The indifference point is the point at which Bill would be
indifferent as to whether he held only the stock or the covered call. This is
figured by adding the strike price of $70 to the 0.85 premium. This is the
effective sale price of the stock if the call is assigned. If Bill wants more
potential for upside profit, he could sell a higher strike. He would have to
sell the April or May 75, since the March 75s are a zero bid. This would
give him a higher indifference point, and the upside profits would
materialize quickly if HOG moved higher, since the covered-call deltas
would be higher with the 75 calls. The April 75 covered-call net delta is
0.796 per contract (the stock delta of 1.00 minus the 0.204 delta of the call).
The May 75 covered-call delta is 0.751.
But Bill is neutral to only slightly bullish. In this case, he’d rather have
the higher premium—high theta is more desirable than high delta in this
situation. Bill buys 1,000 shares of Harley-Davidson at $69 and sells 10
Harley-Davidson March 70 calls at 0.85.
Bill also needs to plan his exit. To exit, he must study two things: an at-
expiration diagram and his greeks. Exhibit 5.8 shows the P&(L) at
expiration of the Harley-Dav


htly bullish. In this case, he’d rather have
the higher premium—high theta is more desirable than high delta in this
situation. Bill buys 1,000 shares of Harley-Davidson at $69 and sells 10
Harley-Davidson March 70 calls at 0.85.
Bill also needs to plan his exit. To exit, he must study two things: an at-
expiration diagram and his greeks. Exhibit 5.8 shows the P&(L) at
expiration of the Harley-Davidson March 70 covered call. Exhibit 5.9
shows the greeks.
EXHIBIT 5.8 Harley-Davidson covered call.
EXHIBIT 5.9 Greeks for Harley-Davidson covered call (per contract).
Delta 0.591
Gamma−0.121
Theta 0.032
Vega −0.066
Taking It Off
If the trade works out perfectly for Bill, 22 days from now Harley-Davidson
will be trading right at $70. He’d profit on both delta and theta. If the trade
isn’t exactly perfect, but still good, Harley-Davidson will be anywhere
above $68.15 in 22 days. It’s the prospect that the trade may not be so good
at March expiration that occupies Bill’s thoughts, but a trader has to hope
for the best and plan for the worst.
If it starts to trend, Bill needs to react. The consequences to the stock’s
trending to the upside are not quite so dire, although he might be somewhat
frustrated with any lost opportunity above the indifference point. It’s the
downside risk that Bill will more vehemently guard against.
First, the same IV/vega considerations exist as they did in the previous
examples. In the event the trade is closed early, IV/vega may help or hinder
profitability. A rise in implied volatility will likely accompany a decline in
the stock price. This can bring Bill to his stop-loss sooner. Delta versus
theta however, is the major consideration. He will plan his exit price in
advance and cover when the planned exit price is reached.
There are more moving parts with the covered call than a naked option. If
Bill wants to close the position early, he can leg out, meaning close only
one leg of the trade (the call or the stock) at a time. If he legs out of the
trade, he’s likely to close the call first. The motivation for exiting a trade
early is to reduce risk. A naked call is hardly less risky than a covered call.
Another tactic Bill can use, and in this case will plan to use, is rolling the
call. When the March 70s expire, if Harley-Davidson is still in the same
range and his outlook is still the same, he will sell April calls to continue
the position. After the April options expire, he’ll plan to sell the Mays.
With this in mind, Bill may consider rolling into the Aprils before March
expiration. If it is close to expiration and Harley-Davidson is trading lower,
theta and delta will both have devalued the calls. At the point when options
are close to expiration and far enough OTM to be offered close to zero, say
0.05, the greeks and the pricing model become irrelevant. Bill must
consider in absolute terms if it is worth waiting until expiration to make
0.05. If there is a lot of time until expiration, the answer is likely to be no.
This is when Bill will be apt to roll into the Aprils. He’ll buy the March 70s
for a nickel, a dime, or maybe 0.15 and at the same time sell the Aprils at
the bid. This assumes he wants to continue to carry the position. If the roll
is entered as a single order, it is called a calendar spread or a time spread.
Covered Put
The last position in the family of basic volatility-selling strategies is the
covered put, sometimes referred to as selling puts and stock. In a covered
put, a trader sells both puts and stock on a one-to-one basis. The term
covered put is a bit of a misnomer, as the strategy changes from limited risk
to unlimited risk when short stock is added to the short put. A naked put can
produce only losses until the stock goes to zero—still a substantial loss.
Adding short stock means that above the strike gains on the put are limited,
while losses on the stock are unlimited. The covered put functions very
much like a naked call. In fact, they are synthetically equal. This concept
will be addressed further in the next chapter.
Let’s looks at another trader, Libby. Libby is an active trader who trades
several positions at once. Libby believes the overall market is in a range
and will continue as such over the next few weeks. She currently holds a
short stock position of 1,000 shares in Harley-Davidson. She is becoming
more neutral on the stock and would consider buying in her short if the
market dipped. She may consider entering into a covered-put position.
There is one caveat: Libby is leaving for a cruise in two weeks and does not
want to carry any positions while she is away. She decides she will sell the
covered put and actively manage the trade until her vacation. Libby will sell
10 Harley-Davidson March (22-day) 70 puts at 1.85 against her short 1,000
shares of Harley-Davidson, which is trading at $69 per share.
She knows that her maximum profit if the stock declines and assignment
occurs will be $850. That’s 0.85 × $100 × 10 contracts. Win or lose, she
will close the position in two weeks when there are only eight days until
expiration. To trade this covered put she needs to watch her greeks.
Exhibit 5.10 shows the greeks for the Harley-Davidson 70-strike covered
put.
EXHIBIT 5.10 Greeks for Harley-Davidson covered put (per contract).
Delta −0.419
Gamma−0.106
Theta 0.031
Vega −0.066
Libby is really focusing on theta. It is currently about $0.03 per day but
will increase if the put stays close-to-the-money. In two weeks, the time
premium will have decayed significantly. A move downward will help, too,
as the −0.419 delta indicates. Exhibit 5.11 displays an array of theoretical
values of the put at eight days until expiration as the stock price changes.
EXHIBIT 5.11 HOG 70 put values at 8 days to expiry.
As long as Harley-Davidson stays below the strike price, Libby can look
at her put from a premium-over-parity standpoint. Below the strike, the
intrinsic value of the put doesn’t matter too much, because losses on
intrinsic value are offse

