Local Volatility: Understanding, Applications, and Market Implications

local volatility, first introduced by economists Emanuel Derman and Iraj Kani, aims to capture the true volatility of options across various strike prices and expiration periods. It diverges from the Black-Scholes model’s uniform volatility assumption by employing a two-factor analysis to derive more accurate volatility readings. In essence, local volatility replaces the constant volatility function of the Black-Scholes model with a dynamic approach that considers asset price and time.

the original Black-Scholes model, though widely used, has been criticized for its oversimplified assumption of constant volatility. The volatility smile phenomenon, observed after the 1987 stock market crash, highlighted the need for more nuanced volatility measures. Local volatility emerged as a solution, allowing for more accurate pricing of complex options and derivatives. It complements implied volatility by providing a granular view of volatility dynamics, which is essential for pricing exotic options accurately.

the volatility surface is a graphical representation of local volatilities across strike prices and time to maturity. Contrary to the flat surface predicted by the Black-Scholes model, the volatility surface exhibits variations influenced by market dynamics. Options with lower strike prices typically display higher implied volatilities, leading to the volatility smile pattern. Additionally, the term structure of volatility highlights changes in volatility levels across different expiration periods. During periods of market stress, such as economic downturns or geopolitical crises, the volatility smile becomes more pronounced, reflecting increased uncertainty in the market.

local volatility takes into account the specific characteristics of individual options, such as strike prices and expiration periods, to determine volatility levels. In contrast, implied volatility assumes a uniform volatility level for all options on the same underlying asset.

local volatility provides a more accurate reflection of option prices by considering the dynamic nature of volatility across different option characteristics. This granularity is essential for pricing complex options accurately and managing risk effectively.

the volatility smile phenomenon, characterized by higher implied volatilities for options with lower strike prices, challenges the uniform volatility assumption of traditional pricing models like the Black-Scholes model. Local volatility helps account for this variation in volatility levels, leading to more accurate option pricing.