# Stochastic volatility

Stochastic volatility is a type of volatility model used to describe the behavior of asset prices that are affected by both deterministic and random components. Stochastic volatility models assume that the volatility of an asset's returns is itself a random variable that follows a stochastic process. This means that the volatility of the asset is not known with certainty, but is instead a random variable that changes over time. This randomness can be captured by a stochastic volatility model, which can be used to estimate the future volatility of an asset and to price financial derivatives such as options.

Stochastic volatility models are typically formulated as a system of stochastic differential equations, which describe the evolution of the asset's underlying return and volatility processes over time. These equations can be used to simulate the evolution of the asset's returns and to estimate the probability of various outcomes. The most common stochastic volatility models are the GARCH and Heston models, which are used to estimate the volatility of stock returns. Other stochastic volatility models, such as the Black-Scholes model, are used to price options.

## Example of Stochastic volatility

A common example of a stochastic volatility model is the GARCH model. This model is used to estimate the volatility of stock returns and is formulated as a system of stochastic differential equations. The GARCH model assumes that the returns of a stock follow a normal distribution and that the volatility of the returns is a random process. This random process is modeled using two parameters, the mean and variance of the returns. The variance of the returns is assumed to follow an autoregressive process, which means that the current volatility is linked to the past volatility of the returns. This autoregressive process captures the changing volatility of the returns over time and can be used to estimate the future volatility of the returns.

## Formula of Stochastic volatility

The most common stochastic volatility models can be formulated as a system of stochastic differential equations. These equations describe the evolution of the asset's underlying return and volatility processes over time. The most common models are the GARCH and Heston models, which are formulated as follows:

{\displaystyle {\begin{aligned}dS_{t}&=\mu S_{t}dt+\sigma _{t}S_{t}dW_{t}^{1}\\d\sigma _{t}^{2}&=\alpha (\beta -\sigma _{t}^{2})dt+\gamma \sigma _{t}dW_{t}^{2}\end{aligned}}}

where St is the asset's underlying return, σt is the volatility of the return process, μ is the expected return, $\alpha$ and β are parameters of the model, γ is the volatility of the volatility process, and Wt1 and Wt2 are two independent Brownian motions. These equations can be used to simulate the evolution of the asset's returns and to estimate the probability of various outcomes.

## When to use Stochastic volatility

Stochastic volatility models are useful when a financial asset's volatility is not constant, but rather can vary over time in an unpredictable manner. This type of model can also be used to price options and other derivatives, as it allows for the incorporation of the uncertainty of volatility into the pricing process. Additionally, stochastic volatility models are useful for estimating the probability of various outcomes in order to make informed decisions about investing in a particular asset.

## Types of Stochastic volatility

Stochastic volatility models come in a variety of forms, depending on the assumptions made about the underlying processes driving the asset's volatility. These models include:

• The GARCH (Generalized Autoregressive Conditional Heteroskedasticity) model, which assumes that the volatility of an asset's returns follows a GARCH process. This model is often used to estimate the volatility of stock returns.
• The Heston Model, which assumes that the volatility of an asset's returns follows a Heston stochastic process. This model is used to price options and is based on the Black-Scholes option pricing model.
• The Black-Scholes Model, which is a deterministic model that assumes that the volatility of an asset's returns is constant over time. This model is used to price options.
• The Merton Model, which is a stochastic model that assumes that the volatility of an asset's returns is a random variable that follows a Merton stochastic process. This model is used to price options and is an extension of the Black-Scholes model.

These models can be used to estimate the future volatility of an asset and to price financial derivatives such as options. By understanding how an asset's volatility changes over time, investors can make more informed decisions about how to manage their portfolios.

## Steps of Stochastic volatility

• Estimation: Estimation of the parameters of a stochastic volatility model involves fitting the model to historical data. This is typically done using maximum likelihood estimation, which involves maximizing the likelihood of the observed data given the model parameters.
• Simulation: Once the model parameters have been estimated, the model can be used to simulate the future evolution of the asset's returns and volatility. This is done by solving the stochastic differential equations that describe the evolution of the asset's return and volatility processes.
• Pricing: Stochastic volatility models can also be used to price financial derivatives, such as options. This is typically done using Monte Carlo simulation, which involves simulating the evolution of the asset's returns and volatility many times and then calculating the expected payoff of the option.

Stochastic volatility models are advantageous because they allow for more accurate pricing of derivatives, such as options. By taking into account the randomness of the asset's volatility, these models can provide more accurate pricing than models that assume a constant volatility. Additionally, stochastic volatility models can account for the fact that asset returns often exhibit clustering behavior, meaning that large returns tend to follow other large returns and small returns tend to follow other small returns.

Furthermore, stochastic volatility models can be used to estimate the probability of different outcomes, including extreme events. By capturing the randomness of an asset's volatility, these models can provide a better estimate of the probability of a large move in the asset price than more traditional models.

## Limitations of Stochastic volatility

Stochastic volatility models have several limitations. Firstly, these models rely on the assumption that the underlying asset's returns and volatility follow a stochastic process. This assumption is often not valid in practice, as many asset returns show significant mean reversion and other non-random behavior. Secondly, these models are not always capable of accurately capturing the dynamics of complex financial instruments, such as options. Finally, the computational complexity of these models can make them difficult to implement in practice.

## Other approaches related to Stochastic volatility

There are a number of related approaches to modeling volatility, such as the jump diffusion model, which captures the effect of large changes in the price of an asset (jumps), and the regime switching model, which captures the effects of different market regimes on asset returns. Additionally, there are a number of hybrid models which combine elements of both stochastic and deterministic volatility models, such as the HAR-RV model.

In summary, Stochastic volatility is a type of volatility model used to describe the behavior of asset prices that are affected by both deterministic and random components. It is typically formulated as a system of stochastic differential equations and is used to simulate the evolution of the asset's returns and to price financial derivatives such as options. There are a number of related approaches to modeling volatility, such as the jump diffusion model and the regime switching model, as well as a number of hybrid models which combine elements of both stochastic and deterministic volatility models.

 Stochastic volatility — recommended articles Autoregressive model — Risk measures — Probability theory — Theory of portfolio — Influence diagram — Random walk theory — Asymmetrical distribution — Statistical hypothesis — Expected utility theory