Understanding Option Theta in Excel

Option’s theta is a measure of how much an option loses its value over time as it approaches its expiration date. It is also known as the time decay of an option. Theta is usually expressed as a negative number for long options and a positive number for short options.

Theta reflects the fact that options are only good for a limited period of time, and their value decreases as the clock ticks away. For example, if you buy an option to buy a stock for $50 that expires in one month, and the stock is currently trading at $55, your option has some intrinsic value (the difference between the strike price and the market price) and some extrinsic value (the amount that reflects other factors such as volatility, interest rates, and dividends). However, as the expiration date gets closer, your option will lose both its intrinsic and extrinsic value due to theta.

The rate of theta decay depends on several factors, such as the strike price, the expiration date, the volatility of the underlying asset, and the interest rate. Generally speaking, longer-term options have higher theta than shorter-term options, because they have less time to benefit from favorable price movements. Also, higher-volatility assets have higher theta than lower-volatility assets, because they have more uncertainty and risk.

Theta can be calculated using various formulas or models that take into account these factors. One simple way to estimate theta is to use historical data on the option’s price and delta (another option Greeks that measures how much an option’s price changes with respect to a change in the underlying asset’s price). By plotting delta against time until expiration for different strike prices and expiration dates of an option contract, one can obtain a curve that shows how much delta changes per day due to theta decay. This curve can then be used to estimate theta by multiplying delta by 0.0175 (the number of days in a month) or by using other methods such as Black-Scholes or binomial trees.

Theta is an important concept for options traders because it affects their profitability and risk management strategies. For long options buyers, theta means that they are losing money every day as their options lose value due to time decay. Therefore, they need to monitor their positions closely and adjust them accordingly to avoid losing too much money or missing out on potential gains if favorable price movements occur before expiration. For short options sellers or writers (also known as hedgers), theta means that they are earning money every day as their options gain value due to time decay. Therefore, they need to monitor their positions closely and adjust them accordingly to avoid losing too much money or missing out on potential gains if unfavorable price movements occur before expiration.

Basic Theory:

Theta, often referred to as time decay, quantifies how much an option’s price decreases with the passage of time. It is a crucial factor in options pricing and trading strategies. Theta is negative for most options, indicating that the value of an option decreases as time passes.

The formula for calculating Theta is as follows:

$\text{Theta} = -\frac{\partial V}{\partial t}$

Where:

$V$ is the option price.
$t$ is time.

Procedures:

Input Data:
- Collect essential information such as the current stock price, option strike price, time to expiration, implied volatility, and risk-free interest rate.
Use Black-Scholes Model:
- The Black-Scholes option pricing model is commonly used to calculate Theta. The formula for a European call option is given by:
  $C = S_0 e^{-qt} N(d_1) - Xe^{-rt} N(d_2)$
- Where:
  - $S_0$ is the current stock price.
  - $X$ is the strike price.
  - $t$ is time to expiration.
  - $N$ is the cumulative distribution function of the standard normal distribution.
  - $d_1$ and $d_2$ are calculated using the Black-Scholes formula.
Calculate Theta:
- The derivative of the Black-Scholes call option formula with respect to time ( $t$ ) gives the Theta for a call option. A similar process applies to put options.
Implement the Formula in Excel:
- Use Excel functions such as NORM.DIST, EXP, and others to implement the Black-Scholes formula and calculate Theta.

Real-World Scenario:

Let’s consider a call option with the following parameters:

Current stock price ( $S_0$ ): $100
Strike price ( $X$ ): $105
Time to expiration ( $t$ ): 30 days
Implied volatility: 0.2
Risk-free interest rate: 0.05

Stock Price	Strike Price	Time to Expiration	Implied Volatility	Risk-Free Rate	Option Price	Delta	Gamma	Vega	Theta
$100	$105	30 days	0.2	0.05

Excel Formula for Option Price:

$C = S_0 e^{-qt} N(d_1) - Xe^{-rt} N(d_2)$

Excel Formula for Theta:

$\text{Theta} = -\frac{\partial C}{\partial t}$

After inputting the data and applying the formulas, we get the option price and Theta for the given scenario.

Results:

For the scenario described, the calculated Theta for the call option is -0.035, indicating that the option’s value is expected to decrease by $0.035 per day due to time decay.

Other Approaches:

Numerical Methods:
- Use numerical methods like finite differences to approximate Theta.
Built-in Functions:
- Some financial modeling software or programming languages (e.g., Python with libraries like NumPy) have built-in functions to calculate Theta.
Theta Profiles:
- Analyze how Theta changes at different time intervals to understand its behavior over time.