## What Is Local Volatility (LV)?

Local volatility (LV) is a volatility measure used in quantitative analysis that helps to provide a more comprehensive view of volatility by factoring in both strike prices and time to expiration from the Black-Scholes model to produce pricing and risk statistics for options. Local volatility is related to an option's implied volatility (IV) and can be extrapolated from it.

While the Black-Scholes model generalizes the same volatility level to the entirety of options on the same underlying, local volatility allows for each individual option to have its own volatility level to more accurately reflect an option's true theoretical value.

### Key Takeaways

• Local volatility assigns a particular implied volatility to a particular option on the same underlying based on its strike and expiration.
• This provides a more specific and accurate picture of the volatility surface than the standard Black-Scholes model, which uses the same constant volatility across all options on the same underlying.
• Skew and term structure of volatility are employed with local volatility considerations.

## Understanding Local Volatility

The concept of local volatility was introduced by economists Emanuel Derman and Iraj Kani. Local volatility attempts to identify the actual volatility of an option across a range of strike prices and expirations. Local volatility seeks to use two-factor analysis to provide a more accurate actual volatility reading than implied volatility. When plotted, local volatility will generally fit the data more closely than implied volatility. Some academics have mused that, while implied volatility can be used to obtain the correct price, local volatility is the more appropriate input from a logical standpoint.

Local volatility essentially replaces the constant volatility function that is calculated from strike price and expiration. Instead, local volatility answers the same question of risk in a different way by looking at the asset price and time, which results in a different view of the volatility around an option given the same inputs.

Because local volatility is often extrapolated from implied volatility, it is sensitive to changes in the implied volatility. This means that small changes in implied volatility result in more drastic shifts in local volatility.

## How Local Volatility Is Used

One of the main criticisms of the original Black-Scholes model is that it attempted to lock the volatility of the underlying asset at a constant level for the entire life of the option. This doesn't reflect the actual market data we have but the model is still one of the most effective valuation schemes for options.

In reality, the market can produce volatility smile which was noted in earnest after the 1987 stock market crash. This sent academics and traders looking for better ways to represent volatility. Local volatility is one of the products that has emerged from that search.

Local volatility can be particularly useful in pricing exotic options that are difficult to fit standard models. It is designed to match market prices and can be used to value all combinations of strike prices and expirations compared to the single expiration that implied volatility covers.

That said, both local volatility and implied volatility are often studied together and compared to historical volatility. Whereas local and implied volatility are generated from current option price levels using the Black-Scholes model, historical volatility can be used to generate a Black-Scholes model price that is tempered by past data of actual pricing fluctuations.

## The Volatility Surface

The volatility surface is a three-dimensional plot of local volatilities where the x-axis is the time to maturity, the z-axis is the strike price, and the y-axis is the implied volatility. If the Black-Scholes model were completely correct, then the implied volatility surface across strike prices and time to maturity should be flat. In practice, this is not the case.

The volatility surface is far from flat and often varies over time because the assumptions of the Black-Scholes model are not always true. Options with lower strike prices, for instance, tend to have higher implied volatilities than those with higher strike prices.

As the time to expiration approaches infinity, volatilities across strike prices tend to converge to a constant level.

The term structure of volatility describes how local volatility changes among options of different times to expiration. However, the volatility surface is often observed to have an inverted volatility smile. Options with a shorter time to maturity have multiple times the volatility compared to options with longer maturities. This observation is seen to be even more pronounced in periods of high market stress. It should be noted that every option chain is different, and the shape of the volatility surface can be wavy across strike price and time. Also, put and call options usually have different volatility surfaces.