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We are interested in the question of characteristic function of time-changed Levy process where the random time T_t has a closed form Laplace transform.  Let Y_t = X_{T_t} and T_t is independent of X_t.  Then the characteristic function of Y_t is the Laplace transform of T_t evaluated at the characteristic exponent of X_t (timechangeLevy_JFE2004) for Negarive Inverse Gaussian, Variance Gamma, and CGMY models provide examples of this technique (CGMY-SVLevy-2003).

We are interested in random time changes based on Mittag-Leffler distributions, so we consider the Laplace transform formula (Haubold-Mathai-Saxena-MLF2009):

L[x^{\beta-1} E^\gamma_{\alpha,\beta}(ax^\alpha)] = s^{-\beta}(1-as^{-\alpha})^{-\gamma}

If we considered time-changed Brownian motion with MLF random times:

Y(t) = \theta T_t + \sigma W(T_t)

we set s= i u \theta + \sigma^2 u^2/2 into the Laplace transform formula.  In order to calculate the Levy density from the characteristic function, we have to use Sebastian Raible’s Theorem (Corollary 2.4):  Sebastian_Raible

\frac{d}{ds} \frac{\phi'(s)}{\phi(s)} = \frac{ (2\alpha\beta+\beta)s^\alpha - (\alpha\beta a + \beta a)}{s - a s^{1-\alpha}}

By Raible’s Theorem, this is the Fourier transform of the measure |x|^2 K(x) dx where K(x) is the Levy density of the time-changed process.

 

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Consider the exact formula for Dirac eigenspinors on spheres computed by Camporesi-Higuchi (Camporesi-Higuchi-EigenfunctionDiracOnSpheres).  Use the Darboux approximation for Jacobi polynomials.  The major issue is that using the Darboux approximation (see this)and separation of variables, we have the approximation of one of the factors (as n\rightarrow \infty)

\phi_{nl} = \frac{2}{\sqrt{\pi n}} \frac{\cos(Kx+\gamma)}{\sin(x)} + O(n^{-3/2})

The \sin(x) in the denominator produces a sharp localization.

One can check this directly even for small n.

 

import matplotlib.pyplot as plt
import numpy as np


x=np.linspace(-5e-300,5e-300,1000)
def f(N,l):
    a=3./2+float(l)-1
    b=3./2+float(l)
    gamma=-(a+0.5)*np.pi/2.
    K=N+(a+b+1.)/2.
    z1=np.sign(np.sin(x))*np.exp(np.log(np.abs(2./np.sin(x)))-0.5*np.log(np.pi*N))
    z=z1*np.cos( K*x + gamma)
    return(z)

plt.plot(x,f(5,1))
plt.show()

diraceigenspinor-closeup-nCarrMadan_OptionValuationUsingtheFastFourierTransform_1999

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Classical electron radius is \Delta x =2.82\times 10^{-15} m (see this).  Let’s say we are in an S^4(1/h) universe, so in terms of diameter of the universe this is \Delta x = 1.86 \times 10^{-48} \cdot R_{universe}.

So we might consider that this corresponds to the n=10^{48}-th eigenspinor of the Dirac operator on S^3(1/h), whose eigenvalue is h n=2.82 \times 10^{-15}.

If we assume that the Dirac eigenvalues are energy, then we can solve for mass:

mc^2 = 2.82\times 10^{-15}

and

m = 3.13 \times 10^{-32} kg

The measured mass of electron is m_e = 9.1\times 10^{-31} kg, which is off by a factor of 30.

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Quantum theory posits some strange features of particles such as ‘wave-particle’ duality.  Here I note that the eigenspinors of the Dirac opertor on a three-sphere have a wavelet-type localization.  In other words, even without a quantum theory, compactness of the space is sufficient to produce localization of Dirac eigenspinors.  From eigenspinors D_{S^3} \psi_0(x) = \lambda \psi_0(x) to the solution of the Dirac wave equation is just multiplication by a phase factor, i.e.