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf


isplays an array of theoretical
values of the put at eight days until expiration as the stock price changes.
EXHIBIT 5.11 HOG 70 put values at 8 days to expiry.
As long as Harley-Davidson stays below the strike price, Libby can look
at her put from a premium-over-parity standpoint. Below the strike, the
intrinsic value of the put doesn’t matter too much, because losses on
intrinsic value are offset by gains on the stock. For Libby, all that really
matters is the time value. She sold the puts at 0.85 over parity. If Harley-
Davidson is trading at $68 with eight days to go, she can buy her puts back
for 0.12 over parity. That’s a 73-cent profit, or $730 on her 10 contracts.
This doesn’t account for any changes in the time value that may occur as a
result of vega, but vega will be small with Harley-Davidson at $68 and
eight days to go. At this point, she would likely close down the whole
position—buying the puts and buying the stock—to take a profit on a
position that worked out just about exactly as planned.
Her risk, though, is to the upside. A big rally in the stock can cause big
losses. From a theoretical standpoint, losses are potentially unlimited with
this type of trade. If the stock is above the strike, she needs to have a mental
stop order in mind and execute the closing order with discipline.
Curious Similarities
These basic volatility-selling strategies are fairly simple in nature. If the
trader believes a stock will not rise above a certain price, the most
straightforward way to trade the forecast is to sell a call. Likewise, if the
trader believes the stock will not go below a certain price he can sell a put.
The covered call and covered put are also ways to generate income on long
or short stock positions that have these same price thresholds. In fact, the
covered call and covered put have some curious similarities to the naked
put and naked call. The similarities between the two pairs of positions are
no coincidence. The following chapter sheds light on these similarities.
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 w


xpensive—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.


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 0.5540.457
Gamma0.0750.078
Theta 0.0200.013
Vega 0.0840.084
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


urchase,
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 o


he
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,


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 th


y 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 f


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.
CHAPTER 7
Rho
Interest is one of the six inputs of an option-pricing model for American
options. Although interest rates can remain constant for long periods, when
interest rates do change, call and put values can be positively or negatively
affected. Some options are more sensitive to changes in the interest rate
than others. To the unaware trader, interest-rate changes can lead to
unexpected profits or losses. But interest rates don’t have to be a wild-card
risk. They’re one that experienced traders watch closely to avoid
unnecessary risk and increase profitability. To monitor the effect of changes
in the interest rate, it is important to understand the quiet greek—rho.
Rho and Interest Rates
Rho is a measurement of the sensitivity of an option’s value to a change in
the interest rate. To understand how and why the interest rate is important to
the value of an option, recall the formula for put-call parity stated in
Chapter 6.
Call + Strike − Interest = Put + Stock 1
From this formula, it’s clear that as the interest rate rises, put prices must
fall and call prices must rise to keep put-call parity balanced. With a little
algebra, the equation can be restated to better illustrate this concept:
and
If interest rates fall,
and
Rho helps quantify this relationship. Calls have positive rho, and puts
have negative rho. For example, a call with a rho of +0.08 will gain $0.08
with each one-percentage-point rise in interest rates and fall $0.08 with
each one-percentage-point fall in interest rates. A put with a rho of −0.08
will lose $0.08 with each one-point rise and gain $0.08 in value with a one-
point fall.
The effect of changes in the interest variable of put-call parity on call and
put values is contingent on three factors: the strike price, the interest rate,
and the number of days until expiration.
Interest = Strike×Interest Rate×(Days to Expiration/365) 2
Interest, for our purposes, is a function of the strike price. The higher the
strike price, the greater the interest and, consequently the more changes in
the interest rate will affect the option. The higher the interest rate is, the
higher the interest variable will be. Likewise, the more time to expiration,
the greater the effect of interest. Rho measures an option’s sensitivity to the
end results of these three influences.
To understand how changes in interest affect option prices, consider a
typical at-the-money (ATM) conversion on a non-dividend-paying stock.
Short 1 May 50 call at 1.92
Long 1 May 50 put at 1.63
Long 100 shares at $50
With 43 days until expiration at a 5 percent interest rate, the interest on
the 50 strike will be about $0.29. Put-call parity ensures that this $0.29
shows up in option prices. After rearranging the equation, we get
In this example, both options are exactly ATM. There is no intrinsic value.
Therefore, the difference between the extrinsic values of the call and the put
must equal interest. If one option were in-the-money (ITM), the intrinsic
value on the left side of the equation would be offset by the Stock − Strike
on the right side. Still, it would be the difference in the time value of the
call and put that equals the interest variable.
This is shown by the fact that the synthetic stock portion of the
conversion is short at $50.29 (call − put + strike). This is $0.29 above the
stock price. The synthetic stock equals the Sto


