Stochastic Volatility

ANATOLIY SWISHCHUK, PhD

Professor of Mathematics and Statistics, University of Calgary

Abstract: Volatility, as measured by the standard deviation, is an important concept in financial modeling because it measures the change in value of a financial instrument over a specific horizon. The higher the volatility, the greater the price risk of a financial instrument. There are different types of volatility: historical, implied volatility, level-dependent volatility, local volatility, and stochastic volatility (e.g., jump-diffusion volatility). Stochastic volatility models are used in the field of quantitative finance. Stochastic volatility means that the volatility is not a constant, but a stochastic process and can explain volatility smile and skew.

Volatility, typically denoted by the Greek letter σ, is the standard deviation of the change in value of a financial instrument over a specific horizon such as a day, week, month, or year. It is often used to quantify the price risk of a financial instrument over that time period. The price risk of a financial instrument is higher the greater its volatility.

Volatility is an important input in option pricing models. The Black-Scholes model for option pricing assumes that the volatility term is a constant. This assumption is not always satisfied in real-world options markets because probability distribution of common stock returns has been observed to have a fatter left tail and thinner right tail than the lognormal ...

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