\psi(t,x) = e^{i\lambda t/\hbar} \psi_0(x)

solves D\psi = i\hbar \partial_t \psi.  With a mass term, one has the Dirac equation

(mc + D)\psi = i\hbar\partial_t \psi

Now the eigenspinors have an explicit expression for spheres given by Camporesi-Higuchi.  They find the Dirac eigenspinors by separation of variables for which the first factor can be used to show the ‘wave-particle duality’ issue.  Approximation using the Darboux formula of this factor gives us

\phi_{nl}(\theta) = 1/\sqrt{n\pi} \cos(N\theta + \gamma)

where a=3/2+l-1

b=3/2+l

N=n+(a+b+1)/2

\gamma = -(a+1/2)\pi/2

Now we can pick a high value say n=10^{12} and look at the graph of \phi_{n0}(\theta) with \theta \in (-\epsilon_0, \epsilon_0).  We can see the localization of the eigenspinor clearly.  Note that no strange theory of wave-particle duality is needed in this case as this is a static eigenspinor of the Dirac operator.  This particular shape of the Dirac eigenspinor is due to the geometry of S^3, the global geometry.  I suggest that much of the complex obfuscation of quantum theory would be clarified if one developed physics directly on compact hypersurfaces of a scaled four-sphere which I claim is the actual shape of the universe.  Mystical interpretations of physics arises from this key missing link that we’re in a compact universe.  We need to have a clearer science of physics.  The Big Bang never happened either.  God does not play dice with the universe and so on.

dirac-eigenspinor-three-sphere

 

 

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Dirac solved the problem of duplexity of point-charge electrons by introducing the matrix square root of the Laplacian operator in 1928.  This was a resolution of experimental problem of observed spectra having twice the expected number of stationary states as the ‘new quantum theory’ following Goudsmit and Uhlenbeck’s introduction of spin angular momentum for the electron.  DiracElectron

DiracIssues

Now the Dirac eigenfunctions on \mathbf{R}^3 are not compactly supported which makes it harder to interpret the Dirac wavefunctions.  Dirac eigenfunctions on a compact manifold have a better chance of being localized.  On the round three-sphere S^3 the eigenspinors can be computed exactly and the localization can be checked directly.  We can make a reasonable conjecture that for perturbations of the great sphere embedded in S^4(1/h) mildly the same localization holds.

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Heston’s original model inverted a characteristic function numerically.  CarrMadan_OptionValuationUsingtheFastFourierTransform_1999 found a way to make FFT practical and Moodley 2005nimalinmoodley2005 provided a Matlab implementation that I adopted to speed up both Heston’s and my time-fractional Heston model.

 

library(Quandl)
library(plyr)
library(rPython)
python.load("/home/user1/ts/mlf.py")


library(fOptions)
Quandl.api_key("CoHxdmas1Pz2t7Hk8maE")



timeStart outlier.threshold]thresh){
            jumps[k]timeEnd || tau<timeStart){
                next
            }
            S<-surface$underlyingpx[rk]
            X<-surface$strike[rk]
            callPrice<-callHestoncf(surface$underlyingpx[rk],
                                    surface$strike[rk],
                                    tau,
                                    rate,
                                    dividendrate,
                                    v0=v0,
                                    vT,
                                    rho,
                                    kappa,
                                    sigma,
                                    implVol=FALSE)
            cp<-callPrice[[1]]
            #mid0.20){
                stars<-'***'
            }
            #print(sprintf("S=%.2f X=%.2f log(MN)=%.3f tau=%.3f mkt=%.2f heston=%.2f relerr=%.4f abserr=%.3f %s",S,X,log(S/X),tau,mktIV,modelIV,relerr,abserr,stars))
            #print(paste('In Heston act mkt iv=',mktIV,' model=',modelIV))

            #weight<-exp(-abs(log(S/x)))
            weight0.0){
                #stimeEnd || tau<timeStart){
            next
        }


        callPrice<-callHestoncf(S,
                                X,
                                tau,
                                rate,
                                dividendrate,
                                v0,
                                vT,
                                rho,
                                kappa,
                                sigma,
                                implVol=FALSE)
        cp<-callPrice[[1]]
                                        #mid0.0){
            stars<-''
            relerr<-abs(mktIV-modelIV)/mktIV
            abserr<-abs(mktIV-modelIV)
            abserrors[rk]0.1){
                stars<-'***'
            }
            print(sprintf("S=%.2f X=%.2f log(MN)=%.3f tau=%.3f mkt=%.2f heston=%.2f relerr=%.4f abserr=%.3f %s",S,X,log(S/X),tau,mktIV,modelIV,relerr,abserr,stars))
                                        #print(paste('S=',S,' X=',X,' tau=',tau,' mkt=',mid,' model=',cp,' relerr=',abs(mid-cp)/mid))
        }
    }
    return(abserrors)
}


