Heston model

In finance, the Heston model, named after Steven L. Heston, is a mathematical model that describes the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process. Basic Heston model The basic Heston model assumes that St, the price of the asset, is determined by a stochastic process, : dS_t = \mu S_t,dt + \sqrt{\nu_t} S_t,dW^S_t, where \nu_t, the instantaneous variance, is given by a Feller square-root or CIR process, : d\nu_t = \kappa(\theta - \nu_t),dt + \xi \sqrt{\nu_t},dW^{\nu}_t, and W^S_t, W^{\nu}_t are Wiener processes (i.e., continuous random walks) with correlation ρ. The model has five parameters:

\nu_0, the initial variance.
\theta, the long variance, or long-run average variance of the price; as t tends to infinity, t

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Unspanned stochastic volatility in the multifactor CIR model

Damir Filipovic, Martin Larsson, Francesco Statti

Empirical evidence suggests that fixed-income markets exhibit unspanned stochastic volatility (USV), that is, that one cannot fully hedge volatility risk solely using a portfolio of bonds. While Collin-Dufresne and Goldstein (2002, Journal of Finance, 57, 1685–1730) showed that no two-factor Cox–Ingersoll–Ross (CIR) model can exhibit USV, it has been unknown to date whether CIR models with more than two factors can exhibit USV or not. We formally review USV and relate it to bond market incompleteness. We provide necessary and sufficient conditions for a multifactor CIR model to exhibit USV. We then construct a class of three-factor CIR models that exhibit USV. This answers in the affirmative the above previously open question. We also show that multifactor CIR models with diagonal drift matrix cannot exhibit USV.

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We introduce a novel stochastic volatility model where the squared volatility of the asset return follows a Jacobi process. It contains the Heston model as a limit case. We show that the joint density of any finite sequence of log-returns admits a Gram–Charlier A expansion with closed-form coefficients. We derive closed-form series representations for option prices whose discounted payoffs are functions of the asset price trajectory at finitely many time points. This includes European call, put and digital options, forward start options, and can be applied to discretely monitored Asian options. In a numerical study, we show that option prices can be accurately and efficiently approximated by truncating their series representations.

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