Furthermore, she realizes that her outlook may be wrong: Johnson &
Johnson may decline. She may have to close the position early—maybe for
aprofit, maybe for aloss. Stacie also needs to study her greeks. Exhibit 5.5
shows the greeks for this trade.
EXHIBIT 5.5 Greeks for short Johnson & Johnson 65 put (per contract).
Delta 0.65
Gamma−0.15
Theta 0.02
Vega −0.07
The first item to note is the delta. This position has adirectional bias. This
bias can work for or against her. With apositive 0.65 delta per contract, this
position has adirectional sensitivity equivalent to being long around 650
shares of the stock. That’sthe delta × 100 shares × 10 contracts.
Stacie’strade is not just abullish version of Brendan’s. Partly because of
the size of the delta, it’sdifferent—specific directional bias aside. First, she
will handle her trade differently if it is profitable.
For example, if over the next week or so Johnson & Johnson rises $1,
positive delta and negative gamma will have anet favorable effect on
Stacie’sprofitability. Theta is small in comparison and won’thave too much
of an effect. Delta/gamma will account for adecrease in the put’stheoretical value of about $0.73. That’sthe estimated average delta times
the stock move, or [0.65 + (–0.15/2)] × 1.00.
Stacie’sactual profit would likely be less than 0.73 because of the bid-ask
spread. Stacie must account for the fact that the bid-ask is 0.05 wide (1.75–
1.80). Because Stacie would buy to close this position, she should consider
the 0.73 price change relative to the 1.80 offer, not the 1.75 trade price—
that is, she factors in anickel of slippage. Thus, she calculates, that the puts
will be offered at 1.07 (that’s 1.80 − 0.73) when the stock is at $65. That is
again of $0.68.
In this scenario, Stacie should consider the Would I Do It Now? rule to
guide her decision as to whether to take her profit early or hold the position
until expiration. Is she happy being short ten 65 puts at 1.07 with Johnson
& Johnson at $65? The premium is lower now. The anticipated move has
already occurred, and she still has 28 days left in the option that could allow

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:174


for the move to reverse itself. If she didn’thave the trade on now, would she
sell ten 65 puts at 1.07 with Johnson & Johnson at $65? Based on her
original intention, unless she believes strongly now that abreakout through
$65 with follow-through momentum is about to take place, she will likely
take the money and run.
Stacie also must handle this trade differently from Brendan in the event
that the trade is aloser. Her trade has ahigher delta. An adverse move in the
underlying would affect Stacie’strade more than it would Brendan’s. If
Johnson & Johnson declines, she must be conscious in advance of where
she will cover.
Stacie considers both how much she is willing to lose and what potential
stock-price action will cause her to change her forecast. She consults astock chart of Johnson & Johnson. In this example, we’ll assume there is
some resistance developing around $64 in the short term. If this resistance
level holds, the trade becomes less attractive. The at-expiration breakeven is
$63.25, so the trade can still be awinner if Johnson & Johnson retreats. But
Stacie is looking for the stock to approach $65. She will no longer like the
risk/reward of this trade if it looks like that price rise won’toccur. She
makes the decision that if Johnson & Johnson bounces off the $64 level
over the next couple weeks, she will exit the position for fear that her
outlook is wrong. If Johnson & Johnson drifts above $64, however, she will
ride the trade out.
In this example, Stacie is willing to lose 1.00 per contract. Without taking
into account theta or vega, that 1.00 loss in the option should occur at astock price of about $63.28. Theta is somewhat relevant here. It helps
Stacie’spotential for profit as time passes. As time passes and as the stock
rises, so will theta, helping her even more. If the stock moves lower (against
her) theta helps ease the pain somewhat, but the further in-the-money the
put, the lower the theta.
Vega can be important here for two reasons: first, because of how implied
volatility tends to change with market direction, and second, because it can
be read as an indication of the market’sexpectations.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:175


