Understanding Option Vega in Excel

Option’s Vega is a measure of how much the price of an option changes when the volatility of the underlying asset changes. Volatility is a measure of how much the price of an asset fluctuates over time. The higher the volatility, the more uncertain the market is about the future price of the asset.

Vega is one of the Greeks, which are mathematical tools that help traders analyze and manage risk in options trading. Vega tells us how sensitive an option’s price is to changes in implied volatility, which is a projection of how much volatility the market expects for a given underlying asset and expiration date.

For example, suppose you buy a call option on stock XYZ with a strike price of $50 and an expiration date of one month from now. The current price of XYZ is $48 and the current implied volatility is 25%. The option’s premium is $1.50.

If you want to know how much your option’s price will change if there is a 1% increase in implied volatility, you can use Vega to calculate it. Vega tells us that for every 1% increase in implied volatility, the option’s price will increase by about 0.25%. So, if implied volatility goes up from 25% to 26%, your option’s price will go up by about $0.25 ($1.50 x 0.25).

On the other hand, if there is a 1% decrease in implied volatility, your option’s price will decrease by about $0.25 ($1.50 x -0.25). This means that if you are long (buying) an option, you want implied volatility to go up and if you are short (selling) an option, you want implied volatility to go down.

Vega also changes as time passes and as the underlying asset moves closer or farther from its strike price. Generally speaking, longer-term options have higher Vega than shorter-term options because they have more time value and are more affected by changes in implied volatility as they approach expiration date.

Similarly, at-the-money options have higher Vega than out-of-the-money or in-the-money options because they have more potential for large price movements as they move closer to their strike price.

Vega can help traders understand how their options positions are exposed to changes in market conditions and how they can adjust their strategies accordingly. For example, if you have a long position in a call option on XYZ with a high Vega, you may want to hedge your risk by buying or selling another option or another asset that has low or negative Vega.

Theory of Vega

Vega quantifies the impact of implied volatility on an option’s price. As implied volatility increases, option prices tend to rise, reflecting the higher perceived risk and potential for larger price swings.

The Vega formula for a European call or put option is:

$\text{Vega} = S \cdot N'(d1) \cdot \sqrt{T}$

Where:

$S$ is the current stock price
$N'(d1)$ is the standard normal distribution function of the option’s Black-Scholes $d1$ value
$T$ is the time to expiration in years

Calculating Vega in Excel

Let’s break down the steps to calculate Vega in Excel using the Black-Scholes model:

Define Variables:
- Enter the current stock price ( $S$ )
- Specify the option’s strike price ( $K$ )
- Input the time to expiration ( $T$ )
- Set the risk-free interest rate ( $r$ )
- Enter the option’s implied volatility ( $\sigma$ )
Calculate $d1$ :
- Use the formula:
  $d1 = \frac{{\ln(S/K) + (r + (\sigma^2)/2) \cdot T}}{{\sigma \cdot \sqrt{T}}}$
Calculate $N'(d1)$ :
- Utilize Excel’s NORM.S.DIST function:
  $N'(d1) = \text{NORM.S.DIST}(d1, \text{TRUE})$
Compute Vega:
- Employ the Vega formula mentioned earlier.

Real-Life Scenario

Let’s consider a scenario:

Stock Price ( $S$ ): $100
Strike Price ( $K$ ): $110
Time to Expiration ( $T$ ): 0.5 years
Risk-Free Interest Rate ( $r$ ): 0.05
Implied Volatility ( $\sigma$ ): 0.2

Excel Calculation

Calculate $d1$ :
$d1 = \frac{{\ln(100/110) + (0.05 + (0.2^2)/2) \cdot 0.5}}{{0.2 \cdot \sqrt{0.5}}}$

$d1 \approx -0.59$
Calculate $N'(d1)$ :
- Using Excel:
  $N'(d1) \approx \text{NORM.S.DIST}(-0.59, \text{TRUE}) \approx 0.277$
Calculate Vega:
$\text{Vega} \approx 100 \cdot 0.277 \cdot \sqrt{0.5} \approx 6.21$

Result

In our scenario, the Vega of the option is approximately 6.21. This implies that for a 1% increase in implied volatility, the option’s price is expected to rise by $6.21.

Other Approaches

While the Black-Scholes model is widely used, alternative models like the Binomial model or advanced numerical methods can be employed for more accurate calculations, especially for complex options or during extreme market conditions. However, these models are more intricate and may require specialized software.