What Is Stochastic Volatility?

Stochastic volatility (SV) refers to the fact that the 澳洲幸运5官方开奖结果体彩网:volatility of asset prices varies and is not constant, as is assumed in the 澳洲幸运5官方开奖结果体彩网:Black Scholes options prജicing model. Stochastic volatility modeling attempts to correct for this problem with Black Scholes𝔍 by allowing volatility to fluctuate over time.

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, 澳洲幸运5官方开奖结果体彩网:stochastic modeling iterates with successive values of a 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网:random walk.

Examples of stochastic models include the 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网:random variable. By allowing the price to vary, the stochastic volatilit꧙y models improved the accuracy of calculations and forecasts.

The Heston Stochastic Volatility Model

The Heston Model is a stochastic volatility mod🔯el 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 distinguꦫish it from other stochastic volatility models:

• It factors in the correlation between an asset's price and its volatility.
• It understands volatility as 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网:lognormal probability distribution.

The Heston Model also incorporates a 澳洲幸运5官方开奖结果体彩网: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 grapꦗhed.