Volatility Models: Heston’s stochastic volatility model
In this section we specify Heston’s stochastic volatility model
(3.8)
(3.9)
To take into account leverage effect, Wiener stochastic processes  should be
correlated  . The stochastic model (3.9) for the volatility is related to the square-root
process of Feller (1951) and Cox, Ingersoll and Ross (1985). For the square-root process (3.9) the volatility is always positive, and if
 then it cannot reach zero. Note that, the deterministic part of proc-ess (3.9) is
asymptotically stable if  . Obviously, that equilibrium point is  .
The attractive features of the Heston stochastic volatility model are:
its volatility updating structure permits analytical solutions to be generated for European op-tions
the form of the Heston stochastic process used to model price dynamics allows for non-lognormal probability distributions
Heston stochastic model takes into account the leverage effect
this model describes important mean-reverting property of volatility
the empirically observed Black-Scholes volatility surfaces are often looking similar to the ones generated by the Heston model
Clearly, that the Heston’s model is a real player in the competition to be a successor of the Black and Scholes model. This model is
very popular among practitioners now.
On the other hand there remain some disadvantages and open questions:
for certain parameter constellations we observed negative option prices or at least prices which were lying below the usual arbitrage
bounds (which makes Black-Scholes volatility in-version impossible !)
the model did not consistently perform well across the various maturity by no means did it eliminate all biases
Heston’s model implicitly takes systematic volatility risk into account by means of a linear specification for the volatility risk premium.
It is worth to note that parameters of Heston stochastic volatility model  after calibration
to market data turn out to be non-constant. This means that at best we can deduce from the prices of derivatives, so called fitting. But
this is far from adequate, the fitting will only work if those who set the prices of derivatives are using the same model and they are
consistent in that the fitted parame-ters do not change when the model is refitted a few days later. Whether we have a deterministic
vola-tility surface or a stochastic volatility model with prescribed or fitted parameters, we will always be faced with how to interpret
refitting. Was the market wrong before but is now right, or was the market correct initially and now there are arbitrage opportunities?
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