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.
Mathematical formulation The Heston model assumes that St, the price of the asset, is determined by a stochastic process,
d S t = μ S t d t + ν t S t d W t S , {\displaystyle dS_{t}=\mu S_{t}\,dt+{\sqrt {\nu _{t}}}S_{t}\,dW_{t}^{S},}
where the volatility ν t {\displaystyle {\sqrt {\nu _{t}}}} is given by a Feller square-root or CIR process,
d ν t = κ ( θ − ν t ) d t + ξ ν t d W t ν , {\displaystyle d\nu _{t}=\kappa (\theta -\nu _{t})\,dt+\xi {\sqrt {\nu _{t}}}\,dW_{t}^{\nu },}
and W t S , W t ν {\displaystyle W_{t}^{S},W_{t}^{\nu }} are Wiener processes (i.e., continuous random walks) with correlation ρ. The value ν t {\displaystyle \nu _{t}} , being the square of the volatility, is called the instantaneous variance. The model has five parameters:
ν 0 {\displaystyle \nu _{0}} , the initial variance.
θ {\displaystyle \theta } , the long variance, or long-run average variance of the price; as t tends to infinity, the expected value of νt tends to θ.
ρ {\displaystyle \rho } , the correlation of the two Wiener processes.
κ {\displaystyle \kappa } , the rate at which νt reverts to θ.
ξ {\displaystyle \xi } , the volatility of the volatility, or 'vol of vol', which determines the variance of νt. If the parameters obey the following condition (known as the Feller condition) then the process ν t {\displaystyle \nu _{t}} is strictly positive
2 κ θ > ξ 2 . {\displaystyle 2\kappa \theta >\xi ^{2}.}
Risk-neutral measure See Risk-neutral measure for the complete article A fundamental concept in derivatives pricing is the risk-neutral measure; this is explained in further depth in the above article. For our purposes, it is sufficient to note the following:
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