The Double Whammy
With the stock around $64, there is anegative vega of about seven cents. As
the stock moves lower, away from the strike, the vega gets abit smaller.
However, the market conditions that would lead to adecline in the price of
Johnson & Johnson would likely cause implied volatility (IV) to rise. If the
stock drops, Stacie would have two things working against her—delta and
vega—adouble whammy. Stacie needs to watch her vega. Exhibit 5.6
shows the vega of Stacie’sput as it changes with time and direction.
EXHIBIT 5.6 Johnson & Johnson 65 put vega.
If after one week passes Johnson & Johnson gaps lower to, say, $63.00 ashare, the vega will be 0.043 per contract. If IV subsequently rises 5 points
as aresult of the stock falling, vega will make Stacie’sputs theoretically
worth 21.5 cents more per contract. She will lose $215 on vega (that’s 0.043
vega × 5 volatility points × 10 contracts) plus the adverse delta/gamma
move.
Agap opening will cause her to miss the opportunity to stop herself out at
her target price entirely. Even if the stock drifts lower, her targeted stop-loss
price will likely come sooner than expected, as the option price will likely
increase both by delta/gamma and vega resulting from rising volatility. This
can cause her to have to cover sooner, which leaves less room for error.
With this trade, increases in IV due to market direction can make it feel as if
the delta is greater than it actually is as the market declines. Conversely, IV
softening makes it feel as if the delta is smaller than it is as the market rises.
The second reason IV has importance for this trade (as for most other
strategies) is that it can give some indication of how much the market thinks
the stock can move. If IV is higher than normal, the market perceives there

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:176


to be more risk than usual of future volatility. The question remains: Is the
higher premium worth the risk?
The answer to this question is subjective. Part of the answer is based on
Stacie’sassessment of future volatility. Is the market right? The other part is
based on Stacie’srisk tolerance. Is she willing to endure the greater price
swings associated with the potentially higher volatility? This can mean
getting whipsawed, which is exiting aposition after reaching astop-loss
point only to see the market reverse itself. The would-be profitable trade is
closed for aloss. Higher volatility can also mean ahigher likelihood of
getting assigned and acquiring an unwanted long stock position.
Cash-Secured Puts
There are some situations where higher implied volatility may be abeneficial trade-off. What if Stacie’smotivation for shorting puts was
different? What if she would like to own the stock, just not at the current
market price? Stacie can sell ten 65 puts at 1.75 and deposit $63,250 in her
trading account to secure the purchase of 1,000 shares of Johnson &
Johnson if she gets assigned. The $63,250 is the $65 per share she will pay
for the stock if she gets assigned, minus the 1.75 premium she received for
the put × $100 × 10 contracts. Because the cash required to potentially
purchase the stock is secured by cash sitting ready in the account, this is
called acash-secured put.
Her effective purchase price if assigned is $63.25—the same as her
breakeven at expiration. The idea with this trade is that if Johnson &
Johnson is anywhere under $65 per share at expiration, she will buy the
stock effectively at $63.25. If assigned, the time premium of the put allows
her to buy the stock at adiscount compared with where it is priced when the
trade is established, $64. The higher the time premium—or the higher the
implied volatility—the bigger the discount.
This discount, however, is contingent on the stock not moving too much.
If it is above $65 at expiration she won’tget assigned and therefore can
only profit amaximum of 1.75 per contract. If the stock is below $63.25 at
expiration, the time premium no longer represents adiscount, in fact, the
trade becomes aloser. In away, Stacie is still selling volatility.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:177


Covered Call
The problem with selling anaked call is that it has unlimited exposure to
upside risk. Because of this, many traders simply avoid trading naked calls.
Amore common, and some would argue safer, method of selling calls is to
sell them covered.
Acovered call is when calls are sold and stock is purchased on ashare-
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 adifferent
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 adelta of one, and all its other greeks are
zero.
The pivotal point for both positions is the strike price. That’sthe 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 afew 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 Iwant 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, amain focus of acovered call is the option

