MATHEMATICAL STUDY OF RISK IN FINANCIAL MARKETS

Abstract

In 1973, a significant breakthrough came in the history of pricing options when Fischer Black and Myron Scholes proposed a valuation formula for the pricing of European call options based on the geometric Brownian motion model of the stock price dynamics. This option pricing model was further developed by Robert C. Merton in the same year, and was named as the “Black-Scholes option pricing model”. Despite its landmark success in the option pricing theory, the model has certain biases like the assumptions of constant volatility and Gaussian distribution of asset log-returns. In practice, the volatility is not constant and the asset log return distributions are non-Gaussian in nature characterized by heavy tails and high peaks. A wide range of research has been done to revise and upgrade classical Black-Scholes model. The relaxation of constant volatility assumption led to the modeling of dynamic volatility. A natural extension is to regard the volatility as a continuous times to chastic process which gave rise to the stochastic volatility modeling. Stochastic volatility models allow the volatility to fluctuate randomly and are able to incorporate many empirical characteristics of volatility namelyvolatilitysmile,mean-reversionandleveragetonameafew. Thesemodels arefurtherextendedtoconsidereithermultiplefactorsofvolatilityortoincludethe jumps in the stock price or volatility process. Volatility is a standard measure of risk, which is a statistical tool to measure the dispersion of asset returns from their mean over a given time period. An alternate measure of risk can be entropy, since it also measures the randomness. Shannon(1948) in his mathematical theory of communication, used entropy as a measure of information which laid the foundation of the field of information theory. Entropy has broad applications in finance too especially in the portfolio selection, asset pricing and time series analysis. The principle of maximum entropy has extensively been used in finance. Furthermore, the concept of entropy is of great help in analysing the stock market since it captures the uncertainty and disorder xiii of the time series without imposing any constraints on the theoretical probability distribution. Thethesisentitled“MathematicalStudyofRiskinFinancialMarkets”isdevoted to the execution of stochastic volatility modeling and entropy approach for the option pricing and the analysis of asset log-return series. We have proposed a multifactor stochastic volatility model with a fast and a slow mean-reverting factorsofvolatility,wheretheslowvolatilityfactorisapproximatedwithaquadratic arc. Using this model, the pricing formulae and the implied volatility smiles are obtained for the European and Asian options. The accuracy of these option pricing formulae is also established. We have also shown that the multifactor stochastic volatility models outperform the stochastic volatility jump diffusion models, by comparing the two extensions of Heston stochastic volatility model. In addition, the option pricing and analysis of asset log-return series is conducted using entropy measures. The concept of approximation of slow volatility factor is infused with the entropy maximization. For this, we have proposed to calibrate the risk-neutral density function of the future asset price by maximizingatwo-parameterentropywithanadditionalvarianceconstraint,where the quadratic arc expression of volatility is considered. The calibrated density function is used to price the European call options for different strikes. We have also proposed a two-parameter permutation entropy and its extensions viz. twoparametermultiscalepermutationentropyandtwo-parameterweightedmultiscale permutation entropy to analyse the asset log-return series