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 --- content/finance/options.md | 218 +++++++++++++++++++++++++++++++++++++ 1 file changed, 218 insertions(+) create mode 100644 content/finance/options.md diff --git a/content/finance/options.md b/content/finance/options.md new file mode 100644 index 0000000..788e1bb --- /dev/null +++ b/content/finance/options.md @@ -0,0 +1,218 @@ +--- +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. -- 2.45.2