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:178


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. Atrader, Bill, is neutral to slightly bullish on Harley-
Davidson over the next three months. Exhibit 5.7 shows aselection 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’san 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 adifferent
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 atheta advantage over the longer-term
choices. The March 70 has atheta of 0.032, while the April 70’stheta is
0.026 and the May 70’sis 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

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:179


percent IV, and the May has a 23 percent IV. March is the cheapest option
by IV standards. This is not necessarily afavorable quality for ashort
candidate. Bill must weigh his assessment of all relevant information and
then decide which trade is best. With this type of astrategy, 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 anet delta of 0.588 per contract.
That’sthe 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 ahigher strike. He would have to
sell the April or May 75, since the March 75s are azero bid. This would
give him ahigher 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’drather 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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:180


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’dprofit on both delta and theta. If the trade
isn’texactly perfect, but still good, Harley-Davidson will be anywhere
above $68.15 in 22 days. It’sthe prospect that the trade may not be so good
at March expiration that occupies Bill’sthoughts, but atrader 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’strending to the upside are not quite so dire, although he might be somewhat
frustrated with any lost opportunity above the indifference point. It’sthe
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. Arise in implied volatility will likely accompany adecline in

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:181


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 anaked 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 atime. If he legs out of the
trade, he’slikely to close the call first. The motivation for exiting atrade
early is to reduce risk. Anaked call is hardly less risky than acovered 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 alot 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 anickel, adime, 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 asingle order, it is called acalendar spread or atime spread.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:182


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 acovered
put, atrader sells both puts and stock on aone-to-one basis. The term
covered put is abit of amisnomer, as the strategy changes from limited risk
to unlimited risk when short stock is added to the short put. Anaked put can
produce only losses until the stock goes to zero—still asubstantial 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 anaked call. In fact, they are synthetically equal. This concept
will be addressed further in the next chapter.
Let’slooks at another trader, Libby. Libby is an active trader who trades
several positions at once. Libby believes the overall market is in arange
and will continue as such over the next few weeks. She currently holds ashort 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 acovered-put position.
There is one caveat: Libby is leaving for acruise 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

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:183


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. Amove 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 apremium-over-parity standpoint. Below the strike, the
intrinsic value of the put doesn’tmatter 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’sa 73-cent profit, or $730 on her 10 contracts.
This doesn’taccount for any changes in the time value that may occur as aresult 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 aprofit on aposition that worked out just about exactly as planned.
Her risk, though, is to the upside. Abig rally in the stock can cause big
losses. From atheoretical standpoint, losses are potentially unlimited with
this type of trade. If the stock is above the strike, she needs to have amental
stop order in mind and execute the closing order with discipline.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:184


Put-Call Parity Essentials
Before the creation of the Black-Scholes model, option pricing was hardly
an exact science. Traders had only afew 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 astock with limited risk can buy amarried put: long stock and along put on ashare-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 amarried
put and along 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
atrader, 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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:187


The stock component of the married put could be purchased on margin.
Buying stock on margin is borrowing capital to finance astock 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’tinvest capital in the stock, the capital will rest in an interest-bearing
account. The trader forgoes that interest when he buys astock. 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 atrader
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 apricing consideration that adds cost to the
married put and not the long call.
So if the married put is amore expensive endeavor than the long call
because of the interest paid on the investment portion that is below the
strike, why would anyone buy amarried put? Wouldn’ttraders 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 awhole 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 arobust market with many savvy traders,
arbitrage opportunities don’texist 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

