Unrealized or Implied Volatility

Unrealized volatility, often referred to as implied volatility, is the market’s forecast of a security’s volatility over a specific period. It’s forward-looking and derived from the prices of financial instruments like options. It reflects the market’s expectations of future volatility in an asset’s prices. Implied volatility differs from historical volatility as it is forward-looking and derived directly from current option prices. Options traders use it to assess expected volatility and calculate probabilities. However, IV is not an exact science and does not predict specific market movements. Different volatility models yield varying results. For example, the Black-Scholes model, which assumes a lognormal distribution of returns, may not accurately reflect market behavior. Using a fat-tailed distribution instead of a normal distribution typically results in higher volatility expectations.

Calculation:

Option Prices

Obtain the current prices of options for the asset.

Option Pricing Model

Use an option pricing model like the Black-Scholes model to derive the implied volatility. The Black-Scholes formula for a call option is:

$$C=S_0\Phi (d_1)-K{e}^{-\mathit{rT}}\Phi (d_2)$$

where:

$$d_1=\frac{\text{ln}<!--\; FUNCTION\; APPLICATION\; -->(S_0∕K)+(r+{\sigma }^2∕2)T}{\sigma \sqrt{T}}$$

$$d_2=d_1-\sigma \sqrt{T}$$

Here, $C$ is the call option price, $S_0$ is the current stock price, $K$ is the strike price, $r$ is the risk-free rate, $T$ is the time to maturity, $\sigma$ is the implied volatility, and $\Phi$ is the cumulative distribution function of the standard normal distribution. The shape of the formula is worth decoding once, so $d_1$ and $d_2$ stop being incantations: the first term is the probability-weighted value of the stock you receive if the option finishes in the money, and the second is the discounted strike you pay in that event — $\Phi (d_2)$ is the model’s (risk-neutral) probability of finishing in the money, and $\Phi (d_1)$ is that probability tilted upward to account for the stock being worth more in exactly the scenarios where you exercise.

Implied Volatility Extraction

Solve for $\sigma$ (implied volatility) by inputting the market price of the option into the Black-Scholes formula. This typically requires numerical methods or iterative algorithms since the formula doesn’t solve for $\sigma$ directly.

Limitations of Black-Scholes model The Black-Scholes model assumes asset returns follow a lognormal distribution, implying constant and normally distributed volatility. This underestimates the probability of extreme price movements (fat tails) observed in real markets. While the model is widely used as an approximation, its limitations are where the unexpected losses come from. The main ones:

Tail Risk

The model underestimates extreme moves. Hedge this with out-of-the-money options.

Liquidity Risk

Assumes instant, cost-less trading, which is unrealistic and hard to hedge.

Volatility Risk

Assumes a stationary process. Hedge this with volatility hedging.

Gap Risk

Assumes continuous time and trading. Hedge this with Gamma hedging.

Pricing Bias

Fed a single at-the-money volatility, the model misprices the wings in both directions — and which way depends on the strike. It underprices out-of-the-money puts, which trade at elevated implied volatility because of the persistent bid for crash protection, and overprices out-of-the-money calls on equity indices, which trade below it. That pattern is the volatility skew, dissected in section “Option Greeks: Impact of Volatility on Options”. Note that by put-call parity a deep in-the-money call carries the same implied volatility as the same-strike out-of-the-money put, so there is no separate “deep ITM” bias to speak of.

The empirical work bears this out: implied volatility and subsequently realized volatility differ in ways that are themselves heavy-tailed, with the ratio of the two best fitted by lognormal or power-law distributions, not anything the model would predict.⁵⁵ In practice, Delta hedging alone is insufficient due to these additional risks.

Better models include:

GARCH (Generalized Autoregressive Conditional Heteroskedasticity)

Captures time-varying volatility, reflecting clustering of volatility over time.

Heston Model

Incorporates stochastic volatility, allowing volatility to fluctuate over time.

Jump-Diffusion Models (e.g., Merton Model)

Adds jumps to the price process to account for sudden, large movements.

These models better capture the heavy tails and volatility clustering observed in financial markets, providing more accurate pricing and risk management.

For all its faults, implied volatility carries information that historical volatility does not: Christensen and Prabhala found that implied volatility is a better predictor of subsequent realized volatility than past realized volatility is, and largely subsumes the information in it.⁵⁶ When the two disagree, weight the implied number — it is a price set by participants with capital at risk, while the historical number is a description of a period that has ended.