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Notes on options trading
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title: "Options"

# Notes on Options Trading

Options are very versatile derivatives which can be used to construct
positions with a variety of risk/reward profiles. An option contract
permits the long side to purchase or sell an underlying security at a
given strike price, and obliges the short side to take the other side of
that trade if the long side exercises that right.

Options come in two flavors, call and put (the right to buy and the
right to sell, respectively), and always have an expiration date (an
"American" option may be exercised at any time before expiration, while
a "European" option may be exercised only at expiration). A single
contract typically covers 100 shares, but prices are typically quoted
per-share - i.e., a call option priced at $4.10 costs $410 to buy.

## The Money

The parameters of an option contract are type (put or call), expiration
date, and strike price. The ultimate intrinsic value of an option comes
at expiry (we're ignoring the option to exercise early in American-style
options, which is more of an edge case) - if the market price of the
underlying security is more (for a call) or less (for a put) than the
option's strike price, exercising the option is profitable - you're
buying below market or selling above market if you exercise.

When the underlying security is trading at a price such that exercise
would be intrinsically profitable, the option is said to be "in the
money"; similarly, the option is "at the money" if the market price of
the underlying is at the strike price, and "out of the money" otherwise.
If an option is in the money at expiry, it has value; if it is out of
the money at expiry, it is worthless. At the moment of expiry, this is
the only value an option has. But what about prior to expiry?

## Premium

An option with more time until expiry has a higher price than one closer
to expiry. This is because the price of an option incorporates a time
premium in addition to intrinsic value, reflecting the fact that the
underlying still has time to move and possibly bring the option strike
into the money. (Generally, "premium" refers to the entire price of the
option contract and "time premium" refers to the portion which is not
intrinsic value, but authors vary in their usage.)

Broadly, time premium is determined by two factors: the length of the
remaining time to expiry, and the market's expectation of volatility in
the underlying. An "implied volatility" can be calculated from the
market price of options, and, at least in theory, reflects the market's
uncertainty about price movements in the underlying driving the price of
options up.

A note about early exercise: we can see why it is an edge case, because
it entails throwing away time premium to capture only intrinsic value,
and in most cases just selling the option and getting both would be
preferable. However, it happens sometimes, most frequently because of
dividends, and the possibility has to be kept in mind.

## Greeks

Options traders calculate (or rather, trading platforms calculate for
them) various factors affecting option price, all of which are partial
derivatives of the option price with respect to something. The overall
PDE model used is the [Black-Scholes
model](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model), which
assumes that prices are a random walk (geometric Brownian motion, i.e.,
log-normally distributed) and ignores the case of early exercise (i.e.,
it assumes European options).  The Black-Scholes PDE is:

{{<katex display>}}\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2
S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S}
- rV = 0{{</katex>}}

where {{<katex>}}V{{</katex>}} is the price of the option,
{{<katex>}}\sigma{{</katex>}} is the volatility of the underlying (a
standard deviation), {{<katex>}}S{{</katex>}} is the price of the
underlying, and {{<katex>}}r{{</katex>}} is the risk-free interest rate,
all at time {{<katex>}}t{{</katex>}}.

The various partial derivatives of interest to traders are known as "the
Greeks", because they are referred to by the names of Greek letters or
fake Greek letters (traders do not seem to know the Greek alphabet as
well as mathematicians).

### Delta and Gamma

Delta ({{<katex>}}\Delta = \frac{\partial V}{\partial S}{{</katex>}})
and Gamma ({{<katex>}}\Gamma = \frac{\partial^2 V}{\partial
S^2}{{</katex>}}) relate to the price of the underlying. In a long
option position, the sign of {{<katex>}}\Delta{{</katex>}} depends on
the type of option (positive for calls, negative for puts); the sign of
{{<katex>}}\Gamma{{</katex>}} is always positive. All these signs are
reversed in a short position.

Having nonzero {{<katex>}}\Delta{{</katex>}} means you are exposed to
directional moves in the price of the underlying. It makes the most
sense to quote {{<katex>}}\Delta{{</katex>}} for the whole contract -
i.e., multiplying per-share {{<katex>}}\Delta{{</katex>}} by 100 - to
keep it comparable across the portfolio. A single share of stock has 1
{{<katex>}}\Delta{{</katex>}} and zero {{<katex>}}\Gamma{{</katex>}} by
definition - it always moves in price exactly the same as 1 share of
itself. Option {{<katex>}}\Delta{{</katex>}} is around 50 when the
option is "at the money", and increases or decreases as the option goes
deeper in or out of the money, as a consequence of
{{<katex>}}\Gamma{{</katex>}} being positive.