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:188


Dividends
Another difference between call and married-put values is dividends. Acall
option does not extend to its owner the right to receive adividend payment.
Traders, however, who are long aput and long stock are entitled to adividend if it is the corporation’spolicy to distribute dividends to its
shareholders.
An adjustment must be made to the put-call parity to account for the
possibility of adividend payment. The equation must be adjusted to account
for the absence of dividends paid to call holders. For adividend-paying
stock, the put-call parity states
The interest advantage and dividend disadvantage of owning acall is
removed from the market by arbitrageurs. Ultimately, that is what is
expressed in the put-call parity. It’saway to measure the point at which the
arbitrage opportunity ceases to exist. When interest and dividends are
factored in, along call is an equal position to along put paired with long
stock. In options nomenclature, along put with long stock is asynthetic
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.
Asynthetic long put is created by buying acall and selling (short) stock.
The at-expiration diagrams in Exhibit 6.2 show identical payouts for these
two trades.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:190


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. Ageneral 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 asynthetic 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.
Asynthetic short put can be created by selling acall of the same month
and strike and buying stock on ashare-for-share basis (i.e., acovered 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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:191


EXHIBIT 6.3 Short put vs. short call + long stock.
Ashort (negative) put is equal to ashort (negative) call plus long stock,
after the basis adjustment. Consider that if the put is sold instead of buying
stock and selling acall, the interest that would otherwise be paid on the cost
of the stock up to the strike price is asavings 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 anet delta of 0.45 (–0.55 + 1.00).
Similarly, asynthetic short call can be created by selling aput and selling
(short) one hundred shares of stock. Exhibit 6.4 shows aconceptual
overview of these two positions at expiration.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:192


Comparing Synthetic Calls and Puts
The common thread among the synthetic positions explained above is that,
for aput-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 acall 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 along 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 asynthetic 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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:194


American-Exercise Options
Put-call parity was designed for European-style options. The early exercise
possibility of American-style options gums up the works abit. Because acall (put) and asynthetic 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 aput-call pair are subtle.
Exhibit 6.5 is acomparison 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 abit more realistic, consider that
because of American exercise, the absolute delta values of put-call pairs
don’talways 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 acombined-
position delta (call delta plus stock delta) that is very close to the put’sdelta. The delta of this synthetic put is −0.446 (0.554 − 1.00). The delta of aput 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

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:195


This holds true whether the options are in-, at-, or out-of-the-money. For
example, with astock 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 anet delta of −0.201.
If long or short stock is added to acall or put to create asynthetic, 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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:196


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 acall and selling aput
of the same strike and time to expiration creates the equivalent of along
stock position. This is called asynthetic stock position, or acombo. 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 along stock position compared with along call
combined with ashort put position.
EXHIBIT 6.6 Long stock vs. long call + short put.
Aquick 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

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:197


to buy the stock at the same strike price. It doesn’tmatter what the strike
price is. As long as the strike is the same for the call and the put, the trader
will have along 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 anet of $2 to get along
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 astock 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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:198


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 apenny 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 abenefit, 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 ayear. 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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:199


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 asynthetic 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 afunction of rounding and early exercise. More on this in the “Conversions
and Reversals” section.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:200


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: aconversion and areversal.
Conversion
Aconversion is athree-legged position in which atrader is long stock, short
acall, and long aput. The options share the same month and strike price.
By most metrics, this is avery flat position. Atrader with aconversion is
long the stock and, at the same time, synthetically short the same stock.
Consider this from the perspective of delta. In aconversion, 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 asimple example of atypical 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 anegative delta to the position, in this case,
−0.63. The long put also contributes anegative 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’sother 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—acall and aput,
respectively—with the same month and strike, the greeks come very close

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:202


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 aposition is
converted. Let’slook at amore detailed example.
Atrader 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 avery 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’sdelta 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:

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:203


or
With American options, aput 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’svalue, 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’sdesire 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 alittle 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’sprice given achange in the interest rate.
The −0.090 rho of the conversion indicates that if the interest rate rises
one percentage point, the position as awhole 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