rinsic
value on the left side of the equation would be offset by the Stock − Strike
on the right side. Still, it would be the difference in the time value of the
call and put that equals the interest variable.
This is shown by the fact that the synthetic stock portion of the
conversion is short at $50.29 (call − put + strike). This is $0.29 above the
stock price. The synthetic stock equals the Stock + Interest, or
Certainly, if the interest rate were higher, the interest on the synthetic
stock would be a higher number. At a 6 percent interest rate, the effective
short price of the synthetic stock would be about $50.35. The call would be
valued at about 1.95, and the put would be 1.60—a net of $0.35.
A one-percentage-point rise in the interest rate causes the synthetic stock
position to be revalued by $0.06—a $0.03 gain in the call value and a $0.03
decline in the put. Therefore, by definition, the call has a +0.03 rho and the
put has a −0.03 rho.
Rho and Time
The time component of interest has a big impact on the magnitude of an
option’s rho, because the greater the number of days until expiration, the
greater the interest. Long-term options will be more sensitive to changes in
the interest rate and, therefore, have a higher rho.
Take a stock trading at about $120 per share. The July, October, and
January ATM calls have the following rhos with the interest rate at 5.5
percent.
Option Rho
July (38-day) 120 calls+0.068
October (130-day) 120 calls+0.226
January (221-day) 120 calls+0.385
If interest rates rise 25 basis points, or a quarter of a percentage point, the
July calls with only 38 days until expiration will gain very little: only
$0.017 (0.068 × 0.25). The October 120 calls with 130 days until expiration
gain more: $0.057 (0.226 × 0.25). The January calls that have 221 days
until they expire make $0.096 theoretically (0.385 × 0.25). If all else is held
constant, the more time to expiration, the higher the option’s rho, and
therefore, the more interest will affect the option’s value.
Considering Rho When Planning
Trades
Just having an opinion on a stock is only half the battle in options trading.
Choosing the best way to trade a forecast can make all the difference to the
success of a trade. Options give traders choices. And one of the choices a
trader has is the month in which to trade. When trading LEAPS—Long-
Term Equity AnticiPation Securities—delta, gamma, theta, and vega are
important, as always, but rho is also a valuable part of the strategy.
LEAPS
Options buyers have time working against them. With each passing day,
theta erodes the value of their assets. Buying a long-term option, or a
LEAPS, helps combat erosion because long-term options can decay at a
slower rate. In environments where there is interest rate uncertainty,
however, LEAPS traders have to think about more than the rate of decay.
Consider two traders: Jason and Susanne. Both are bullish on XYZ Corp.
(XYZ), which is trading at $59.95 per share. Jason decides to buy a May 60
call at 1.60, and Susanne buys a LEAPS 60 call at 7.60. In this example,
May options have 44 days until expiration, and the LEAPS have 639 days.
Both of these trades are bullish, but the traders most likely had slightly
different ideas about time, volatility, and interest rates when they decided
which option to buy. Exhibit 7.1 compares XYZ short-term at-the-money
calls with XYZ LEAPS ATM calls.
EXHIBIT 7.1 XYZ short-term call vs. LEAPS call.
To begin with, it appears that Susanne was allowing quite a bit of time for
her forecast to be realized—almost two years. Jason, however, was looking
for short-term price appreciation. Concerns about time decay may have
been a motivation for Susanne to choose a long-term option—her theta of
0.01 is half Jason’s, which is 0.02. With only 44 days until expiration, the
theta of Jason’s May call will begin to rise sharply as expiration draws near.
But the trade-off of lower time decay is lower gamma. At the current
stock price, Susanne has a higher delta. If the XYZ stock price rises $2, the
gamma of the May call will cause Jason’s delta to creep higher than
Susanne’s. At $62, the delta for the May 60s would be about 0.78, whereas
the LEAPS 60 call delta is about 0.77. This disparity continues as XYZ
moves higher.
Perhaps Susanne had implied volatility (IV) on her mind as well as time
decay. These long-term ATM LEAPS options have vegas more than three
times the corresponding May’s. If IV for both the May and the LEAPS is at
a yearly low, LEAPS might be a better buy. A one- or two-point rise in
volatility if IV reverts to its normal level will benefit the LEAPS call much
more than the May.
Theta, delta, gamma, and vega are typical considerations with most
trades. Because this option is long term, in addition to these typical
considerations, Susanne needs to take a good hard look at rho. The LEAPS
rho is significantly higher than that of its short-term counterpart. A one-
percentage-point change in the interest rate will change Susanne’s P&(L) by
$0.64—that’s about 8.5 percent of the value of her option—and she has
nearly two years of exposure to interest rate fluctuations. Certainly, when
the Federal Reserve Board has great concerns about growth or inflation,
rates can rise or fall by more than one percentage point in one year’s time.
It is important to understand that, like the other greeks, rho is a snapshot
at a particular price, volatility level, interest rate, and moment in time. If
interest rates were to fall by one percentage point today, it would cause
Susanne’s call to decline in value by $0.64. If that rate drop occurred over
the life of the option, it would have a much smaller effect. Why? Rate
changes closer to expiration have less of an effect on option values.
Assume that on the trade date, when the LEAPS has 639 days until
expiration, interest rates fall by 25 basis points. The effect will be a decline
in the value of the call of 0.16—one-fourth of the 0.638 rho. If the next rate
cut occurs si


