A contract with a delta of 1.0 (100Δ ) has maximal directional exposure and is maximally ITM. 100Δ options behave like the stock price, as a $1 increase in the underlying creates a $1 increase in the option's price per share. A contract with a delta of 0.0 has no directional exposure and is maximally OTM. A 50Δ contract is defined as having the ATM strike.14
Because delta is a measure of directional exposure, it plays a large role when hedging directional risks. For instance, if a trader currently has a 50Δ position on and wants the position to be relatively insensitive to directional moves in the underlying, the trader could offset that exposure with the addition of 50 negative deltas (e.g., two 25Δ long puts). The composite position is called delta neutral.
Gamma
(1.16)
As with delta, the sign of gamma depends on the type of position:
● Long call and long put:
● Short call and short put:
In other words, if there is a $1 increase in the underlying price, then the delta for all long positions will become more positive, and the delta for all short positions will become more negative. This makes intuitive sense because a $1 increase in the underlying pushes long calls further ITM, increasing the directional exposure of the contract, and it pushes long puts further OTM, decreasing the inverse directional exposure of the contract and bringing the negative delta closer to zero. The magnitude of gamma is highest for ATM positions and lower for ITM and OTM positions, meaning that delta is most sensitive to underlying price movements at –50Δ and 50Δ .
Awareness of gamma is critical when trading options, particularly when aiming for specific directional exposure. The delta of a contract is typically transient, so the gamma of a position gives a better indication of the long‐term directional exposure. Suppose traders wanted to construct a delta neutral position by pairing a short call (negative delta) with a short put (positive delta), and they are considering using 20Δ or 40Δ contracts (all other parameters identical). The 40Δ contracts are much closer to ATM (50Δ ) and have more profit potential than the 20Δ positions, but they also have significantly more gamma risk and are less likely to remain delta neutral in the long term. The optimal choice would then depend on how much risk traders are willing to accept and their profit goals. For traders with high profit goals and a large enough account to handle the large P/L swings and loss potential of the trade, the 40Δ contracts are more suitable.
Theta
(1.17)
where V is the price of the option (a call or a put) and t is time. The sign of theta depends on the type of position and is opposite gamma:
● Long call and long put:
● Short call and short put:
In other words, the time decay of the extrinsic value decreases the value of the long position and increases the value of the short position. For instance, a long call with a theta of –5 per one lot is expected to decline in value by $5 per day. This makes intuitive sense because the holders of the contract take gradual losses as their asset depreciates with time, a result of the value of the option converging to its intrinsic value as uncertainty dissipates. Because the extrinsic value of a contract decreases with time, the short side of the position profits with time and experiences positive time decay. The magnitude of theta is highest for ATM options and lower for ITM and OTM positions, all else constant.
There is a trade‐off between the gamma and theta of a position. For instance, a long call with the benefit of a large, positive gamma will also be subjected to a large amount of negative time decay. Consider these examples:
● Position 1: A 45 DTE, 16Δ call with a strike price of $50 is trading on a $45 underlying. The long position has a gamma of 5.4 and a theta of –1.3.
● Position 2: A 45 DTE, 44Δ call with a strike price of $50 is trading on a $49 underlying. The long position has a gamma of 7.9 and a theta of –2.2.
Compared to the first position, the second position has more gamma exposure, meaning that the contract delta (and the contract price) is more sensitive to changes in the underlying price and is more likely to move ITM. However, this position also comes with more theta decay, meaning that the extrinsic value also decreases more rapidly with time.
To conclude this discussion of the Black‐Scholes model and its risk measures, note that the outputs of all options pricing models should be taken with a grain of salt. Pricing models are founded on simplified assumptions of real financial markets. Those assumptions tend to become less representative in highly volatile market conditions when potential profits and losses become much larger. The assumptions and Greeks of the Black‐Scholes model can be used to form reasonable expectations around risk and return in most market conditions, but it's also important to supplement that framework with model‐free statistics.
Covariance and Correlation
Up until now we have discussed trading with respect to a single position, but quantifying the relationships between multiple positions is equally important. Covariance quantifies how two signals move relative to their means with respect to one another. It is an effective way to measure the variability between two variables. For one signal, X, with observations
(1.18)
Represented in terms of random variables X and Y,
In practice, the strike and underlying prices for 50Δ contracts tend to differ