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:204


finance the long stock. This is proven mathematically by put-call parity.
Negative rho indicates abearish position on the interest rate; the trader
wants it to go lower. Positive rho is abullish interest rate position.
But aone-percentage-point change in the interest rate in one day is abig
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’sjust $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. Amarket maker with a
5,000-lot conversion would stand to make or lose $11,250, given aquarter-
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, amarket maker may set out to sell calls and in turn buy stock to
hedge the call’sdelta risk (this will be covered in Chapters 12 and 17), then
buy puts and the rest of the stock to create abalanced conversion: one call
to one put to one hundred shares. The trader may try to put on the
conversion in the previous example for atotal of $0.50 over the price of the
long stock instead of the $0.46 it’sworth. 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
Areversal, or reverse conversion, is simply the opposite of the conversion:
buy call, sell put, and sell (short) stock. Areversal can be executed to close
aconversion, or it can be an opening transaction. Using the same stock and
options as in the previous example, atrader could establish areversal as
follows:

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:205


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 ashort position in the stock at $51.54 and along
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 abullish position on
interest rates, which is indicated by apositive 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 ahigher interest rate. If rates fall one percentage
point, the synthetic long stock loses $0.09. The trader earns less interest
being short stock given alower interest rate.
With the reversal, the fact that the put can be exercised early is arisk.
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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:206


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’srevisit the mind of amarket 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 abroker), 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 amarketable order is
sent to the exchange.
Managing risk can get abit 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 areversal and thereby have very little risk. If they can lock in
the reversal for asmall 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

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:207


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 along delta position after expiration. If the puts
are not assigned, they should exercise the calls to get delta flat. It’salso
possible that only some of the puts will be assigned.
Because they don’tknow 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 adifferent strike
price, the resulting position is known as abox. With abox, atrader 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. Astudy of the greeks shows that the
delta is close to zero. Gamma, theta, vega, and rho are also negligible.
Here’san 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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:208


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 abox?
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 aconversion or areversal. Often, they have aconversion at one strike and areversal at another. The stock positions for
these cancel each other out and the trader is left with only the four option
legs—that is, abox. They can eliminate pin risk on both strikes by trading
the box as asingle trade to close all four legs. Another reason for trading abox has to do with capital.
Borrowing and Lending Money
The first thing to consider is how this spread is priced. Let’slook at another
example of abox, 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

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:209


Atrader 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 atrader 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. Arational 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—Alosing
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’sfree 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.
Atrader 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
Ajelly roll, or simply aroll, is also aspread with four legs and acombination of two synthetic stock trades. In abox, the difference between
the synthetics is the strike price; in aroll, it’sthe contract month. Here’san
example:
Long 1 April 50 call
Short 1 April 50 put
Short 1 May 50 call
Long 1 May 50 put

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:210


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 amostly flat position. Delta, gamma,
theta, vega, and even rho have only small effects on ajelly roll, but like the
others, this spread serves apurpose.
Atrader with aconversion or reversal can roll the option legs of the
position into amonth with alater expiration. For example, atrader 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 amonth farther out.
Another reason for trading aroll 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 atrader’sexpectations of future changes in interest rates,
aposition can be constructed to exploit opportunities in interest.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:211


Rho and Interest Rates
Rho is ameasurement of the sensitivity of an option’svalue to achange 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’sclear that as the interest rate rises, put prices must
fall and call prices must rise to keep put-call parity balanced. With alittle
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, acall with arho 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. Aput with arho of −0.08
will lose $0.08 with each one-point rise and gain $0.08 in value with aone-
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 afunction 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,

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:217


the greater the effect of interest. Rho measures an option’ssensitivity to the
end results of these three influences.
To understand how changes in interest affect option prices, consider atypical at-the-money (ATM) conversion on anon-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 Stock + Interest, or
Certainly, if the interest rate were higher, the interest on the synthetic
stock would be ahigher 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—anet of $0.35.
Aone-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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:218


