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STOCHASTIC VOLATILITY MODELING

Volatility of stochastic process S(t) measures the variability of S(t) over fixed intervals of time. Volatility is defined separately for discrete time processes (time series) and continuous time processes.

1] Suppose S(t) is a time series of the form

$S(t) = f(S(t-1), S(t-2), ...) + \epsilon(t),$

where ε(t)
is a new random shock at time t, independent of everything in the past. Then the volatility of S(t) is the standard deviation of ε(t).

2] Suppose S(t) is a jump-diffusion defined by the following stochastic differential equation
:

$dS(t) = A(t,S(t)) dt + B(t,S(t)) dW(t) + C(t,S(t)) dN(t),$

where
A(t,s), B(t,s) and C(t,s) are deterministic functions or stochastic processes, W(t) is a Brownian motion and N(t) is a counting process. Then the volatility of S(t) is defined as either B(t,S(t)) (absolute change) or B(t,S(t)) / S(t) (relative change). The latter definition is more common, especially in the world of finance. In the simple case of the Black-Scholes model

$dS(t) = r S(t) dt + \sigma S(t) dW(t),$

the (relative) volatility is deterministic and is equal to σ.

It was established that most processes of interest in engineering, biology, economics and finance do not exhibit constant or even deterministic volatility. That is the case with no regard to whether the processes are modeled in discrete time or continuous time. The standard deviation of the new members of a certain bird population depends on the weather, populations of other species in the food chain and some other factors. Those factors are stochastic. A relative change in a stock price in a given one-minute window depends on the trading volume and open interest in the market. Those variables are stochastic.

The case of deterministic volatility is often nicer because certain calculations and simulations are easier in that case. Sometimes, when the time horizon is not big, stochastic volatility can be approximated by deterministic volatility reasonably well. Sometimes the key variables can be transformed in such a nonlinear way that the resulting transformed versions have deterministic volatility. Then the modeling is done in terms of the transformed variables. Unfortunately, not always do these simple tricks work. If the volatility is stochastic and cannot be captured by various approximations and transformations, then special stochastic volatility models come to use. Selected examples in discrete time: ARCH, GARCH and EGARCH. Selected examples in continuous time: Cox-Ingersoll-Ross, Heston, Chen, local volatility model and Black-Karasinski. For detailed overview of the most popular stochastic volatility models, their properties and applications, see the references.

STOCHASTIC VOLATILITY MODELING REFERENCES

Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. New York: Springer.

Oksendal, B. K. (2002). Stochastic Differential Equations: An Introduction with Applications (5th ed). Springer-Verlag Berlin Heidelberg.

Duffie, D. (2001), Dynamic Asset Pricing Theory (3rd ed), Princeton University Press.

Bjork, T. (2009), Arbitrage Theory in Continuous Time (3rd ed), Oxford University Press.

Lipton, A. (2001). Mathematical Methods for Foreign Exchange: A Financial Engineer's Approach. Singapore: World Scientific.

Greene, W. H. (2011). Econometric Analysis (7th ed). Upper Saddle River, NJ: Prentice Hall.

Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press.

Brockwell, P.J., & Davis, R. A. (1991). Time Series: Theory and Methods (2nd ed). New York: Springer.

Tsay, R. S. (2005). Analysis of Financial Time Series. New Jersey: Wiley-Interscience.

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