The Greeks — Delta, Gamma, Theta, Vega, Rho

The greeks are the partial derivatives of an option's price with respect to the underlying variables. There are five first-order greeks: delta, gamma, theta, vega, and rho. Each greek measures the sensitivity of the option's price to a specific change in the underlying, the time, the implied volatility, or the interest rate. The journal monitors the greeks of every position at entry and during the life of the position.

Delta

Delta is the rate of change of the option's price with respect to a $1 change in the underlying. A call option has a positive delta between 0 and 1; a put option has a negative delta between -1 and 0. The delta is also the approximate probability that the option will expire in-the-money, which is why the delta is sometimes called the "probability of profit" for a single option.

For a portfolio of options, the delta is the sum of the individual deltas, weighted by the position size. A portfolio with a positive delta is long the underlying; a portfolio with a negative delta is short the underlying. The journal's portfolio delta is the net exposure to the underlying's price movement.

The journal monitors the delta at entry to ensure that the position's directional exposure is consistent with the directional bias. A bullish call spread should have a positive delta between 0.20 and 0.50; a bearish put spread should have a negative delta between -0.20 and -0.50. The delta of an iron condor should be approximately zero, with the long wings offsetting the short body.

Gamma

Gamma is the rate of change of the delta with respect to a $1 change in the underlying. Gamma is the second derivative of the option's price with respect to the underlying. The gamma is highest for at-the-money options with the longest time to expiration, and the gamma decreases as the option moves in-the-money or out-of-the-money.

The gamma is important for the position's risk profile. A long option position has positive gamma; the delta increases as the underlying moves toward the position and decreases as the underlying moves away. A short option position has negative gamma; the delta becomes more negative as the underlying moves against the position, which is a self-reinforcing loss.

The journal's short-premium structures (credit spreads, iron condors, iron butterflies) have negative gamma. The negative gamma means that the position's directional exposure increases as the underlying moves against the position, which is a feature, not a bug: the structure is defined-risk, so the directional exposure is bounded by the structure's max-loss. The journal monitors the gamma at entry to ensure that the position's max-loss is consistent with the structure's max-loss at expiration.

The journal's long-premium structures (debit spreads, long calls, long puts) have positive gamma. The positive gamma means that the position's directional exposure increases as the underlying moves in the direction of the thesis. The journal monitors the gamma at entry to ensure that the position's directional exposure is consistent with the thesis's expected move.

Theta

Theta is the rate of change of the option's price with respect to a 1-day decrease in the time to expiration. Theta is negative for long options and positive for short options. The theta is the source of the "time decay" that the journal's short-premium structures exploit.

The theta is not constant; it accelerates as the option approaches expiration. The theta of an at-the-money option is largest at about 30-45 DTE; the theta of an out-of-the-money option is largest at about 14-21 DTE. The theta behavior is the basis for the journal's preference for 30-45 DTE options on short-premium structures.

The journal monitors the theta at entry to ensure that the position's daily time decay is consistent with the position's expected holding period. A short-premium position with a 30-45 DTE entry should have a theta of about 5-10% of the max-profit per day, which is the expected daily gain if the position is held to the target.

The journal's long-premium structures have negative theta. The negative theta means that the position loses value every day, which is a cost the journal pays for the directional exposure. The journal's long-premium positions are sized to account for the negative theta: the position is sized to be profitable on the directional move, not on the time decay.

Vega

Vega is the rate of change of the option's price with respect to a 1% change in the implied volatility. Vega is positive for long options and negative for short options. The vega is the source of the "volatility exposure" that the journal's structures carry.

The vega is largest for at-the-money options with the longest time to expiration. The vega of an option decreases as the option approaches expiration; the vega of an option is essentially zero at expiration. The vega behavior is the basis for the journal's preference for 30-45 DTE options on short-premium structures: the position has meaningful vega exposure during the first half of the position's life, and the vega exposure decreases as the position approaches expiration.

The journal's short-premium structures have negative vega. The negative vega means that the position loses value if the implied volatility increases. The journal's IV rank signal is designed to identify the implied volatility regimes where the negative vega is most profitable: when the IV rank is high, the negative vega is a tailwind because the implied volatility is likely to decrease.

The journal's long-premium structures have positive vega. The positive vega means that the position gains value if the implied volatility increases. The journal's long-premium positions are typically opened when the IV rank is low, where the positive vega is most likely to be profitable.

Rho

Rho is the rate of change of the option's price with respect to a 1% change in the risk-free interest rate. Rho is the smallest of the greeks for equity options, and the journal typically does not monitor rho explicitly. The reason is that equity options are short-dated relative to the interest rate cycle, and the rho exposure is small compared to the other greeks.

The journal's positions are typically held for 30-90 days, during which the interest rate is unlikely to change by enough to move the option's price. The journal's preference for SPX/XSP options over equity options is partly driven by the rho exposure: the rho of SPX/XSP options is smaller than the rho of equity options because the cash-settled structure has a different dividend treatment.

The journal's rule for the rho is that the position's rho exposure should be less than 5% of the position's max-loss. The rule is rarely binding, but it is a check on the position's overall risk profile.

The greeks in a position

The journal's positions are multi-leg structures, and the position's greeks are the sum of the individual legs' greeks. The position's delta is the weighted sum of the long and short deltas; the position's theta is the weighted sum of the long and short thetas; and so on.

The journal's check at entry is that the position's greeks are consistent with the directional bias and the IV regime. A bullish call spread should have a positive delta, a positive theta (because the short leg's theta is larger than the long leg's theta), and a negative vega (because the short leg's vega is larger than the long leg's vega). An iron condor should have a delta near zero, a positive theta, and a negative vega.

The journal also monitors the position's greeks during the life of the position. The position's delta changes as the underlying moves; the position's theta changes as the time to expiration decreases; the position's vega changes as the implied volatility changes. The journal's adjustment rules are based on the position's greeks at the time of adjustment, not on the position's greeks at entry.

The greeks in the portfolio

The journal's portfolio is the sum of the positions' greeks. The portfolio's delta is the net exposure to the underlying's price movement; the portfolio's theta is the net daily time decay; the portfolio's vega is the net exposure to the implied volatility.

The journal's portfolio construction rules are designed to keep the portfolio's greeks at a sustainable level. The 6% rule per underlying limits the portfolio's delta exposure to any single underlying; the 8% rule per broad-market factor limits the portfolio's delta exposure to the broad market. The journal's rule for the portfolio's theta is to keep the net theta positive (the portfolio is a net beneficiary of time decay); the portfolio's vega is monitored to ensure that the portfolio's net vega is consistent with the IV regime.