# Stochastic Volatility

Last Updated :

21 Aug, 2024

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Dheeraj Vaidya

Table Of Contents

## What Is Stochastic Volatility (SV)?

Stochastic Volatility (SV) is a financial term that describes the dynamics of stock prices. It explains how a stock price can change or fluctuate with time. Incorporating this volatility helps to understand the unpredictable or random fluctuations in the volatility of the stock price.

The origins of the modern stochastic volatility models date back to the 1993 paper of Heston. It talks particularly about options pricing and stock prices. Also, it forms a major component of risk management and its dynamics.

##### Table of contents

- Stochastic volatility refers to the popular concept in the financial markets that describes the random fluctuations and dynamics of stock prices.
- The modern SV model was first presented in 1993 by Heston. It helps to understand how the frequent changes in the market can influence the asset value.
- This model is further applied in the Heston Model, SARCH, and GARCH models. Here, the first two are used for determining option prices. The GARCH model is used for analyzing time series.
- It differs from local volatility as the latter uses the two-factor analysis for an accurate outcome.

### Stochastic Volatility Explained

Stochastic volatility is a vital concept of financial modules that measures the random fluctuations and changes in stock prices. Here, the term stochastic defines how frequent changes in stock prices can lead to a volatile situation. In contrast to others that assume constant volatility, it allows the volatility to change over time. It gives a solution to the problems faced in the Black Scholes model. Thus, it is possible to get accurate results on why a stock price changes.

In the process of implementing SV models, the major component remains to be volatility. While the Black Scholes believes in keeping the volatility the same, it is the reverse for stochastic volatility models. The former assumes that the stock price has no change in the volatility factor. However, the SV model keeps the price volatility as a random variable. Therefore, any changes we notice in the stock prices will be random and fluctuate. Thus, the forecasts made with SV turn out to be on par with the current scenarios. Hence, the use of this model is visible in options pricing as well.

The stochastic volatility inspired models include the Heston and Stochastic Alpha Beta Rho (SABR) models. They are majorly used as a tool of stochastic volatility for options pricing. However, the Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) is used to analyze the time series. Yet, all these models aim to find the associated volatility with the product.

### Examples

Let us look at some examples of stochastic volatility to comprehend the concept in a better manner:

#### Example #1

Suppose XYZ Ltd is a company listed on the popular stock exchange. They have been on this exchange for many years. At the time of listing, the stock price was $2, and after three years, the price reached $10. However, the traders were still skeptical of the stock. Therefore, a trader named John tried to find the volatility associated with this stock. At first, he assumed the volatility to be fixed for a month and saw a positive relationship between the price and the returns generated. However, it was still not justifying the fluctuations happening in the market. Therefore, John used the stochastic volatility inspired models to compete with it.

In the later stages, John was able to find an indirect correlation (relationship) between the price and volatility. So, as there was a negative segment in the market, the price of XYZ stock rose. The very next week, the market dragged the stock down with massive volatility and selling by Foreign Institutional Investors (FII). Likewise, similar patterns were noticed for the entire month. As a result, we can see a chain of dynamics in the stock prices.

#### Example #2

Another example of this concept would be the application of the stochastic volatility model in option pricing. Suppose there is a sudden war-related or political news in circulation in the market. At this point, if John, from the above example, wants to monitor the pricing of options, he will use the Heston model. This model will basically capture the risk associated with the stock price. So, if there is high volatility regarding the stock, the prices will also lead to high premiums on options. Also, the risk associated with it will rise.

### Effects On Asset Value

The stochastic volatility can have an extreme impact on the asset prices and the contracts associated with it. And some of the major influence is visible in the stochastic volatility options pricing, portfolios, and market. Let us study them in detail:

#### #1 - Option Pricing

The prime usage of SV models is in the derivatives instruments. Traders use this model to determine the prices of the option contracts on par with the volatility generated around them. As the definition says, option contracts allow the right to buy or sell assets at a price but do not oblige (force) to perform so. However, volatility does exist in this space, too. So, as the market fluctuates, the option's value also behaves the same. Hence, a higher volatility will lead to a rise in the value of the option contracts and vice versa.

#### #2 - Portfolio Management

To a greater extent, stochastic volatility also influences the assets within a portfolio. While different assets lie in the same portfolio, any market fluctuations strongly hamper the risk associated with it. In short, if the volatility rises, the risk will also rise, and it will indirectly reduce the portfolio's net asset value (NAV).

#### #3 - Risk Management

Stochastic mainly refers to random fluctuations in asset value. However, if there are frequent changes, the complexity of risk also rises. This creates additional levels to the risk management strategy, which further influences the asset value. Thus, traders must manage risk with dynamic changes in the market.

In the same manner, stochastic volatility can also influence the market behavior and the liquidity of the asset. With the uncertainty of the market, assets become illiquid, and it becomes tough to sell them in the market. As a result, traders and investors may respond abruptly to such situations.

### Stochastic Volatility vs Local Volatility

Although volatility is a core component of stochastic and local volatility, they behave and work differently. So, let us look at their differences and prime features that distinguish them from each other:

Basis | Stochastic Volatility | Local Volatility |
---|---|---|

Meaning | It refers to the financial concept explaining the random fluctuations in stock prices over a period. | Local volatility is a simplified version of SV that relates to current asset price and time. |

Assumption | SV assumes the volatility to be random and not constant over a time. | In contrast, local volatility changes run parallel with the current market conditions. |

Method | It uses a stochastic process to determine the asset value. | LV uses a two-factor analysis to provide more accurate results. |

Examples or application of this model | Heston Model, SARCH, and GARCH models. | Dupire's local volatility model and Black-Scholes model. |

### Frequently Asked Questions (FAQs)

**1. What is Heston stochastic volatility?**

The Heston model was named after the mathematician Steven L. Heston, who explains how volatility evolves with the underlying asset. It follows the principles of random volatility that is not constant yet follows a stochastic process. It is majorly used in the pricing of European options.

**2. What is the CEV stochastic volatility model?**

The constant elasticity of variance (or CEV) SV model is a popular concept used in financial modelling, including equities and commodities. It utilizes the stochastic volatility and the leverage associated with it. It helps the traders to understand the relationship between the stock price and the variance.

**3. What is the 3/2 stochastic volatility model?**

This model is a non-affine SV model that is an extension of the Heston stochastic model. The main reason for this variant is that it allows a power-law term of "3/2" on the volatility equation. It is utilized in option pricing and studying the random dynamics of the asset models.

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