Implied Volatility and Volatility Smile › Defining Implied Volatility

Recall from Chapter 6 that the Black-Scholes formula for a European call option is:

This elegant formula expresses the theoretical value of a European call option as the difference between two probability-weighted terms. The first term, $S \cdot N(d_1)$, represents the expected present value of receiving the stock upon exercise, accounting for the probability that the option finishes in the money under the risk-neutral measure and weighted by the stock price's drift. The second term, $K e^{-rT} \cdot N(d_2)$, represents the present value of the payment you would make if you exercise the option, weighted by the risk-neutral probability that you exercise. The notation and components deserve careful attention.

• $S$: current spot price of the underlying asset
• $K$: strike price of the option
• $T$: time to expiration in years
• $r$: risk-free interest rate
• $\sigma$: volatility of the underlying asset returns
• $N(\cdot)$: cumulative standard normal distribution function
• $d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$: standardized moneyness adjusted for volatility and time
• $d_2 = d_1 - \sigma\sqrt{T}$: $d_1$ adjusted for the volatility over the option's life
• $S \cdot N(d_1)$: the expected value of the asset at expiration, conditional on the option being in-the-money
• $K e^{-rT} \cdot N(d_2)$: the present value of the strike price, weighted by the probability of exercise

The parameter $d_1$ captures how far the current stock price is from the strike in standardized units. It adjusts for the expected drift and the accumulation of uncertainty over the option's life. The numerator contains the log-moneyness $\ln(S/K)$ (measuring how far the current price stands from the strike in percentage terms) plus an adjustment $(r + \sigma^2/2)T$ that accounts for the expected appreciation under the risk-neutral measure. The denominator $\sigma\sqrt{T}$ normalizes this measure by the total expected standard deviation of returns over the option's life, converting everything into units of standard deviations.

The inputs $S$, $K$, $T$, and $r$ are all observable market variables. The current stock price is available on any trading screen, the strike price is specified in the option contract, the time to expiration follows from the contract's expiration date, and the risk-free rate can be approximated from Treasury bill yields or other money market instruments. Volatility $\sigma$ is the only unobservable parameter. It describes how much the underlying asset's returns fluctuate, but this quantity cannot be directly observed because it pertains to the future. Given a market price $C^{mkt}$, the implied volatility $\sigma_{IV}$ is the value of $\sigma$ that satisfies:

This equation establishes a relationship between the theoretical Black-Scholes world and the actual market. The left side represents the theoretical price that emerges from the model when we use a particular volatility assumption. The right side represents the price at which actual buyers and sellers agree to trade. When these two quantities match, we have found the volatility that the market is implicitly using to price the option. The components of this equation deserve explicit identification:

• $C^{BS}$: theoretical Black-Scholes option price
• $\sigma_{IV}$: implied volatility parameter to be solved for
• $C^{mkt}$: observed market price of the option
• $S, K, T, r$: observable market inputs (spot, strike, time, rate)

The concept of implied volatility rests on one mathematical property: the Black-Scholes price is strictly increasing in volatility. When volatility is zero, the call option price equals its discounted intrinsic value (or zero if out of the money). As volatility increases without bound, the call option price approaches the current stock price. Because the price increases monotonically and continuously between these bounds, for any market price within this range, there exists exactly one volatility value that produces that price. This uniqueness guarantees that implied volatility is well-defined.