Stochastic Volatility

What Is Stochastic Volatility?

Stochastic volatility (SV) refers to the fact that the volatility of asset prices varies and is not constant, as is assumed in the Black Scholes options pricing model. Stochastic volatility modeling attempts to correct for this problem with Black Scholes by allowing volatility to fluctuate over time.

Key Takeaways

  • Stochastic volatility is a concept that allows for the fact that asset price volatility varies over time and is not constant.
  • Many fundamental options pricing models such as Black Scholes assumes constant volatility, which creates inefficiencies and errors in pricing.
  • Stochastic models that let volatility vary randomly such as the Heston model attempt to correct for this blind spot.

Stochastic Modeling Definition

Understanding Stochastic Volatility

The word "stochastic" means that some variable is randomly determined and cannot be predicted precisely. However, a probability distribution can be ascertained instead. In the context of financial modeling, stochastic modeling iterates with successive values of a random variable that are non-independent from one another. What non-independent means is that while the value of the variable will change randomly, its starting point will be dependent on its previous value, which was hence dependent on its value prior to that, and so on; this describes a so-called random walk.

Examples of stochastic models include the Heston model and SABR model for pricing options, and the GARCH model used in analyzing time-series data where the variance error is believed to be serially autocorrelated.

The volatility of an asset is a key component to pricing options. Stochastic volatility models were developed out of a need to modify the Black Scholes model for pricing options, which failed to effectively take the fact that the volatility of the price of the underlying security can change into account. The Black Scholes model instead makes the simplifying assumption that the volatility of the underlying security was constant. Stochastic volatility models correct for this by allowing the price volatility of the underlying security to fluctuate as a random variable. By allowing the price to vary, the stochastic volatility models improved the accuracy of calculations and forecasts.

The Heston Stochastic Volatility Model

The Heston Model is a stochastic volatility model created by finance scholar Steven Heston in 1993. The Model uses the assumption that volatility is more or less random and has the following characteristics that distinguish it from other stochastic volatility models:

  • It factors in the correlation between an asset's price and its volatility.
  • It understands volatility as reverting to the mean.
  • It gives a closed-form solution, meaning that the answer is derived from an accepted set of mathematical operations.
  • It does not require that stock price follow a lognormal probability distribution.

The Heston Model also incorporates a volatility smile, which allows for more implied volatility to be weighted to downside strike relative to upside strikes. The "smile" name is due to the concave shape of these volatility differentials when graphed.

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