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Concept
Heston model
Summary
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.
The basic Heston model assumes that St, the price of the asset, is determined by a stochastic process,
where , the instantaneous variance, is given by a Feller square-root or CIR process,
and are Wiener processes (i.e., continuous random walks) with correlation ρ.
The model has five parameters:
the initial variance.
the long variance, or long-run average variance of the price; as t tends to infinity, the expected value of νt tends to θ.
the correlation of the two Wiener processes.
the rate at which νt reverts to θ.
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 is strictly positive
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:
To price a derivative whose payoff is a function of one or more underlying assets, we evaluate the expected value of its discounted payoff under a risk-neutral measure.
A risk-neutral measure, also known as an equivalent martingale measure, is one which is equivalent to the real-world measure, and which is arbitrage-free: under such a measure, the discounted price of each of the underlying assets is a martingale. See Girsanov's theorem.
In the Black-Scholes and Heston frameworks (where filtrations are generated from a linearly independent set of Wiener processes alone), any equivalent measure can be described in a very loose sense by adding a drift to each of the Wiener processes.
By selecting certain values for the drifts described above, we may obtain an equivalent measure which fulfills the arbitrage-free condition.
Official source
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