Rho and Time
The time component of interest has abig impact on the magnitude of an
option’srho, 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 ahigher rho.
Take astock 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 aquarter of apercentage 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’srho, and
therefore, the more interest will affect the option’svalue.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:219


LEAPS
Options buyers have time working against them. With each passing day,
theta erodes the value of their assets. Buying along-term option, or a
LEAPS, helps combat erosion because long-term options can decay at aslower 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 abit 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 amotivation for Susanne to choose along-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 ahigher delta. If the XYZ stock price rises $2, the
gamma of the May call will cause Jason’sdelta to creep higher than
Susanne’s. At $62, the delta for the May 60s would be about 0.78, whereas

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:221


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
ayearly low, LEAPS might be abetter buy. Aone- 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 agood hard look at rho. The LEAPS
rho is significantly higher than that of its short-term counterpart. Aone-
percentage-point change in the interest rate will change Susanne’s P&(L) by
$0.64—that’sabout 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’stime.
It is important to understand that, like the other greeks, rho is asnapshot
at aparticular price, volatility level, interest rate, and moment in time. If
interest rates were to fall by one percentage point today, it would cause
Susanne’scall to decline in value by $0.64. If that rate drop occurred over
the life of the option, it would have amuch 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 adecline
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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:222


Pricing in Interest Rate Moves
In the same way that volatility can get priced in to an option’svalue, 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 apace 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). Atrader, Kyle, enters parameters into the model for ABC options and notices
that the prices don’tline 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 ashare, 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:

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:223


The theoretical values, in bold type, are those that don’tline 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’sexpectations 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 arevision of option values. In long-
term options that have higher rhos, this is abona fide risk. Short-term

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:224


Trading Rho
While it’spossible 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 along-term option to be 5 to 8 percent
of the option’svalue. For example, Exhibit 7.2 shows atwo-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’sonly about 1.5 percent of the call’svalue. On one hand, 1.5 percent is
not avery big profit on atrade. 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 asecond 25-basis-point rate increase, other influences will diminish rho’ssignificance. If over the six-week period between Fed meetings, the

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:226


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 aloss 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 aconcern. 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 apractical way to
speculate on interest rates.
To take aposition 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 ahurdle to trading
these strategies for non–market makers. Generally, rho is agreek that for
most traders is important to understand but not practical to trade.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:227


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’ssensitivity 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 atrader’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 achange 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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:229


Dividends and Option Pricing
The preceding discussion demonstrated how dividends affect stock traders.
There’sone problem: we’re option traders! Option holders or writers do not
receive or pay dividends, but that doesn’tmean dividends aren’trelevant to
the pricing of these securities. Observe the behavior of aconversion or areversal 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’slook at an example to explore
why.
At the close on the day before the ex-date of astock paying a $0.25
dividend, atrader 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’sclose. 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

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:232


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—aright 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’slook at an example of areversal 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, atrader 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 anegligible delta, will remain unchanged. But what about the call?
With no dividend left in the stock, the put call-parity states
In this case,

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:234


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’sagain of $0.25 on the call. That sounds
good, but because the trader is short stock, if he hasn’texercised, he will
owe the $0.40 dividend—anet 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 aprofit nor aloss. 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 atrade in this manner, there is the risk of
slippage. If the call is sold first, the stock can move before the trader has achance 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 afairly flat position (short
stock/long synthetic stock) and ends with ashort 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?

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:235


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’sappropriate 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 avaluation 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 areversal, 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 acostly 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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:236


Dividend Plays
The day before an ex-dividend date in astock, option volume can be
unusually high. Tens of thousands of contracts sometimes trade in names
that usually have average daily volumes of only acouple thousand. This
spike in volume often has nothing to do with the market’sopinion 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 apotential liability and apotential benefit.
The potential liability is that they can forget to exercise. This is aliability
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
adifferent 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’tget assigned the better.
This usually occurs in options that have high open interest, meaning there
are alot 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.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:237


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 asubstitute 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 asmall 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. Alittle common sense should override what the computer
spits out.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:238