e drop occurred over
the life of the option, it would have a much smaller effect. Why? Rate
changes closer to expiration have less of an effect on option values.
Assume that on the trade date, when the LEAPS has 639 days until
expiration, interest rates fall by 25 basis points. The effect will be a decline
in the value of the call of 0.16—one-fourth of the 0.638 rho. If the next rate
cut occurs six months later, the rho of the LEAPS will be smaller, because it
will have less time until expiration. In this case, after six months, the rho
will be only 0.46. Another 25-basis-point drop will hurt the call by $0.115.
After another six months, the option will have a 0.26 rho. Another quarter-
point cut costs Susanne only $0.065. Any subsequent rate cuts in ensuing
months will have almost no effect on the now short-term option value.
Pricing in Interest Rate Moves
In the same way that volatility can get priced in to an option’s value, so can
the interest rate. When interest rates are expected to rise or fall, those
expectations can be reflected in the prices of options. Say current interest
rates are at 8 percent, but the Fed has announced that the economy is
growing at too fast of a pace and that it may raise interest rates at the next
Federal Open Market Committee meeting. Analysts expect more rate hikes
to follow. The options with expiration dates falling after the date of the
expected rate hikes will have higher interest rates priced in. In this situation,
the higher interest rates in the longer-dated options will be evident when
entering parameters into the model.
Take options on Already Been Chewed Bubblegum Corp. (ABC). A
trader, Kyle, enters parameters into the model for ABC options and notices
that the prices don’t line up. To get the theoretical values of the ATM calls
for all the expiration months to sit in the middle of the actual market values,
Kyle may have to tinker with the interest rate inputs.
Assume the following markets for the ATM 70-strike calls in ABC
options:
Calls Puts
Aug 70 calls1.75–1.851.30–1.40
Sep 70 calls2.65–2.751.75–1.85
Dec 70 calls4.70–4.902.35–2.45
Mar 70 calls6.50–6.702.65–2.75
ABC is at $70 a share, has a 20 percent IV in all months, and pays no
dividend. August expiration is one month away.
Entering the known inputs for strike price, stock price, time to expiration,
volatility, and dividend and using an 8 percent interest rate yields the
following theoretical values for ABC options:
The theoretical values, in bold type, are those that don’t line up in the
middle of the call and put markets. These values are wrong. The call
theoretical values are too low, and the put theoretical values are too high.
They are the product of an interest rate that is too low being applied to the
model. To generate values that are indicative of market prices, Kyle must
change the interest input to the pricing model to reflect the market’s
expectations of future interest rate changes.
Using new values for the interest rate yields the following new values:
After recalculating, the theoretical values line up in the middle of the call
and put markets. Using higher interest rates for the longer expirations raises
the call values and lowers the put values for these months. These interest
rates were inferred from, or backed out of, the option-market prices by use
of the option-pricing model. In practice, it may take some trial and error to
find the correct interest values to use.
In times of interest rate uncertainty, rho can be an important factor in
determining which strategy to select. When rates are generally expected to
continue to rise or fall over time, they are normally priced in to the options,
as shown in the previous example. When there is no consensus among
analysts and traders, the rates that are priced in may change as economic
data are made available. This can cause a revision of option values. In long-
term options that have higher rhos, this is a bona fide risk. Short-term
options are a safer play in this environment. But as all traders know, risk
also implies opportunity.
Trading Rho
While it’s possible to trade rho, most traders forgo this niche for more
dynamic strategies with greater profitability. The effects of rho are often
overshadowed by the more profound effects of the other greeks. The
opportunity to profit from rho is outweighed by other risks. For most
traders, rho is hardly ever even looked at.
Because LEAPS have higher rho values than corresponding short-term
options, it makes sense that these instruments would be appropriate for
interest-rate plays. But even with LEAPS, rho exposure usually pales in
comparison with that of delta, theta, and vega.
It is not uncommon for the rho of a long-term option to be 5 to 8 percent
of the option’s value. For example, Exhibit 7.2 shows a two-year LEAPS on
a $70 stock with the following pricing-model inputs and outputs:
EXHIBIT 7.2 Long 70-strike LEAPS call.
The rho is +0.793, or about 5.8 percent of the call value. That means a 25-
basis-point rise in rates contributes to only a 20-cent profit on the call.
That’s only about 1.5 percent of the call’s value. On one hand, 1.5 percent is
not a very big profit on a trade. On the other hand, if there are more rate
rises at following Fed meetings, the trader can expect further gains on rho.
Even if the trader is compelled to wait until the next Fed meeting to make
another $0.20—or less, as rho will get smaller as time passes—from a
second 25-basis-point rate increase, other influences will diminish rho’s
significance. If over the six-week period between Fed meetings, the
underlying declines by just $0.60, the $0.40 that the trader hoped to make
on rho is wiped out by delta loss. With the share price $0.60 lower, the
0.760 delta costs the trade about $0.46. Furthermore, the passing of six
weeks (42 days) will lead to a loss of about $0.55 from time decay because
of the −0.013 theta. There is also the risk from the fat vegas associated with
LEAPS. A 1.5 percent drop in impl


