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## ZULF MODEL SUPERIOR TO HESTON MODEL IN 80% OF CASES FOR CVX

### zulfikar.ahmed@gmail.com<zulfikar.ahmed@gmail.com>

11:38 AM (1 hour ago)

 to jfrenkel, jhansen, jharrington, jharris, jhp, jhricko_4, jhs, jhutasoi, jianjunp, jinha, jjbrehm, jlind, jlondon, jlw, jmateo, jmerseth, jmetcalf, jmg, jmogel, joel, joelms, john.aldrich, john.beatty, johncrawford53, jose.oliveira

The Zulf option pricing model is a time-fractional Heston model, a Heston model extended to address long memory in volatility.  The Heston model is one of the most accurate option pricing models in the world.  Our method for extending the Heston model consists of inserting into the Heston model what is essentially the waiting time distribution of a fractional Poisson process (there are other ways of considering our extension).  We introduce into the Heston model

psi(t) = kappa*delta^(-alpha)*mlf(-kappa*delta^alpha, alpha, alpha)

as a deflator to Heston’s D(t).  The R code attached is one I used to compare the fitting of Heston model versus Zulf model on a liquid option, the oil stock CVX.  Summary of results is statistically significant improvement in fits in N=49 samples

>>> from scipy.stats import ttest_rel
>>> ttest_rel(x.ix[:,2],x.ix[:,3])
(array(5.192283183607352), 4.1753688416565105e-06)

>>> sum(x.ix[:,2]>x.ix[:,3])
39
>>> len(x)
49
>>> 39./49.
0.7959183673469388

The full table of comparison is below.  Now the important thing here is that at least at this stage of establishing what I am sure is the world’s best option pricing model, it is best to stick to Nelder-Mead algorithm on R since there is a delicate issue of when the Black-Scholes implied volatility computation fails.  For our results it was also important to set the INITIAL PARAMETERS so that the convergence of fits were relatively smooth.  So for the waiting time

psi(t, alpha, delta) = kappa*delta(-alpha) * mlf(-kappa*t^alpha,alpha,alpha)

you want initial parameters for optimization alpha=2.0 and delta=1.0.  In fact, I suspect that delta could be eliminated altogether so that a single-parameter extension of the Heston model could handle long memory.

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Preview attachment hestonVzulf.R