The Greeks have given us feta cheese, philosophy, mathematics, and
the Oedipal complex. They also tell us how much risk our option
positions have.
There are ways of estimating the risks associated with options, such
as the risk of the stock price moving up or down, implied volatility
moving up or down, or how much money is made or lost as time passes.
They are numbers generated by mathematical formulas. Collectively, they
are known as the "greeks", because most use Greek letters as names. Each
greek estimates the risk for one variable: delta measures the change in
the option price due to a change in the stock price, gamma measures the
change in the option delta due to a change in the stock price, theta
measures the change in the option price due to time passing, vega
measures the change in the option price due to volatility changing, and
rho measures the change in the option price due to a change in interest
rates.
Delta
The first and most commonly used greek is "delta". For the record, and
contrary to what is frequently written and said about it, delta is NOT
the probability that the option will expire ITM. Simply, delta is a
number that measures how much the theoretical value of an option will
change if the underlying stock moves up or down $1.00. Positive delta
means that the option position will rise in value if the stock price
rises, and drop in value if the stock price falls. Negative delta means
that the option position will theoretically rise in value if the stock
price falls, and theoretically drop in value if the stock price rises.
The delta of a call can range from 0.00 to 1.00; the delta of a put
can range from 0.00 to –1.00. Long calls have positive delta; short
calls have negative delta. Long puts have negative delta; short puts
have positive delta. Long stock has positive delta; short stock has
negative delta. The closer an option's delta is to 1.00 or –1.00, the
more the price of the option responds like actual long or short stock
when the stock price moves.
So, if the XYZ Aug 50 call has a value of $2.00 and a delta of +.45
with the price of XYZ at $48, if XYZ rises to $49, the value of the XYZ
Aug 50 call will theoretically rise to $2.45. If XYZ falls to $47, the
value of the XYZ Aug 50 call will theoretically drop to $1.55.
If the XYZ Aug 50 put has a value of $3.75 and a delta of -.55 with
the price of XYZ at $48, if XYZ rises to $49, the value of the XYZ Aug
50 put will drop to $3.20. If XYZ falls to $47, the value of the XYZ Aug
50 put will rise to $4.30.
Now, these numbers assume that nothing else changes, such as a rise or
fall in volatility or interest rates, or time passing. Changes in any
one of these can change delta, even if the price of the stock doesn't
change.
Note that the delta of the XYZ Aug 50 call is .45 and the delta of the
Aug 50 put is -.55. The sum of their absolute values is 1.00 (|.45| +
|-.55| = 1.00). This is true for every call and put at every strike. The
intuition behind this is that long stock has a delta of +1.00.
Synthetic long stock is long a call and short a put at the same strike
in the same month. Therefore, the delta of a long call plus the delta of
a short put must equal the delta of long stock. In the case of the XYZ
Aug call and put, .45 + .55 = 1.00. Remember, a short put has a positive
delta. (Note: delta can be calculated with different formulas, which
won't be discussed here. Using the Black-Scholes model for
European-style options, the sum of the absolute values of the call and
put is 1.00. But using other models for American-style options and under
certain circumstances, the sum of the absolute values of the call and
put can be slightly less or slightly more than 1.00.)
You can add, subtract, and multiply deltas to calculate the delta of a
position of options and stock. The position delta is a way to see the
risk/reward characteristics of your position in terms of shares of
stock, and it's how thinkorswim presents it to you on the Position
Statement on the Monitor page. The calculation is very straightforward.
Position delta = option theoretical delta * quantity of option contracts
* number of shares of stock per option contract. (The number of shares
of stock per option contract in the U.S. is usually 100 shares. But it
can be more or less, due to stock splits or mergers.) thinkorswim
performs this calculation for each option in your position, then adds
them together for each stock.
So, if you are long 5 of the XYZ Aug 50 calls, each with a delta of
+.45, and short 100 shares of XYZ stock, you will have a position delta
of +125. (Short 100 shares of stock = -100 deltas, long 5 calls with
delta +.45, with 100 shares of stock per contract = +225. –100 + 225 =
+125)
A way to interpret this delta is that if the price of XYZ rises $1,
you will theoretically make $125. If XYZ falls $1, you will
theoretically lose $125. IMPORTANT: These numbers are theoretical. In
reality, delta is accurate for only very small changes in the stock
price. Nevertheless, it is still a very useful tool for a $1.00 change,
and is a good way to evaluate your risk.
An ATM option has a delta close to .50. The more ITM an option is, the
closer its delta is to 1.00 (for calls) or –1.00 (for puts). The more
OTM and option is, the closer its delta is to 0.00.
Delta is sensitive to changes in volatility and time to expiration.
The delta of ATM options is relatively immune to changes in time and
volatility. This means an option with 120 days to expiration and an
option with 20 days to expiration both have deltas close to .50. But the
more ITM or OTM an option is, the more sensitive its delta is to
changes in volatility or time to expiration. Fewer days to expiration or
a decrease in volatility push the deltas of ITM calls closer to 1.00
(-1.00 for puts) and the deltas of OTM options closer to 0.00. So an ITM
option with 120 days to expiration and a delta of .80 could see its
delta grow to .99 with only a couple days to expiration without the
stock moving at all.
