From 8d647008bd92eeaa5bb48095ab60629d4b8c5966 Mon Sep 17 00:00:00 2001
From: rep
Date: Sun, 26 Apr 2020 13:11:09 -0400
Subject: [PATCH] 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:
+
+{{}}\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{{}}
+
+where {{}}V{{}} is the price of the option,
+{{}}\sigma{{}} is the volatility of the underlying (a
+standard deviation), {{}}S{{}} is the price of the
+underlying, and {{}}r{{}} is the risk-free interest rate,
+all at time {{}}t{{}}.
+
+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 ({{}}\Delta = \frac{\partial V}{\partial S}{{}})
+and Gamma ({{}}\Gamma = \frac{\partial^2 V}{\partial
+S^2}{{}}) relate to the price of the underlying. In a long
+option position, the sign of {{}}\Delta{{}} depends on
+the type of option (positive for calls, negative for puts); the sign of
+{{}}\Gamma{{}} is always positive. All these signs are
+reversed in a short position.
+
+Having nonzero {{}}\Delta{{}} means you are exposed to
+directional moves in the price of the underlying. It makes the most
+sense to quote {{}}\Delta{{}} for the whole contract -
+i.e., multiplying per-share {{}}\Delta{{}} by 100 - to
+keep it comparable across the portfolio. A single share of stock has 1
+{{}}\Delta{{}} and zero {{}}\Gamma{{}} by
+definition - it always moves in price exactly the same as 1 share of
+itself. Option {{}}\Delta{{}} 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
+{{}}\Gamma{{}} 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 {{}}\Delta{{}}, 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
+{{}}\Delta{{}}.
+
+{{}}\Delta{{}} is sometimes used as a "cowboy
+probability" by traders - i.e., a 50 {{}}\Delta{{}}
+contract is considered to have a 50% probability of expiring in the
+money. This approximates working because the
+{{}}\Delta{{}} curve is generally fairly close to the
+random walk CDF, but it is not a real probability.
+
+### Vega
+
+"Vega" ({{}}\mathcal{V} = \frac{\partial V}{\partial
+\sigma}{{}}) (a fake Greek letter invented by traders, available
+in LaTeX typesetting as `\mathcal{V}`) relates to the volatility of the
+underlying. {{}}\mathcal{V}{{}} 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 ({{}}\Theta = \frac{\partial V}{\partial t}{{}})
+relates to the passage of time. {{}}\Theta{{}} 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 ({{}}\rho = \frac{\partial V}{\partial
+R}{{}}) relates to changes in the risk-free interest rate.
+{{}}\rho{{}} 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
+link](https://en.wikipedia.org/wiki/Greeks_(finance)).
+
+## 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.
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