Introduction

Implied volatility plays a pivotal role in option pricing and risk assessment. It represents the market’s expectation of future volatility and is a key input in the Black-Scholes-Merton model. In this chapter, we will define implied volatilities and provide a comprehensive understanding of how to compute implied volatilities from market prices of options using the Black-Scholes-Merton model.

Implied Volatility

Implied volatility is a measure of the expected future volatility of an underlying asset’s price, as implied by the current market prices of options. It is essentially the market’s collective forecast of how much an asset’s price is likely to fluctuate over a specific period. Implied volatility is expressed as a percentage and is a crucial parameter in option pricing models like Black-Scholes-Merton.

Importance of Implied Volatility

Implied volatility is a critical element in option pricing for several reasons:

• It reflects market sentiment and expectations regarding future price movements.
• Higher implied volatility typically leads to higher option premiums, as options become more valuable in uncertain markets.
• Traders and investors use implied volatility to assess the risk associated with options and make informed trading decisions.

Computing Implied Volatility

Implied volatility is not directly observable in the market; instead, it is derived from option prices using the Black-Scholes-Merton model. To compute implied volatility, we reverse-engineer the Black-Scholes-Merton formula to solve for the implied volatility value.

Black-Scholes-Merton Formula for European Call Options

The Black-Scholes-Merton formula for European call options is as follows:

$$C=S_0 e^{-q t} N\left(d_1\right)-K e^{-r t} N\left(d_2\right)$$

Where:

• $C=$ Call option price
• $S_0=$ Current stock price
• $K$ = Strike price of the option
• $t$ = Time to expiration
• $N\left(d_1\right)$ and $N\left(d_2\right)=$ Cumulative standard normal distribution functions
• $r$ = Risk-free interest rate
• $q$ = Dividend yield (if applicable)
• $d_1$ and $d_2$ are calculated as per the Black-Scholes-Merton formula.

Solving for Implied Volatility

To calculate implied volatility, we rearrange the Black-Scholes-Merton formula and solve for $\sigma$, the implied volatility:

$$C=S_0 e^{-q t} N\left(d_1\right)-K e^{-r t} N\left(d_2\right)$$

This equation is nonlinear, and finding the implied volatility requires numerical methods such as the Newton-Raphson method or software tools designed for this purpose.