The delta of an option depends largely on the price of the stock
relative to the strike price. Therefore, when the stock price changes,
the delta of the option changes. That's why gamma is important.
Gamma
Gamma is an estimate of how much the delta of an option changes when
the price of the stock moves $1.00. As a tool, gamma can tell you how
"stable" your delta is. A big gamma means that your delta can start
changing dramatically for even a small move in the stock price.
Long calls and long puts both always have positive gamma. Short calls
and short puts both always have negative gamma. Stock has zero gamma
because its delta is always 1.00 – it never changes. Positive gamma
means that the delta of long calls will become more positive and move
toward +1.00 when the stock prices rises, and less positive and move
toward 0.00 when the stock price falls. It means that the delta of long
puts will become more negative and move toward –1.00 when the stock
price falls, and less negative and move toward 0.00 when the stock price
rises. The reverse is true for short gamma.
For example, the XYZ Aug 50 call has a delta of +.45, and the XYZ Aug
50 put has a delta of -.55, with the price of XYZ at $48.00. The gamma
for both the XYZ Aug 50 call and put is .07. If XYZ moves up $1.00 to
$49.00, the delta of the XYZ Aug 50 call becomes +.52 (+.45 + ($1 *
.07), and the delta of the XYZ Aug 50 put becomes -.48 (-.55 + ($1 *
.07). If XYZ drops $1.00 to $47.00, the delta of the XYZ Aug 50 call
becomes +.38 (+.45 + (-$1 * .07), and the delta of the XYZ Aug 50 put
becomes -.62 (-.55 + (-$1 * .07).
Position gamma measures how much the delta of a position changes when
the stock price moves $1.00. Position gamma is calculated much in the
same way as position delta. In the Position Statement on the Monitor
page, thinkorswim takes the gamma of each option in your position,
multiplies it by the number of contracts and the number of shares of
stock per option contract, then adds them together.
Just as delta changes, so does gamma. If you were to look at a graph
of gamma versus the strike prices of the options, it would look like a
hill, the top of which is very near the ATM strike. Gamma is highest for
ATM options, and is progressively lower as options are ITM and OTM.
This means that the delta of ATM options changes the most when the stock
price moves up or down. Let's look at a deep ITM call option (delta
near 1.00), an ATM call option (delta near .50), and an OTM call option
(delta near .10). If the stock rises, the value of the ITM call will
increase the most because it acts most like stock. Even though the ITM
call has positive gamma, its delta really doesn't get much closer to
1.00 than before the stock rose. The value of the OTM call will also
increase, and its delta will probably increase as well, but it will
still be a long way from 1.00. The value of the ATM option increases,
and its delta changes the most. That is, its delta is moving closer to
1.00 much quicker than the delta of the OTM call. Practically speaking,
the ATM call can provide a good balance of potential profit if the stock
rises versus loss if the stock falls. The OTM call will not make as
much money if the stock rises, and the ITM will lose more money if the
stock falls.
Judging how gamma changes as time passes and volatility changes
depends on whether the option is ITM, ATM or OTM. Time passing or a
decrease in volatility acts as if it's "pulling up" the top of the hill
on the graph of gamma, and making the slope away from the top steeper.
What happens is that the ATM gamma increases, but the ITM and OTM gamma
decreases. The gamma of ATM options is higher when either volatility is
lower or there are fewer days to expiration. But if an option is
sufficiently OTM or ITM, the gamma is also lower when volatility is
lower or there are fewer days to expiration.
What this all means to the option trader is that a position with
positive gamma is relatively safe, that is, it will generate the deltas
that benefit from an up or down move in the stock. But a position with
negative gamma can be dangerous. It will generate deltas that will hurt
you in an up or down move in the stock. But all positions that have
negative gamma are not all dangerous. For example, a short straddle and a
long ATM butterfly both have negative gamma. But the short straddle
presents unlimited risk if the stock price moves up or down. The long
ATM butterfly will lose money if the stock price moves up or down, but
the losses are limited to the total cost of the butterfly.
Gamma is a good reason to look at a profit/loss graph of your position
over a wide range of possible stock prices. The thinkorswim Analysis
page will help you see how risky a negative gamma position might be.
Theta
Theta, a.k.a. time decay, is an estimate of how much the theoretical
value of an option decreases when 1 day passes and there is no move in
either the stock price or volatility. Theta is used to estimate how much
an option's extrinsic value is whittled away by the always-constant
passage of time. The theta for a call and put at the same strike price
and the same expiration month are not equal. Without going into detail,
the difference in theta between calls and puts depends on the cost of
carry for the underlying stock. When the cost of carry for the stock is
positive (i.e. dividend yield is less than the interest rate) theta for
the call is higher than the put. When the cost of carry for the stock is
negative (i.e. dividend yield is greater than the interest rate) theta
for the call is lower than the put.
