# Volatility Modelling

Volatility models are usually used to forecast volatility. In finance, volatility models virtually entail forecasting aspects of future returns. They are used to forecast the absolute magnitude of returns, predict quantiles, or the entire density. Volatility forecasts are usually applied in risk management, hedging, and derivative pricing, market making, portfolio selection, and market timing among other financial activities. It is the predictability of volatility that is required in these applications.

Risk managers must be well-versed in the likelihood of their portfolios declining in the future. Additionally, someone who trades in options will want to know the volatility expected over the future life of the contract. He/she will also want to know how volatile the forecast volatility is in order to hedge this contract. It will be in the best interest of a portfolio manager to sell a stock or portfolio before it becomes too volatile. When the future is believed to be more volatile, a market-maker may want to set the bid-ask spread wider.

Enormous research is now being conducted on volatility models. Several scholars have written articles on the same that have proven to be useful to both practitioners and academicians. It is essential to formulate the properties that volatility models should satisfy, as new approaches are proposed and tested. It is also important to discuss properties that standard volatility models appear not to satisfy.

Types of volatility models

Volatility models are classified into two general categories in widespread use:

1. The first class is used to formulate the conditional variance directly as a function of observables. Some of the simplest examples of this category are ARCH and GARCH models.
2. The second class is used to formulate volatility models that are not purely the functions of observables. They are also known as latent volatility or stochastic volatility models

Latent volatility models are known to be elaborate with structural breaks at random times and with random amplitudes, jumps and fat-tailed shocks, multiple factors, fractals and multifractals, and general types of nonlinearities. Although these types of models can typically be simulated, they are difficult to estimate and forecast.

Features of asset price volatility processes

An excellent volatility model must capture and reflect on the following features:

• Persistence

The first documented features of the volatility process of asset prices are the clustering of large moves and small moves in the price process. Large changes in the price of an asset are often followed by small changes. Fama (1965) and Mandelbrot (1963) both reported evidence of this. So what is the implication of such volatility clustering? Well, the volatility shocks today will influence the expectation of volatility in many periods in the future.

Volatility models should have significant cumulative persistence for at least a year in the future. This is as suggested by the response of long-term option prices to volatility shocks. The ‘half-life’ of volatility is a further measure of the persistence in a volatility model. This refers to the time that volatility takes to move halfway back towards its unconditional mean following a deviation from it.

• Mean-reverting

When we say volatility clustering, what we are implying is that volatility comes and goes. In other words, a period of high volatility will eventually give way to more normal volatility. In the same way, a period of low volatility will be followed by a rise. In volatility, mean reversion can be interpreted to mean that there is a normal level of volatility to which volatility will eventually return. Regardless of when they are made, the long-run volatility forecast should all converge to this same normal level of volatility. Most practitioners and scholars believe that this is a characteristic of volatility. However, the models might differ on the normal level of volatility and whether it is constant overall time and institutional changes. In basic language, volatility mean reversion means that the current information does not affect the long-run forecast.

• Innovations may asymmetrically impact volatility

The assumption that the conditional volatility of the asset is affected symmetrically by positive and negative innovations is suitably imposed by many proposed volatility models. For example, the GARCH model allows the variance to be affected only by the square of the lagged innovations. It completely disregards the sign of that innovation. Moreover, it is particularly unlikely that negative and positive shocks have the same impact on the volatility for equity returns. Sometimes, this asymmetry is attributed to a

• Risk premium effect – Increasing volatility reduces the demand for a stock because of risk aversion.
• Leverage effect – The debt to equity ratio rises as the price of stock falls. This increases the volatility of returns to equity holders.

Under simple derivative pricing assumptions, the asymmetric structure of volatility generates skewed distributions of forecast prices. This results in option-implied volatility surfaces that have a skew.

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