calibrate.lmheston.model<-function(surface){
    ratetimeEnd || tau<timeStart){
                next
            } 
            S<-surface$underlyingpx[rk]
            X<-surface$strike[rk]
            #print(c(S,X,tau,rate,dividendrate,v0,vT,rho,kappa,sigma,alpha))



            callPrice<-callLMHestoncf(S,
                                    X,
                                    tau,
                                    rate,
                                    dividendrate,
                                    v0,
                                    vT,
                                    rho,
                                    kappa,
                                    sigma,
                                    alpha,
                                    implVol=FALSE)
            cp<-callPrice[[1]]
                                        #mid0.20){
                stars<-'***'
            }            

            #print(sprintf("S=%.2f X=%.2f log(MN)=%.3f tau=%.3f mkt=%.2f heston=%.2f relerr=%.4f abserr=%.3f %s",S,X,log(S/X),tau,mktIV,modelIV,relerr,abserr,stars))

            #weight<-exp(-abs(log(S/X)))
            weight0.0){
                #stimeEnd || tau<timeStart){
            next
        }


        #callPrice<-lmheston.price(lmheston.params,secparams)  
        callPrice<-callLMHestoncf(S,
                     X,
                     tau,
                     rate,
                     dividendrate,
                     v0,
                     vT,
                     rho,
                     kappa,
                     sigma,
                     alpha,
                     implVol=FALSE)


        cp0.0){
            stars<-''
            relerr<-abs(mktIV-modelIV)/mktIV
            abserr<-abs(mktIV-modelIV)
            abserrors[rk]0.2){
                stars<-'***'
            }
            print(sprintf("S=%.2f X=%.2f log(MN)=%.3f tau=%.3f mkt=%.2f fracheston=%.2f relerr=%.4f abserr=%.3f %s",S,X,log(S/X),tau,mktIV,modelIV,relerr,abserr,stars))
                                        #print(paste('S=',S,' X=',X,' tau=',tau,' mkt=',mid,' model=',cp,' relerr=',abs(mid-cp)/mid))
        }
    }
    return(abserrors)
    
}

NDays6){
            print(paste('Stock:',symbol))
        } else {
            return
        }
        
  
        if (1){
            lmhparms<-calibrate.lmheston.model(Snew)
            #print('Before Heston calibration')
            hparms<-calibrate.heston.model(Snew)
            #print('Completed Heston calibration')
            
            #print(paste('Completed calibration for',symbol))
        }

        
        heston.errors<-compare.heston.versus.market(hparms,Snew)
        lmheston.errors<-compare.lmheston.versus.market(lmhparms,Snew)
        HestonErr<-append(HestonErr,c(mean(heston.errors)))
        LMHestonErr<-append(LMHestonErr,c(mean(lmheston.errors)))
        PValues<-append(PValues,t.test(heston.errors,lmheston.errors)$p.value)
        print(paste(symbol,dates[k],mean(heston.errors),mean(lmheston.errors),sep=","))
    }

    comp<-data.frame(dates[1:length(HestonErr)],HestonErr,LMHestonErr,PValues)
    colnames(comp)<-c("Date","Heston","fracHeston","PValue")
    #tt<-t.test(comp$Heston,comp$fracHeston)
    print(paste(stock,dates[1],dates[NDays],mean(comp$Heston),mean(comp$fracHeston),mean(comp$PValue),sep=","))

    compdata.fn<-sprintf("compdata-%s.csv",stock)
    write.csv(comp,file=compdata.fn)
    #print(comp)
}

files0<-list.files(path="/home/user1/Dropbox/options/bystock/")
files<-sample(files0,1000)
#files<-c("IBM.csv")
for (f in files){
    fs<-strsplit(f,"[.]")
    stock<-fs[[1]][1]
    possibleError<-tryCatch({
        heston.vs.lmheston.analysis(stock)
    },error=function(e)e)

    if (inherits(possibleError,"error")){
        next
    }
}

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ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON MODEL IN MORE THAN 90% IN 56 CASES

