Options and Greeks
The primary Greeks in options trading are Delta, Vega, Gamma, and Theta. Additional Greeks include Rho and secondary sensitivities such as Vomma and Charm.
Time Decay (Theta)
Theta quantifies the daily erosion of an option’s time value.
• As time passes, the option ages and gradually loses value.
• Specifically, it loses its time premium.
• Long vs. short options: who benefits or suffers from the passage of time?
• We need to measure how much value decays day by day.
By definition, Theta represents the daily rate of time decay of an option.
Option Pricing Models
The theoretical value of an option is derived from quantitative models such as:
• Black-Scholes
• Binomial Tree
• Bjerksund-Stensland
• Jump Diffusion
Key inputs include:
• Strike Price
• Spot Price (Delta measures sensitivity to changes in the underlying asset)
• Volatility
• Interest Rate
• Dividends
• Time to Maturity
Directional Sensitivity (Delta)
Delta measures how much an option’s value changes in response to movements in the underlying asset.
• What is the directional exposure (risk)?
• What is the probability the option will expire in-the-money (ITM)?
The sign of Delta indicates the directional bias:
• Positive Delta: the option gains value as the underlying rises.
• Negative Delta: the option gains value as the underlying falls.
Delta ranges between -1 and +1.
Delta also serves as a proxy for the probability of expiring ITM.
Example: Delta = ±0.30
• ~30% chance of expiring ITM
• ~70% chance of expiring out-of-the-money (OTM)
Although Delta is technically a decimal (e.g., 0.30), traders often refer to it in percentage terms (e.g., “Delta 30”).
Share Equivalency
Delta expresses the equivalent number of shares represented by the option position:
• Delta +30 = equivalent to holding 30 long shares
• Delta -15 = equivalent to holding 15 short shares
ITM Probability Revisited
Delta approximates the likelihood of an option expiring ITM:
• ~30% ITM
• ~70% OTM
Why We Use Delta 0.70 in Our Models
In our pricing and payoff models, we often use a Delta of 0.70.
Why? Selling a 0.70 Delta option implies a theoretical probability of 93% for a favorable outcome.
If theoretical and actual probabilities were perfectly aligned, trading would be futile—expected gains and losses would net to zero.
However, the market demands compensation for risk.
Thus, longer-dated options are priced higher to reflect the risk premium—this is the spread between implied volatility and realized volatility.
Delta is derived from implied volatility, whereas historical volatility reflects actual market behavior.
Market makers price options based on historical execution data. For instance, over a 3-year period with 260–270 trades, only one resulted in a loss (99.3% win rate).
This explains why options are often overpriced relative to their real-world risk.