ng declines by just $0.60, the $0.40 that the trader hoped to make
on rho is wiped out by delta loss. With the share price $0.60 lower, the
0.760 delta costs the trade about $0.46. Furthermore, the passing of six
weeks (42 days) will lead to a loss of about $0.55 from time decay because
of the −0.013 theta. There is also the risk from the fat vegas associated with
LEAPS. A 1.5 percent drop in implied volatility completely negates any
hopes of rho profits.
Aside from the possibility that delta, theta, and vega may get in the way
of profits, the bid-ask spread with these long-term options tends to be wider
than with their short-term counterparts. If the bid-ask spread is more than
$0.40 wide, which is often the case with LEAPS, rho profits are canceled
out by this cost of doing business. Buying the offer and selling the bid
negative scalps away potential profits.
With LEAPS, rho is always a concern. It will contribute to prosperity or
peril and needs to be part of the trade plan from forecast to implementation.
Buying or selling a LEAPS call or put, however, is not a practical way to
speculate on interest rates.
To take a position on interest rates in the options market, risk needs to be
distilled down to rho. The other greeks need to be spread off. This is
accomplished only through the conversions, reversals, and jelly rolls
described in Chapter 6. However, the bid-ask can still be a hurdle to trading
these strategies for non–market makers. Generally, rho is a greek that for
most traders is important to understand but not practical to trade.
Notes
1 . Please note, for simplification, dividends are not included.
2 . Note, for simplicity, simple interest is used in the calculation.
CHAPTER 8
Dividends and Option Pricing
Much of this book studies how to break down and trade certain components
of option prices. This chapter examines the role of dividends in the pricing
structure. There is no greek symbol that measures an option’s sensitivity to
changes in the dividend. And in most cases, dividends are not “traded” by
means of options in the same way that volatility, interest, and other option
price influences are. Dividends do, though, affect option prices, and
therefore a trader’s P&(L), so they deserve attention.
There are some instances where dividends provide ample opportunity to
the option trader, and there some instances where a change in dividend
policy can have desirable, or undesirable, effects on the bottom line.
Despite the fact that dividends do not technically involve greeks, they need
to be monitored in much the same way as do delta, gamma, theta, vega, and
rho.
Dividend Basics
Let’s start at the beginning. When a company decides to pay a dividend,
there are four important dates the trader must be aware of:
1. Declaration date
2. Ex-dividend date
3. Record date
4. Payable date
The first date chronologically is the declaration date. This date is when
the company formally declares the dividend. It’s when the company lets its
shareholders know when and in what amount it will pay the dividend.
Active traders, however, may buy and sell the same stock over and over
again. How does the corporation know exactly who collects the dividend
when it is opening up its coffers?
Dividends are paid to shareholders of record who are on the company’s
books as owning the stock at the opening of business on another important
date: the record date. Anyone long the stock at this moment is entitled to the
dividend. Anyone with a short stock position on the opening bell on the
record date is required to make payment in the amount of the dividend.
Because the process of stock settlement takes time, the important date is
actually not the record date. For all intents and purposes, the key date is two
days before the record date. This is called the ex-dividend date, or the ex-
date.
Traders who have earned a dividend by holding a stock in their account
on the morning of the ex-date have one more important date they need to
know—the date they get paid. The date that the dividend is actually paid is
called the payable date. The payable date can be a few weeks after the ex-
date.
Let’s walk through an example. ABC Corporation announces on March
21 (the declaration date) that it will pay a 25-cent dividend to shareholders
of record on April 3 (the record date), payable on April 23 (the payable
date). This means market participants wishing to receive the dividend must
own the stock on the open on April 1 (the ex-date). In practice, they must
buy the stock before the closing bell rings on March 31 in order to have it
for the open the next day.
This presents a potential quandary. If a trader only needs to have the stock
on the open on the ex-date, why not buy the stock just before the close on
the day before the ex-date, in this case March 31, and sell it the next
morning after the open? Could this be an opportunity for riskless profit?
Unfortunately, no. There are a couple of problems with that strategy. First,
as far as the riskless part is concerned, stock prices can and often do change
overnight. Yesterday’s close and today’s open can sometimes be
significantly different. When they are, it is referred to as a gap open.
Whenever a stock is held (long or short), there is risk. The second problem
with this strategy to earn riskless profit is with the profit part. On the ex-
date, the opening stock price reflects the dividend. Say ABC is trading at
$50 at the close on March 31. If the market for the stock opens unchanged
the next morning—that is, a zero net change on the day on—ABC will be
trading at $49.75 ($50 minus the $0.25 dividend). Alas, the quest for
riskless profit continues.
Dividends and Option Pricing
The preceding discussion demonstrated how dividends affect stock traders.
There’s one problem: we’re option traders! Option holders or writers do not
receive or pay dividends, but that doesn’t mean dividends aren’t relevant to
the pricing of these securities. Observe the behavior of a conversion or a
reve


