Stochastic implied volatility. A factor-based model.
The most elusive input parameter in the Black-Scholes formula for the price of an option is the volatility of the underlying stock. But many options are traded on and priced by the market. The price of such an option is called its market price. The implied volatility is the volatility value that makes the Black-Scholes price of an option agree with its market price. According to the Black-Scholes model the implied volatility should remain constant for any option on the same stock regardless of the strike price and the expiration date. But that is not the case in the real world. The structure of implied volatility can be broken down into two components: the volatility smile and the term structure of volatility. The first accounts for the changes of implied volatility with strike prices for options of a fixed expiration. The term structure describes the way at-the-money implied volatility varies with time until expiration. The combination of both factors is captured by the volatility surface. Overall, this monograph is an enjoyable reading on the topic of stochastic implied volatility. The work is well organized and developed, presented in a clear way with abundant and up-to-date references. It is structured in two parts: a theoretical introduction to the subject and the proposed stochastic implied volatility model, and an empirical application to the DAX index options. The theoretical part of this monograph presents a general mathematical model of a financial market in continuous time where implied volatility is an exogenous variable. The primary securities are a non-dividend paying stock, a money market account and a continuum of standard European call options on the stock. The dynamics of the volatility surface is supposed to be driven by a small number of risk factors. In this stochastic implied volatility environment, a theory for the consistent pricing and hedging, risk management and trading of equity index derivatives as well as volatility derivatives is developed. The empirical part of the work focuses on the DAX index derivatives. First, the main properties of DAX implied volatilities are identified and then a four-factor model for the stochastic evolution of the DAX volatility surface is proposed. The four risk factors may be interpreted as the level of implied volatility, the slope of the volatility smile, the curvature of the volatility smile and the slope of the at-the-money term structure of volatility. Naturally, the model is tested and calibrated to market data. Finally, some applications of the proposed factor-based stochastic implied volatility model are presented: the pricing and hedging of selected exotic derivatives, including volatility derivatives; the computation of the value at risk for options portfolios; and a discussion and testing of some volatility trading strategies.
factor model for DAX implied volatilities
scientific article; zbMATH DE number 2104445