Long calls and long puts always have negative theta. Short calls and
short puts always have positive theta. Stock has zero theta – its value
is not eroded by time. All other things being equal, an option with more
days to expiration will have more extrinsic value than an option with
fewer days to expiration. The difference between the extrinsic value of
the option with more days to expiration and the option with fewer days
to expiration is due to theta. Therefore, it makes sense that long
options have negative theta and short options have positive theta. If
options are continuously losing their extrinsic value, a long option
position will lose money because of theta, while a short option position
will make money because of theta.
But theta doesn't reduce an option's value in an even rate. Theta has
much more impact on an option with fewer days to expiration than an
option with more days to expiration. For example, the XYZ Oct 75 put is
worth $3.00, has 20 days until expiration and has a theta of -.15. The
XYZ Dec 75 put is worth $4.75, has 80 days until expiration and has a
theta of -.03. If one day passes, and the price of XYZ stock doesn't
change, and there is no change in the implied volatility of either
option, the value of the XYZ Oct 75 put will drop by $0.15 to $2.85, and
the value of the XYZ Dec 75 put will drop by $0.03 to $4.72.
Theta is highest for ATM options, and is progressively lower as
options are ITM and OTM. This makes sense because ATM options have the
highest extrinsic value, so they have more extrinsic value to lose over
time than an ITM or OTM option. The theta of options is higher when
either volatility is lower or there are fewer days to expiration. If you
think about gamma in relation to theta, a position of long options that
has the highest positive gamma also has the highest negative theta.
There is a trade-off between gamma and theta. Think of long gamma as the
stuff that provides the power to a position to make money if the stock
price starts to move big (think of a long straddle). But theta is the
price you pay for all that power. The longer the stock price does not
move big, the more theta will hurt your position.
Position theta measures how much the value of a position changes when
one day passes. Position theta is calculated much in the same way as
position delta, but instead of using the number of shares of stock per
option contract, theta uses the dollar value of 1 point for the option
contract. (The dollar value of 1 point in an option contract for U.S.
equities is usually $100, but can be different due to stock splits.)
thinkorswim takes the theta of each option in your position, multiplies
it by the number of contracts and the value of 1 point for the option
contract, then adds them together.
Vega
Vega (the only greek that isn't represented by a real Greek letter) is
an estimate of how much the theoretical value of an option changes when
volatility changes 1.00%. Higher volatility means higher option prices.
The reason for this is that higher volatility means a greater price
swings in the stock price, which translates into a greater likelihood
for an option to make money by expiration.
Long calls and long puts both always have positive vega. Short calls
and short puts both always have negative vega. Stock has zero vega –
it's value is not affected by volatility. Positive vega means that the
value of an option position increases when volatility increases, and
decreases when volatility decreases. Negative vega means that the value
of an option position decreases when volatility increases, and increases
when volatility decreases.
Let's look at the XYZ Aug 50 call again. It has a value of $2.00 and a
vega of +.20 with the volatility of XYZ stock at 30.00%. If the
volatility of XYZ rises to 31.00%, the value of the XYZ Aug 50 call will
rise to $2.20. If the volatility of XYZ falls to 29.00%, the value of
the XYZ Aug 50 call will drop to $1.80.
Vega is highest for ATM options, and is progressively lower as options
are ITM and OTM. This means that the value of ATM options changes the
most when the volatility changes. The vega of ATM options is higher when
either volatility is higher or there are more days to expiration.
Position vega measures how much the value of a position changes when
volatility changes 1.00%. Position vega is calculated much in the same
way as position theta. thinkorswim takes the vega of each option in your
position, multiplies it by the number of contracts and the dollar value
of 1 point for the option contract, then adds them together.
Rho
Rho is an estimate of how much the theoretical value of an option
changes when interest rates move 1.00%. The rho for a call and put at
the same strike price and the same expiration month are not equal. Rho
is one of the least used greeks. When interest rates in an economy are
relatively stable, the chance that the value of an option position will
change dramatically because of a drop or rise in interest rates is
pretty low. Nevertheless, we'll describe it here for your edification.
Long calls and short puts have positive rho. Short calls and long puts
have negative rho. How does this happen? The cost to hold a stock
position is built into the value of an option. It all has to do with the
idea of an option being a substitute of sorts for a stock position. For
example, if you think the stock of XYZ is going to rise, you could buy
100 shares of XYZ for $4800, or you could buy 2 of the XYZ Aug 50 calls
for $400. (2 XYZ Aug 50 calls would give me a position delta of +90 —
pretty close to the XYZ stock position delta of +100.) As you can see,
you would have to spend about 12X the amount spent on the options that
you would spend on the stock. That means that you would have to borrow
money or take cash out of an interest-bearing account to buy the stock.
That interest cost is built into the call option's value.
The more expensive it is to hold a stock position, the more expensive
the call option. An increase in interest rates increases the value of
calls and decreases the value of puts. A decrease in interest rates
decreases the value of calls and increases the value of puts.
Back to the XYZ Aug 50 calls. They have a value of $2.00 and a rho of
+.02 with XYZ at $48.00 and interest rates at 5.00%. If interest rates
increase to 6.00%, the value of the XYZ Aug 50 calls would increase to
$2.02. If interest rates decrease to 4.00%, the value of the XYZ Aug 50
calls would decrease to $1.98.
Source : https://www.thinkorswim.com