$0.25 dividend). Alas, the quest for
riskless profit continues.
Dividends and Option Pricing
The preceding discussion demonstrated how dividends affect stock traders.
There’s one problem: we’re option traders! Option holders or writers do not
receive or pay dividends, but that doesn’t mean dividends aren’t relevant to
the pricing of these securities. Observe the behavior of a conversion or a
reversal before and after an ex-dividend date. Assuming the stock opens
unchanged on the ex-date, the relationship of the price of the synthetic stock
to the actual stock price will change. Let’s look at an example to explore
why.
At the close on the day before the ex-date of a stock paying a $0.25
dividend, a trader has an at-the-money (ATM) conversion. The stock is
trading right at $50 per share. The 50 puts are worth 2.34, and the 50 calls
are worth 2.48. Before the ex-date, the trader is
Long 100 shares at $50
Long one 50 put at 2.34
Short one 50 call at 2.48
Here, the trader is long the stock at $50 and short stock synthetically at
$50.14—50 + (2.48 − 2.34). The trader is synthetically short $0.14 over the
price at which he is long the stock.
Assume that the next morning the stock opens unchanged. Since this is
the ex-date, that means the stock opens at $49.75—$0.25 lower than the
previous day’s close. The theoretical values of the options will change very
little. The options will be something like 2.32 for the put and 2.46 for the
call.
After the ex-date, the trader is
Long 100 shares at $49.75
Long one 50 put at 2.32
Short one 50 call at 2.46
Each option is two cents lower. Why? The change in the option prices is
due to theta. In this case, it’s $0.02 for each option. The synthetic stock is
still short from an effective price of $50.14. With the stock at $49.75, the
synthetic short price is now $0.39 over the stock. Incidentally, $0.39 is
$0.25 more than the $0.14 difference before the ex-date.
Did the trader who held the conversion overnight from before the ex-date
to after it make or lose money? Neither. Before the ex-date, he had an asset
worth $50 per share (the stock) and he shorted the asset synthetically at
$50.14. After the ex-date, he still has assets totaling $50 per share—the
stock at $49.75 plus the 0.25 dividend—and he is still synthetically short
the stock at $50.14. Before the ex-date, the $0.14 difference between the
synthetic and the stock is interest minus the dividend. After the ex-date, the
$0.39 difference is all interest.
Dividends and Early Exercise
As the ex-date approaches, in-the-money (ITM) calls on equity options can
often be found trading at parity, regardless of the dividend amount and
regardless of how far off expiration is. This seems counterintuitive. What
about interest? What about dividends? Normally, these come into play in
option valuation.
But option models designed for American options take the possibility of
early exercise into account. It is possible to exercise American-style calls
and exchange them for the underlying stock. This would give traders, now
stockholders, the right to the dividend—a right for which they would not be
eligible as call holders. Because of the impending dividend, the call
becomes an exercise just before the ex-date. For this reason, the call can
trade for parity before the ex-date.
Let’s look at an example of a reversal on a $70 stock that pays a $0.40
dividend. The options in this reversal have 24 days until expiration, which
makes the interest on the 60 strike roughly $0.20, given a 5 percent interest
rate. The day before the ex-date, a trader has the following position at the
stated prices:
Short 100 shares at $70
Long one 60 call at 10.00
Short one 60 put at 0.05
To understand how American calls work just before the ex-date, it is
helpful first to consider what happens if the trader holds the position until
the ex-date. Making the assumption that the stock is unchanged on the ex-
dividend date, it will open at $69.60, lower by the amount of the dividend—
in this case, $0.40. The put, being so far out-of-the-money (OTM) as to
have a negligible delta, will remain unchanged. But what about the call?
With no dividend left in the stock, the put call-parity states
In this case,

Before the ex-date, the model valued the call at parity. Now it values the
same call at $0.25 over parity (9.85 − [69.60 − 60]). Another way to look at
this is that the time value of the call is now made up of the interest plus the
put premium. Either way, that’s a gain of $0.25 on the call. That sounds
good, but because the trader is short stock, if he hasn’t exercised, he will
owe the $0.40 dividend—a net loss of $0.15. The new position will be
Short 100 shares at $69.60
Owe $0.40 dividend
Long one 60 call at 9.85
Short one 60 put at 0.05
At the end of the trading day before the ex-date, this trader must exercise
the call to capture the dividend. By doing so, he closes two legs of the trade
—the call and the stock. The $10 call premium is forfeited, the stock that is
short at $70 is bought at $60 (from the call exercise) for a $10 profit. The
transaction leads to neither a profit nor a loss. The purpose of exercising is
to avoid the $0.15 loss ($0.25 gain in call time value minus the $0.40 loss in
dividends owed).
The other way the trader could achieve the same ends is to sell the long
call and buy in the short stock. This is tactically undesirable because the
trader may have to sell the bid in the call and buy the offer in the stock.
Furthermore, when legging a trade in this manner, there is the risk of
slippage. If the call is sold first, the stock can move before the trader has a
chance to buy it at the necessary price. It is generally better and less risky to
exercise the call rather than leg out of the trade.
In this transaction, the trader begins with a fairly flat position (short
stock/long synthetic stock) and ends with a short put that is significantly
out-of-the-money. For all intents and purposes, exercising the call in this
trade is like sy


