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ACTA WASAENSIA 513.3 Modelling Time-Serial Dependence of Returns3.3.1 Stochastic Volatility ModelsThe Geometric Brownian Motion introduced in section 3.1 may be written in differentialform equivalently as:dP t = μP t dt + σP t dW t (3.14)d ln P t =μ − σ2dt + σ dW t (3.15)2where μ and σ denote the instantaneous drift and (constant) volatility, and W t standsfor standard Brownian motion. GBM has become a very popular model of asset returnsdue to its analytical tractability. The famous option pricing theory by Black & Scholes(1973) for example, assumes stock prices to follow geometric Brownian motion.Allowing the volatility parameter σ to become a random variable, one obtaines so calledStochastic Volatility (SV) models. A discrete time formulation (ignoring drift) is thengiven byr t = σ t · t , t ∼ N (0, 1) (3.16)where r t denotes the logreturn over one period, and the instantaneous volatility σ t isa strictly stationary process–often assumed but not necessarily– independent of theiid symmetric noise process t . 78The first stochastic volatility model has been introduced by Taylor (1986), who assumedln σ t to follow an AR(1) process. The earliest continuous time formulation of stochasticvolatility is due to Hull & White (1987), who choose the following stochastic processesfor the stock price P t and its instantaneous variance V t = σt 2:dP t = φ(P t , σ t ,t)P t dt + σ t P t dW (1)t (3.17)dV t = μ(σ t ,t)V t dt + ξ(σ t ,t)V t dW (2)t (3.18)where W (1)t and W (2)t denote (possibly correlated) Wiener processes. The fact, thatthe parameter μ is allowed to depend upon σ t , allows the inclusion of mean-reverting78 see Taylor (1994) and Mikosch (2003b).