At The Money

Calculating At-The-Money (ATM) Volatility

At-The-Money (ATM) volatility is a crucial metric in options pricing, representing the implied volatility when the strike price of the option is equal to the current price of the underlying asset. It reflects the market's expectations of future volatility over the life of the option.

Calculation Method

ATM volatility can be derived using various market models and data, typically from observed market prices of options. The common methods include:

1. Black-Scholes Model: One of the most widely used models for calculating ATM volatility. The implied volatility is extracted by solving the Black-Scholes formula for volatility, given the market price of an ATM option.

2. Interpolation of Market Data: Often, ATM volatility is interpolated from a volatility surface that is constructed using market data of options with different strikes and maturities.

For simplicity, we will illustrate the Black-Scholes Model method.

Black-Scholes Model

The Black-Scholes formula for a call option price ( C ) is given by:$C = S_0 \cdot N(d_1) - K \cdot e^{-rT} \cdot N(d_2)$

Where:

• $S_O$ is the current price of the underlying asset.

• $K$ is the strike price of the option.

• $r$ is the risk-free interest rate.

• $T$ is the time to maturity.

• $N(\cdot)$ is the cumulative distribution function of the standard normal distribution.

• $d_1$ and $d_2$ are calculated as:

$d_1 = \frac{\ln(S_0 / K) + (r + \sigma^2 / 2) T}{\sigma \sqrt{T}}$ and $d_2 = d_1 - \sigma \sqrt{T}$

Here, $\sigma$ represents the volatility, which we solve for given the market price of the option.

Steps to Calculate ATM Volatility

1. Obtain Market Data:

   • Current price of the underlying asset $(S_O)$.

   • Strike price of the ATM option $(K)$, which is equal to $(S_O)$​.

   • Market price of the ATM option $(C)$.

   • Risk-free interest rate $(r)$.

   • Time to maturity of the option $(T)$

2. Calculate the Option Price Using the Black-Scholes Formula:

   • Use the market price of the ATM option and input it into the Black-Scholes formula.

3. Solve for Implied Volatility $(\sigma)$:

   • The implied volatility is found by solving the Black-Scholes formula for σσ such that the calculated option price matches the market price.