before the trader has a
chance to buy it at the necessary price. It is generally better and less risky to
exercise the call rather than leg out of the trade.
In this transaction, the trader begins with a fairly flat position (short
stock/long synthetic stock) and ends with a short put that is significantly
out-of-the-money. For all intents and purposes, exercising the call in this
trade is like synthetically selling the put. But at what price? In this case, it’s
$0.15. This again is the cost benefit of saving $0.40 by avoiding the
dividend obligation versus the $0.25 gain in call time value. Exercising the
call is effectively like selling the put at 0.15 in this example. If the dividend
is lower or the interest is higher, it may not be worth it to the trader to
exercise the call to capture the dividend. How do traders know if their calls
should be exercised?
The traders must do the math before each ex-dividend date in option
classes they trade. The traders have to determine if the benefit from
exercising—or the price at which the synthetic put is essentially being sold
—is more or less than the price at which they can sell the put. The math
used here is adopted from put-call parity:
This shows the case where the traders can effectively synthetically sell the
put (by exercising) for more than the current put value. Tactically, it’s
appropriate to use the bid price for the put in this calculation since that is
the price at which the put can be sold.
In this case, the traders would be inclined to not exercise. It would be
theoretically more beneficial to sell the put if the trader is so inclined.
Here, the traders, from a valuation perspective, are indifferent as to whether
or not to exercise. The question then is simply: do they want to sell the put
at this price?
Professionals and big retail traders who are long (ITM) calls—whether as
part of a reversal, part of another type of spread, or because they are long
the calls outright—must do this math the day before each ex-dividend date
to maximize profits and minimize losses. Not exercising, or forgetting to
exercise, can be a costly mistake. Traders who are short ITM dividend-
paying calls, however, can reap the benefits of those sleeping on the job. It
works both ways.
Traders who are long stock and short calls at parity before the ex-date
may stand to benefit if some of the calls do not get assigned. Any shares of
long stock remaining on the ex-date will result in the traders receiving
dividends. If the dividends that will be received are greater in value than the
interest that will subsequently be paid on the long stock, the traders may
stand reap an arbitrage profit because of long call holders’ forgetting to
exercise.
Dividend Plays
The day before an ex-dividend date in a stock, option volume can be
unusually high. Tens of thousands of contracts sometimes trade in names
that usually have average daily volumes of only a couple thousand. This
spike in volume often has nothing to do with the market’s opinion on
direction after the dividend. The heavy trading has to do with the
revaluation of the relationship of exercisable options to the underlying
expected to occur on the ex-dividend date.
Traders that are long ITM calls and short ITM calls at another strike just
before an ex-dividend date have a potential liability and a potential benefit.
The potential liability is that they can forget to exercise. This is a liability
over which the traders have complete control. The potential benefit is that
some of the short calls may not get assigned. If traders on the other side of
the short calls (the longs) forget to exercise, the traders that are short the
call make out by not having to pay the dividend on short stock.
Professionals and big retail traders who have very low transaction costs
will sometimes trade ITM call spreads during the afternoon before an ex-
dividend date. This consists of buying one call and selling another call with
a different strike price. Both calls in the dividend-play strategy are ITM and
have corresponding puts with little or no value (to be sure, the put value is
less than the dividend minus the interest). The traders trade the spreads,
fairly indifferent as to whether they buy or sell the spreads, in hope of
skating—or not getting assigned—on some of their short calls. The more
they don’t get assigned the better.
This usually occurs in options that have high open interest, meaning there
are a lot of outstanding contracts already. The more contracts in existence,
the better the possibility of someone forgetting to exercise. The greatest
volume also tends to occur in the front month.
Strange Deltas
Because American calls become an exercise possibility when the ex-date is
imminent, the deltas can sometimes look odd. When the calls are trading at
parity, they have a 1.00 delta. They are a substitute for the stock. They, in
fact, will be stock if and when they are exercised just before the ex-date.
But if the puts still have some residual time value, they may also have a
small delta, of 0.05 or perhaps more.
In this unique scenario, the delta of the synthetic can be greater than
+1.00 or less than −1.00. It is not uncommon to see the absolute values of
the call and put deltas add up to 1.07 or 1.08. When the dividend comes out
of the options model on the ex-date, synthetics go back to normal. The delta
of the synthetic again approaches 1.00. Because of the out-of-whack deltas,
delta-neutral traders need to take extra caution in their analytics when ex-
dates are near. A little common sense should override what the computer
spits out.
Inputting Dividend Data into the
Pricing Model
Often dividend payments are regular and predictable. With many
companies, the dividend remains constant quarter after quarter. Some
corporations have a track record of incrementally increasing their dividends
every year. Some companies pay dividends in a very irregular fashion, by
paying special dividends that are often announced as a surprise


