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
References
[1]
K. Demeterfi, E. Derman, M. Kamal, and J. Zou, “A guide to volatility and variance swaps,” The Journal of Derivatives, vol. 6, no. 4, pp. 9–32, 1999.
[2]
J. Hull, Options, Futures and Other Derivatives, Prentice Hall, Upper Saddle River, NJ, USA, 4th edition, 2000.
[3]
A. Swishchuk, “Pricing of variance and volatility swaps with semi-Markov volatilities,” Canadian Applied Mathematics Quarterly. Accepted.
[4]
F. Black and M. Scholes, “The pricing of options and corporate liabilities,” Journal of Political Economy, vol. 81, pp. 637–654, 1973.
[5]
J. C. Cox, S. A. Ross, and M. Rubinstein, “Option pricing: a simplified approach,” Journal of Financial Economics, vol. 7, no. 3, pp. 229–263, 1979.
[6]
B. Dupire, “Pricing with a smile,” Risk, vol. 7, no. 1, pp. 18–20, 1994.
[7]
E. Derman and I. Kani, “Riding on a smile,” Risk, vol. 7, no. 2, pp. 32–39, 1994.
[8]
P. Wilmott, S. Howison, and J. Dewynne, Option Pricing: Mathematical Models and Computations, Oxford Financial Press, Oxford, UK, 1995.
[9]
R. C. Merton, “Theory of rational option pricing,” The Rand Journal of Economics, vol. 4, pp. 141–183, 1973.
[10]
J. Hull and A. White, “The pricing of options on assets with stochastic volatilities,” The Journal of Finance, vol. 42, pp. 281–300, 1987.
[11]
S. Heston, “A closed-form solution for options with stochastic volatility with applications to bond and currency options,” Review of Financial Studies, vol. 6, pp. 327–343, 1993.
[12]
R. J. Elliott and A. V. Swishchuk, “Pricing options and variance swaps in Markov-modulated Brownian markets,” in Hidden Markov Models in Finance, R. Mamon and R. Elliott, Eds., vol. 104 of Internat. Ser. Oper. Res. Management Sci., pp. 45–68, Springer, New York, NY, USA, 2007.
[13]
A. Swishchuk, Random Evolutions and Their Applications. New Trends, vol. 504 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
[14]
M. Avellaneda, A. Levy, and A. Paras, “Pricing and hedging derivative securities in markets with uncertain volatility,” Applied Mathematical Finance, vol. 2, pp. 73–88, 1995.
[15]
Y. Kazmerchuk, A. Swishchuk, and J. Wu, “A continuous-time Garch model for stochastic volatility with delay,” Canadian Applied Mathematics Quarterly, vol. 13, no. 2, pp. 123–149, 2005.
[16]
T. Bollerslev, “Generalized autoregressive conditional heteroskedasticity,” Journal of Econometrics, vol. 31, no. 3, pp. 307–327, 1986.
[17]
A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models, Theory, vol. 3 of Advanced Series on Statistical Science & Applied Probability, World Scientific, River Edge, NJ, USA, 1999.
[18]
David G. Hobson and L. C. G. Rogers, “Complete models with stochastic volatility,” Mathematical Finance, vol. 8, no. 1, pp. 27–48, 1998.
[19]
A. Javaheri, P. Wilmott, and E. Haug, “GARCH and volatility swaps,” Quantitative Finance, vol. 4, no. 5, pp. 589–595, 2004.
[20]
O. Brockhaus and D. Long, “Volatility swaps made simple,” Risk, vol. 2, no. 1, pp. 92–96, 2000.
[21]
J. Janssen, R. Manca, and E. Volpe, Mathematical Finance: Deterministic and Stochastic Models, Wiley, New York, NY, USA, 2008.
[22]
V. Korolyuk and A. Swishchuk, Semi-Markov Random Evolutions, vol. 308 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995.
[23]
E. B. Dynkin, Markov Processes. Vols. I, II, Die Grundlehren der Mathematischen Wissenschaften, B？nde 121-122, Academic Press Publishers, New York, NY, USA, 1965.
[24]
P. Carr and D. Madan, “Towards a Theory of Volatility Trading,” in Volatility: New Estimation Techniques for Pricing Derivatives, pp. 417–427, Risk Book Publications, London, UK, 1998.
[25]
G. D'Amico, J. Janssen, and R. Manca, “Valuing credit default swap in a non-homogeneous semi-Markovian rating based model,” Computational Economics, vol. 29, no. 2, pp. 119–138, 2007.
