# Do Path-Independent Volatilities Exist?

Volatility of an asset is a measure of how much the price of that asset varies over time. In other words, it is a measure of how “risky” an investment in that asset is. The higher the volatility, the greater the risk.

There are two main types of volatility: historical and implied. Historical volatility is simply a measure of how much an asset’s price has varied over a certain period of time. Implied volatility is a measure of how much the market thinks the asset’s price will vary in the future. Implied volatility is usually derived from observable options prices using a pricing model.

In the Black-Scholes world, volatility is considered path-independent. Path dependency is manifested in a stochastic volatility model in which the volatility is fed into the stochastic differential equation for the asset price.

Reference [1] proposed a framework in which the asset price is fed into the equation for volatility. This is exactly the opposite of the stochastic volatility model. Under the proposed framework, all volatilities are path-dependent.

The article provided an intuitive explanation of the path-dependent nature of volatility,

A simple prediction exercise best explains path-dependent volatility. Assume that the SPX value is currently 4,000. What is your best guess of the VIX value in one year if the SPX is worth 5,000 at that time? That would mean that the SPX gained 25% in one year, and due to the leverage effect (the negative link between an asset price and its volatility), a natural guess is a low VIX value, say, 12. Now, if we add the information that two weeks before the one year horizon, the SPX was worth 5,500, would you update your best guess for the VIX? You certainly would, as this means that the market crashed, with the SPX losing 9% of its value in just two weeks, in which case the VIX always shoots up, as SPX puts are more in demand and get more expensive out of fear of a deeper market crash; a best guess could be around 40 for example. This immediately shows that volatility is best explained by the path of the asset price, rather than by its current value; or, stated otherwise, by PDV models rather than by the LV model.

The authors developed a path-dependent volatility model that is capable of capturing

• Leverage effect,
• Volatility clustering and volatility bursts,
• Time-reversal asymmetry.

We believe that this path-dependent volatility model is worth implementing into the pricing and risk-management framework as it is more intuitive than other econometric models such as GARCH, and ARIMA.

Let us know what you think in the comments below.

References

[1]  Guyon, Julien and Lekeufack Sopze, Jordan, Volatility Is (Mostly) Path-Dependent (2022). https://ssrn.com/abstract=4174589

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