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?


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.


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, 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” (\(\mathcal{V} = \frac{\partial V}{\partial \sigma}\)) (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 (\(\Theta = \frac{\partial V}{\partial t}\)) 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).


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.


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.


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.