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 atrack record of incrementally increasing their dividends
every year. Some companies pay dividends in avery irregular fashion, by
paying special dividends that are often announced as asurprise to investors.
In atruly 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 acompany has aconstant, reasonably predictable dividend, there is
not alot of guesswork. Take Exelon Corp. (EXC). From November 2008 to
the time of this writing, Exelon has paid aregular quarterly dividend of
$0.525. During that period, atrader 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’sexamine an interesting case study:
General Electric (GE).
For along time, GE was acompany that has had ahistory 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’sdividend 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 apretty

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:239


Good and Bad Dates with Models
Using an incorrect date for the ex-date in option pricing can lead to
unfavorable results. If the ex-dividend date is not known because it has yet
to be declared, it must be estimated and adjusted as need be after it is
formally announced. Traders note past dividend history and estimate the
expected dividend stream accordingly. Once the dividend is declared, the
ex-date is known and can be entered properly into the pricing model. Not
executing due diligence to find correct known ex-dates can lead to trouble.
Using abad date in the model can yield dubious theoretical values that can
be misleading or worse—especially around the expiration.
Say acall is trading at 2.30 the day before the ex-date of a $0.25
dividend, which happens to be thirty days before expiration. The next day,
of course, the stock may have moved higher or lower. Assume for
illustrative purposes, to compare apples to apples as it were, that the stock is
trading at the same price—in this case, $76.
If the trader is using the correct date in the model, the option value will
adjust to take into account the effect of the dividend expiring, or reaching
its ex-date, when the number of days to expiration left changes from 30 to
29. The call trading postdividend will be worth more relative to the same
stock price. If the dividend date the trader is using in the model is wrong,
say one day later than it should be, the dividend will still be an input of the
theoretical value. The calculated value will be too low. It will be wrong.
Exhibit 8.1 compares the values of a 30-day call on the ex-date given the
right and the wrong dividend.
EXHIBIT 8.1 Comparison of 30-day call values

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:243


At the same stock price of $76 per share, the call is worth $0.13 more
after the dividend is taken out of the valuation. Barring any changes in
implied volatility (IV) or the interest rate, the market prices of the options
should reflect this change. Atrader using an ex-date in the model that is
farther in the future than the actual ex-date will still have the dividend as
part of the generated theoretical value. With the ex-date just one day later,
the call would be worth 2.27. The difference in option value is due to the
effect of theta—in this case, $0.03.
With abad date, the value of 2.27 would likely be significantly below
market price, causing the market value of the option to look more expensive
than it actually is. If the trader did not know the date was wrong, he would
need to raise IV to make the theoretical value match the market. This option
has avega of 0.08, which translates into adifference of about two IV points
for the theoretical values 2.43 and 2.27. The trader would perceive the call
to be trading at an IV two points higher than the market indicates.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:244


Dividend Size
It’snot just the date but also the size of the dividend that matters. When
companies change the amount of the dividend, options prices follow in step.
In 2004, when Microsoft (MSFT) paid aspecial dividend of $3 per share,
there were unexpected winners and losers in the Microsoft options. Traders
who were long calls or short puts were adversely affected by this change in
dividend policy. Traders with short calls or long puts benefited. With long-
term options, even less anomalous changes in the size of the dividend can
have dramatic effects on options values.
Let’sstudy an example of how an unexpected rise in the quarterly
dividend of astock affects along call position. Extremely Yellow Zebra
Corp. (XYZ) has been paying aquarterly dividend of $0.10. After asteady
rise in stock price to $61 per share, XYZ declares adividend payment of
$0.50. It is expected that the company will continue to pay $0.50 per
quarter. Atrader, James, owns the 528-day 60-strike calls, which were
trading at 9.80 before the dividend increase was announced.
Exhibit 8.2 compares the values of the long-term call using a $0.10
quarterly dividend and using a $0.50 quarterly dividend.
EXHIBIT 8.2 Effect of change in quarterly dividend on call value.
This $0.40 dividend increase will have abig effect on James’scalls. With
528 days until expiration, there will be six dividends involved. Because

