%0 Journal Article
%T Modeling and Pricing of Variance and Volatility Swaps for Local Semi-Markov Volatilities in Financial Engineering
%A Anatoliy Swishchuk
%A Raimondo Manca
%J Mathematical Problems in Engineering
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/537571
%X We consider a semi-Markov modulated security market consisting of a riskless asset or bond with constant interest rate and risky asset or stock, whose dynamics follow gemoetric Brownian motion with volatility that depends on semi-Markov process. Two cases for semi-Markov volatilities are studied: local current and local semi-Markov volatilities. Using the martingale characterization of semi-Markov processes, we find the minimal martingale measure for this incomplete market. Then we model and price variance and volatility swaps for local semi-Markov stochastic volatilities. 1. Introduction Financial engineering is a multidisciplinary field involving financial theory, the methods of financing, using tools of mathematics, computation, and the practice of programming to achieve the desired end results. The financial engineering methodologies usually apply engineering methodologies and quantitative methods to finance. It is normally used in the securities, banking, and financial management and consulting industries, or by quantitative analysts in corporate treasury and finance departments of general manufacturing and service firms. One of the recent and new financial products are variance and voaltility swaps, which are useful for volatility hedging and speculation. The market for variance and volatility swaps has been growing, and many investment banks and other financial institutions are now actively quoting volatility swaps on various assets: stock indexes, currencies, as well as commodities. A stock's volatility is the simplest measure of its riskiness or uncertainty. Formally, the volatility is the annualized standard deviation of the stock's returns during the period of interest, where the subscript denotes the observed or ¡°realized¡± volatility. Why trade volatility or variance? As mentioned in [1], ¡°just as stock investors think they know something about the direction of the stock market so we may think we have insight into the level of future volatility. If we think current volatility is low, for the right price we might want to take a position that profits if volatility increases¡±. The Black-Scholes model for option pricing assumes that the volatility term is a constant. This assumption is not always satisfied by real-life options as the probability distribution of an equity has a fatter left tail and thinner right tail than the lognormal distribution (see Hull [2]), and the assumption of constant volatility in financial model (such as the original Black-Scholes model) is incompatible with derivatives prices observed in the market. Stochastic
%U http://www.hindawi.com/journals/mpe/2010/537571/