It is possible to construct a "synthetic long stock" position by buying
a call at the money and selling a put at the money, each of which has
positive 50 {{<katex>}}\Delta{{</katex>}}, resulting in a position which
moves with the price of the underlying in the same way as 100 shares -
but is much cheaper to enter. A synthetic short position may be created
by switching puts and calls in the above to get -100

{{<katex>}}\Delta{{</katex>}} is sometimes used as a "cowboy
probability" by traders - i.e., a 50 {{<katex>}}\Delta{{</katex>}}
contract is considered to have a 50% probability of expiring in the
money. This approximates working because the
{{<katex>}}\Delta{{</katex>}} curve is generally fairly close to the
random walk CDF, but it is not a real probability.

### Vega

"Vega" ({{<katex>}}\mathcal{V} = \frac{\partial V}{\partial
\sigma}{{</katex>}}) (a fake Greek letter invented by traders, available
in LaTeX typesetting as `\mathcal{V}`) relates to the volatility of the
underlying. {{<katex>}}\mathcal{V}{{</katex>}} is positive for long
positions and negative for short positions. Because we can only know the
actual volatility of a symbol in a historical sense, an "implied
volatility" is back-calculated from option prices to get a sense of what
portion of the premium is due to volatility expectations.

The rule to follow with volatility is to either buy it low and sell it
high, or sell it high and buy it low. A trader who wants to enter a long
options position should try to do so when implied volatility is low, and
a trader who wants to enter a short options position should try to do so
when implied volatility is high. In general, realized volatility ends up
being less than implied volatility, and this difference is the option
seller's profit.

Look for "IV Rank" in your trading platform to determine whether
volatility is high or low for a given underlying - this compares current
implied volatilities of options to their historical range. The direct
implied volatility number itself isn't that useful without something to
compare it to.

### Theta

Theta ({{<katex>}}\Theta = \frac{\partial V}{\partial t}{{</katex>}})
relates to the passage of time. {{<katex>}}\Theta{{</katex>}} is always
negative for a long position - when there is less time until expiry, the
time premium necessarily decreases.

Time decay mostly accelerates closer to expiry. This is why purchasing
very short-dated options is generally a bad idea. The existence of time
decay is a nice perk of short positions, although in my experience
volatility regressing is the chief moneymaker as an option seller. Time
going forward is the only trend that can't unexpectedly reverse on you
(and if it does, you have (well, had) bigger problems).

### Rho

The widely-ignored Rho ({{<katex>}}\rho = \frac{\partial V}{\partial
R}{{</katex>}}) relates to changes in the risk-free interest rate.
{{<katex>}}\rho{{</katex>}} is positive for long calls and negative for
long puts, and isn't that important except for very long-term contracts.

### Others

Other partial derivatives can of course be computed, but their names are
even more offensively fake than "Vega" - rather than discuss them, I
will just drop a [Wikipedia

## Positions

Options can be combined in various ways to construct positions exposed
to certain risks and rewards. A fairly simple example is the iron condor
\- a short position (and therefore best entered when IV rank is high)
with defined maximum profit and loss that profits when the price of the
underlying remains within a certain range.

The position is simple: sell an out of the money call and an out of the
money put, then buy a call and a put each further out of the money. The
individual options of a strategy like this are called "legs". The long
legs will have cost less to buy than you were paid to sell the short
legs, so this is a net credit position (note that your broker will hold
this credit as collateral, along with enough cash to cover the maximum
loss (or whatever your margin requirement is) - you did receive cash
right away when you entered the position, but it's locked up in the
position until it is closed).

The short legs are the moneymakers here - you were paid for them, and
the theoretical goal is that they both expire worthless, resulting in
you keeping the entire premium. The long legs are protective, and are
the reason this strategy has a defined maximum loss - without them, you
would potentially have an unlimited loss from a short leg going in the
money. With the long legs for protection, if assigned on a short leg,
you could exercise the corresponding long leg to cover it, limiting your
losses to the difference in strikes. (In practice, actually going
througn an assignment is not what you want to do - you would close the
position for a loss, or "roll" it into the future in the hope of making
the loss back.)

Since this is a short position, time decay and volatility regression are
on your side, and the risks are that the price moves outside your short
legs or that volatility goes even higher. Holding a position until
expiry to achieve theoretical max profit is not usually a good idea - it
is best to close successful trades at a percentage of max profit,
freeing up money for the next (hopefully successful) trade. However,
closing a losing trade is not the best way to manage it, and leaving
those open until much closer to expiry and possibly managing them into a
win or at least a break even makes sense. This goes against people's
natural instinct to let a winner ride and win more, while getting rid of
a loser. The fact is, winners and losers can both turn around, so you
want to give losers that chance, while not giving winners that chance.