s out.
Inputting Dividend Data into the
Pricing Model
Often dividend payments are regular and predictable. With many
companies, the dividend remains constant quarter after quarter. Some
corporations have a track record of incrementally increasing their dividends
every year. Some companies pay dividends in a very irregular fashion, by
paying special dividends that are often announced as a surprise to investors.
In a truly capitalist society, there are no restrictions and no rules on when,
whether, or how corporations pay dividends to their shareholders.
Unpredictability of dividends, though, can create problems in options
valuation.
When a company has a constant, reasonably predictable dividend, there is
not a lot of guesswork. Take Exelon Corp. (EXC). From November 2008 to
the time of this writing, Exelon has paid a regular quarterly dividend of
$0.525. During that period, a trader has needed simply to enter 0.525 into
the pricing calculator for all expected future dividends to generate the
theoretical value. Based on recent past performance, the trader could feel
confident that the computed analytics were reasonably accurate. If the
trader believed the company would continue its current dividend policy,
there would be little options-related dividend risk—unless things changed.
When there is uncertainty about when future dividends will be paid in
what amounts, the level of dividend-related risk begins to increase. The
more uncertainty, the more risk. Let’s examine an interesting case study:
General Electric (GE).
For a long time, GE was a company that has had a history of increasing
its dividends at fairly regular intervals. In fact, there was more than a 30-
year stretch in which GE increased its dividend every year. During most of
the first decade of the 2000s, increases in GE’s dividend payments were
around one to six cents and tended to occur toward the end of December,
after December expiration. The dividends were paid four times per year but
not exactly quarterly. For several years, the ex-dates were in February, June,
September, and December. Option traders trading GE options had a pretty
easy time estimating their future dividend streams, and consequently
evaded valuation problems that could result from using wrong dividend
data. Traders would simply adjust the dividend data in the model to match
their expectations for predictably increasing future dividends in order to
achieve an accurate theoretical value. Let’s look back at GE to see how a
trader might have done this.
The following shows dividend-history data for GE.
Ex-DateDividend*
12/27/02$0.19
02/26/03$0.19
06/26/03$0.19
09/25/03$0.19
12/29/03$0.20
02/26/04$0.20
06/24/04$0.20
09/23/04$0.20
12/22/04$0.22
02/24/05$0.22
06/23/05$0.22
09/22/05$0.22
12/22/05$0.25
02/23/06$0.25
06/22/06$0.25
09/21/06$0.25
12/21/06$0.28
02/22/07$0.28
06/21/07$0.28
* These data are taken from the following Web page on GE’s web site:
www.ge.com/investors/stock_info/dividend_history.html .
At the end of 2006, GE raised its dividend from $0.25 to $0.28. A trader
trading GE options at the beginning of 2007 would have logically
anticipated the next increase to occur again in the following December
unless there was reason to believe otherwise. Options expiring before this
anticipated next dividend increase would have the $0.28 dividend priced
into their values. Options expiring after December 2007 would have a
higher dividend priced into them—possibly an additional three cents to 0.31
(which indeed it was). Calls would be adversely affected by this increase,
and puts would be favorably affected. A typical trader would have
anticipated those changes. The dividend data a trader pricing GE options
would have entered into the model in January 2007 would have looked
something like this.
Ex-DateDividend*
02/22/07$0.28
06/21/07$0.28
09/20/07$0.28
12/20/07$0.31
02/21/08$0.31
06/19/08$0.31
09/18/08$0.31
* These data are taken from the following Web page on GE’s web site:
www.ge.com/investors/stock_info/dividend_history.html .
The trader would have entered the anticipated future dividend amount in
conjunction with the anticipated ex-dividend date. This trader projection
goes out to February 2008, which would aid in valuing options expiring in
2007 as well as the 2008 LEAPS. Because the declaration dates had yet to
occur, one could not know with certainty when the dividends would be
announced or in what amount. Certainly, there would be some estimation
involved for both the dates and the amount. But traders would probably get
it pretty close—close enough.
Then, something particularly interesting happened. Instead of raising the
dividend going into December 2008 as would be a normal pattern, GE kept
it the same. As shown, the 12/24/08 ex-dated dividend remained $0.31.
Ex-DateDividend*
02/22/07$0.28
06/21/07$0.28
09/20/07$0.28
12/20/07$0.31
02/21/08$0.31
06/19/08$0.31
09/18/08$0.31
12/24/08$0.31
* These data are taken from the following Web page on GE’s web site:
www.ge.com/investors/stock_info/dividend_history.html .
The dividend stayed at $0.31 until the June 2009 dividend, which held
another jolt for traders pricing options. Around this time, GE’s stock price
had taken a beating. It fell from around $42 a share in the fall of 2007
ultimately to about $6 in March 2009. GE had its first dividend cut in more
than three decades. The dividend with the ex-date of 06/18/09 was $0.10.
12/24/08$0.31
02/19/09$0.31
06/18/09$0.10
09/17/09$0.10
12/23/09$0.10
02/25/10$0.10
06/17/10$0.10
09/16/10$0.12
12/22/10$0.14
02/24/11$0.14
06/16/11$0.15
09/15/11$0.15
Though the company gave warnings in advance, the drastic dividend
change had a significant impact on option prices. Call prices were helped by
the dividend cut (or anticipated dividend cut) and put prices were hurt.
The break in the pattern didn’t stop there. The dividend policy remained
$0.10 for five quarters until it rose to $0.12 in September 2010, then to
$0.14 in December 2010, then to $0.15 in June 2011. These irregula