Theta (Θ) is a measure of an option's sensitivity to time decay. It represents the rate at which the option's price decreases over time, assuming all other factors remain constant. It is usually expressed as a negative value because the time decay reduces the option's value as expiration approaches. Typically expressed in terms of price change per day (e.g., $ per day).

Factors affecting Theta

Several factors influence the magnitude and behaviour of theta for an option:

1. Moneyness

   1. Call Option
      • At-the-money (ATM): Theta is most negative at-the-money because time decay is most significant when the option is near its intrinsic value threshold. Small movements in price near S=K (strike price) significantly impact the premium.
      • In-the-money (ITM): Theta is less negative than ATM because the intrinsic value dominates, and time decay has less relative impact.
      • Out-of-the-money (OTM): Theta is also less negative than ATM because the premium is small, being largely extrinsic value, which decays slower or the probability of expiring profitable is low (OTM)
   2. Put Option: The behaviour is similar to call options:
      • ATM: Theta is most negative, as time decay significantly impacts premium here.
      • ITM: Theta is less negative because intrinsic value dominates.
      • OTM: Theta is less negative than ATM but still affected by time decay.

   General Relationship

   The theta (Θ) as a function of stock price (S) for both call and put options often resembles a curve that peaks in negativity at-the-money and diminishes as you move further in-the-money or out-of-the-money.

   For a European option, the Black-Scholes formula provides the theta:

   

   Let us understand it with an example:

   Lets plot a graph showing how theta varies with stock price for both calls and puts under specific parameters

   K=100, T=30 days, r=0.05, σ=0.2

   

   Key observations:

   1. Call Option Theta (dashed line):
      • Most negative near the strike price (K) where the option is at-the-money (ATM).
      • Becomes less negative as the option moves deep in-the-money (ITM) or out-of-the-money (OTM).
   2. Put Option Theta (solid line):
      • Similarly, most negative near the ATM region.
      • Less negative (closer to zero) in ITM or OTM scenarios.
   3. The gray dashed line marks the strike price (K=100). This behaviour reflects the time decay dynamics of options pricing.
   • Expand to see python code to generate the graph

2. Time to Expiration

   • Shorter Time to Expiration: Theta increases as expiration approaches, with the time decay accelerating significantly in the last 30 days (known as "gamma decay").
   • Longer Time to Expiration: Theta is lower for long-dated options because most of the premium comes from time value that decays slowly over time.

   

   Key observations:

   • Call Options (dashed line): Theta becomes more negative as time to expiration decreases, with the steepest decline occurring close to expiration.
   • Put Options (solid line): Similar behavior, with theta becoming more negative as expiration approaches.
   • Expand to see python code to generate the graph

3. Implied Volatility

   • Higher Volatility: Options with higher implied volatility have higher premiums, leading to higher theta (time decay is more significant when the option is expensive).
   • Lower Volatility: Options with lower implied volatility have less extrinsic value and lower theta.

   

   Key observations:

   • Call Options (dashed line): Theta becomes more negative as implied volatility increases because higher volatility increases the extrinsic value, accelerating time decay.
   • Put Options (solid line): Similar behaviour, with theta becoming more negative as implied volatility rises.
   • Expand to see python code to generate the graph

4. Risk-Free Interest Rate: Higher interest rates slightly increase call theta and decrease put theta because the cost of carrying the underlying asset changes.

   

   Key observations:

   • Call Options (dashed line): Theta slightly increases with higher risk-free rates. Higher rates increase the cost of carrying the stock, impacting the time decay of calls.
   • Put Options (solid line): Theta slightly decreases with increasing risk-free rates, as the higher rates reduce the relative attractiveness of holding puts.
   • Expand to see python code to generate the graph

5. Dividend Payments : For options on dividend-paying stocks:

   • Call options may have lower theta due to the anticipated drop in stock price when dividends are paid.
   • Put options may have higher theta for the same reason.

Why Theta Accelerates Near Expiration?

Theta decays rapidly as an option approaches expiration primarily due to rapid decay of extrinsic value. Here are some more details:

1. Time value dominance:

   • The total price (premium) of an option consists of two components:
     • Intrinsic Value: The in-the-money (ITM) portion of the option price (if any).
     • Extrinsic Value: The portion attributable to time, volatility, and market expectations.
   • Near expiration, extrinsic value decreases rapidly because the time remaining for the option to move in favor of the holder diminishes, leading to faster time decay.

2. Probability Convergence:

   As expiration nears, the probability of significant price movements decreases because there’s less time for the underlying asset to make meaningful moves.

   • For at-the-money (ATM) options, the premium erodes fastest because they are most sensitive to small changes in price and time.
   • For in-the-money (ITM) or out-of-the-money (OTM) options, the premium decreases less rapidly because these options have less extrinsic value to begin with.

3. Gamma and Delta Senstivities:

   • Gamma, which measures the rate of change of delta, is highest near expiration, especially for ATM options. This heightened sensitivity to price changes leads to a larger impact of time decay on premium.
   • As delta stabilizes near ITM or OTM regions, theta slows down for those options.

4. Exponential Decay:

   The time decay of extrinsic value follows a non-linear (exponential) pattern:

   • Far from expiration: Time decay is slow because there’s ample time for the option to retain its value based on market uncertainties.
   • Near expiration: Time decay accelerates significantly because each passing moment reduces the likelihood of favorable movements in the underlying asset.