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:245


James is long the calls, he loses 1.52 per option. If, however, he were short
the calls, 1.52 would be his profit on each option.
Put traders are affected as well. Another trader, Marty, is long the 60-
strike XYZ puts. Before the dividend announcement, Marty was running his
values with a $0.10 dividend, giving his puts avalue of 5.42. Exhibit 8.3
compares the values of the puts with a $0.10 quarterly dividend and with a
$0.50 quarterly dividend.
EXHIBIT 8.3 Effect of change in quarterly dividend on put value.
When the dividend increase is announced, Marty will benefit. His puts
will rise because of the higher dividend by $0.66 (all other parameters held
constant). His long-term puts with six quarters of future expected dividends
will benefit more than short-term XYZ puts of the same strike would. Of
course, if he were short the puts, he would lose this amount.
The dividend inputs to apricing model are best guesses until the dates
and amounts are announced by the company. How does one find dividend
information? Regularly monitoring the news and press releases on the
companies one trades is agood way to stay up to date on dividend
information, as well as other company news. Dividend announcements are
widely disseminated by the major news services. Most companies also have
an investor-relations phone number and section on their web sites where
dividend information can be found.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:246


CHAPTER 9
Vertical Spreads
Risk—it is the focal point around which all trading revolves. It may seem as
if profit should be occupying this seat, as most important to trading options,
but without risk, there would be no profit! As traders, we must always look
for ways to mitigate, eliminate, preempt, and simply avoid as much risk as
possible in our pursuit of success without diluting opportunity. Risk must be
controlled. Trading vertical spreads takes us one step further in this quest.
The basic strategies discussed in Chapters 4 and 5 have strengths when
compared with pure linear trading in the equity markets. But they have
weaknesses, too. Consider the covered call, one of the most popular option
strategies.
Acovered call is best used as an augmentation to an investment plan. It
can be used to generate income on an investment holding, as an entrance
strategy into astock, or as an exit strategy out of astock. But from atrading
perspective, one can often find better ways to trade such aforecast.
If the forecast on astock is neutral to moderately bullish, accepting the
risk of stock ownership is often unwise. There is always the chance that the
stock could collapse. In many cases, this is an unreasonable risk to assume.
To some extent, we can make the same case for the long call, short put,
naked call, and the like. In certain scenarios, each of these basic strategies is
accompanied with unwanted risks that serve no beneficial purpose to the
trader but can potentially cause harm. In many situations, avertical spread
is abetter alternative to these basic spreads. Vertical spreads allow atrader
to limit potential directional risk, limit theta and vega risk, free up margin,
and generally manage capital more efficiently.

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:248


Bull Call Spread
Abull call spread is along call combined with ashort call that has ahigher
strike price. Both calls are on the same underlying and share the same
expiration month. Because the purchased call has alower strike price, it
costs more than the call being sold. Establishing the trade results in adebit
to the trader’saccount. Because of this debit, it’scalled adebit spread.
Below is an example of abull call spread on Apple Inc. (AAPL):
In this example, Apple is trading around $391. With 40 days until
February expiration, the trader buys the 395–405 call spread for anet debit
of $4.40, or $440 in actual cash. Or one could simply say the trader paid
$4.40 for the 395–405 call.
Consider the possible outcomes if the spread is held until expiration.
Exhibit 9.1 shows an at-expiration diagram of the bull call spread.
EXHIBIT 9.1 AAPL bull call spread.
Before discussing the greeks, consider the bull call spread from an at-
expiration perspective. Unlike the long call, which has two possible

eBooks\Trading Options Greeks_ How Time, Volatility, and Other Pricing Factors Drive Profits (Bloomberg Financial).pdf#page:250