Difference between Heston model and "Schöbel & Zhu" stochastic volatility model

[Popi]

Hello every body,

I'm working on implementing a stochastic volatility model using Matlab to price exotic options. The model was developed by two people Schöbel & Zhu.

I'm not very good in mathematics and I'm trying to learn as much as possible from this work so I would like to know the main difference between this model and Heston.

The main dynamics of the model are :
* dx(t) = (r-0.5v(t)²)dt + v(t) dws(t)
* dv(t) = k(o-v(t)) dt + sigma dwv(t)

where
* x(t) = ln(s(t)) (log of the spot price);
* ws, wv two correlated brownian motions;
* v(t) instantaneous volatility;
* k, o, sigma : constant parameters.

link for the full model :
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=100831

any help please ?

Thanks.

 

[Alexei]

First of all, the first model is the extension of the Ornstein–Uhlenbeck process - a mean reverting process. With Heston you have a drifting process with some dynamics for a volatility. With the O-U model you have a reverting process that does not take a look at current values of s(t) while the Heston does account for that; it means that jumps of s(t) will be of a different scale. The volatility dynamics looks the same.

 

[Adrián Facchi]

With the O-U model you have a reverting process that does not take a look at current values of s(t) while the Heston does account for that.

I don't see how the Heston model takes in account the values of the underlying to obtain the instantaneous volatility. I rather would say that this process is an extension of the Heston model in the sense that it can be reduced to that by chosing the parameters accordingly. In the other hand, the OU process and its square is mean reverting while in the Heston model only the square of the volatility is mean reverting.

The volatility dynamics will look more or less the same since finally all depends on the parameters and there is no unique way to calibrate any stochastic volatility model.

For option pricing proposes and from a heuristic point of view I would say that both models are pretty much equally valid in the sense that if the calibration conditions are the same (say that are calibrated to match the prices of a bunch of vanilla options both satisfying the non arbitrage conditions imposed by relations on the parameters) then prices of exotics will be valid. In practice there will be nothing such as a correct price and all will finally depend on the preferences on the trader and the calibration conditions.

 

[Timmi]

There are key differences. From the first equation you have provided for x(t) = ln(s(t)), I can deduce from Ito's Lemma that your model can be expressed as

ds(t) = r*s(t)dt + v(t)*s(t)*dws(t)
dv(t) = k(o-v(t)) dt + sig*dwv(t)

I express it in this format so I can compare with the Heston Model:

ds(t) = r*s(t)dt + sqrt[v(t)]*s(t)*dws(t)
dv(t) = k(o-v(t))dt + sig*sqrt[v(t)]*dwv(t)

Differences
In your model, v(t) is an Ornstein-Uhlenbeck process, i.e. v(t) can be negative or positive.
In the Heston model, v(t) is a CIR affine process, i.e. v(t) is strictly positive.

In your model, the stochastic volatility which drives the s(t) process is v(t).
In the Heston model, the stochastic volatility which drives the s(t) process is sqrt(v(t)).

In practice, the fact that v(t) can assume negative or positive values will not dramatically change the stock price behaviour, since by the Levy characterisation Theorem, dw(t) = signum(v(t))dws(t) is still a standard Brownian motion. I'm therefore guessing that the differences in the models will lie in the difference between the probability distributions of sqrt[v(t)] in the Heston model, and |v(t)| in your model. In the end, the extent to which the prices differ between both models will depend on the calibrated values for k, o and sigma.

 

[quotes]

I don't see how the Heston model takes in account the values of the underlying to obtain the instantaneous volatility. I rather would say that this process is an extension of the Heston model in the sense that it can be reduced to that by chosing the parameters accordingly. In the other hand, the OU process and its square is mean reverting while in the Heston model only the square of the volatility is mean reverting.

How about using B-S framework directly to dynamically hedge an option?
Is the tracking error too large to satisfy investors?
Can Heston Model reduce the tracking error by a great deal?