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

Attachments11:09 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
 

​Ladies and Gentlemen,

I am pleased to announce that our time-fractional Heston model fits the volatility surfaces determined by the LAST TRADE of call options statistically significantly better than the Heston model in 51/55 of the cases (stock/day) that I have checked last night.  The model differs from the Heston model by a single extra parameter 0<alpha<2.  The comparison is to calibrate both the Heston and Zulf models on a volatility surface and record the error of fit (sum of absolute difference of implied volatilities). The paired t-test of these errors give:

>>> ttest_rel(x.ix[:,2],x.ix[:,3])
(array(4.27139160907284), 7.7558339944524236e-05)

Here is the actual table of errors

    UPL  2003-01-02  0.0957002682349182  0.0773492111916326
0   UPL  2003-01-03            0.078657        6.652813e-02
1   UPL  2003-01-06            0.069270        5.893240e-02
2   KMI  2002-02-08            0.007780        6.266435e-03
3   KMI  2002-02-11            0.008966        5.994275e-03
4   KMI  2002-02-12            0.008309        5.424351e-03
5   KMI  2002-02-13            0.007713        5.128947e-03
6   KMI  2002-02-14            0.005948        5.272464e-03
7   PAC  2006-11-21            0.000290        1.745138e-07
8   PAC  2006-11-22            0.000558        3.908786e-07
9   NFG  2003-01-02            0.088293        8.326848e-02
10  NFG  2003-01-03            0.092200        1.840495e-02
11  NFG  2003-01-06            0.082446        6.947264e-02
12  NFG  2003-01-07            0.089481        1.946255e-02
13  NFG  2003-01-08            0.091463        8.470383e-02
14  OXY  2002-02-08            0.034763        3.187211e-02
15  OXY  2002-02-11            0.029676        3.690711e-02
16  OXY  2002-02-12            0.022510        1.772629e-02
17  OXY  2002-02-13            0.017657        2.116841e-02
18  OXY  2002-02-14            0.024343        3.202859e-02
19  AEM  2002-02-08            0.123033        1.217102e-01
20  AEM  2002-02-11            0.110734        6.377745e-02
21  AEM  2002-02-12            0.100758        9.943284e-02
22  AEM  2002-02-13            0.088986        8.803400e-02
23  AEM  2002-02-14            0.103508        9.958264e-02
24  NWY  2006-10-11            0.005319        4.450291e-08
25  NWY  2006-10-12            0.001743        1.144806e-07
26  NWY  2006-10-13            0.004078        2.571337e-06
27  NWY  2006-10-16            0.004726        3.099954e-03
28  TEF  2002-02-08            0.010447        5.958070e-03
29  TEF  2002-02-11            0.009764        8.375140e-03
30  CIB  2006-08-18            0.001899        6.670807e-07
31  CIB  2006-08-21            0.001459        8.779171e-09
32  CIB  2006-08-22            0.004307        1.318464e-08
33  CIB  2006-08-23            0.006848        4.271603e-07
34  SKO  2002-02-08            0.059888        4.294716e-02
35  SKO  2002-02-11            0.061167        4.417510e-02
36  SKO  2002-02-12            0.061014        4.901067e-02
37  SKO  2002-02-13            0.055221        4.730380e-02
38  SKO  2002-02-14            0.061628        4.659908e-02
39  PAG  2007-07-03            0.005586        3.715073e-08
40  PAG  2007-07-05            0.008058        1.047600e-06
41  PAG  2007-07-06            0.007521        5.312208e-07
42  PAG  2007-07-09            0.007519        4.822871e-07
43  PAG  2007-07-10            0.006269        7.045978e-07
44  RBK  2002-02-08            0.119141        1.042945e-01
45  RBK  2002-02-11            0.128607        1.035408e-01
46  RBK  2002-02-12            0.131435        1.201102e-01
47  WGO  2002-02-08            0.109837        9.434335e-02
48  WGO  2002-02-11            0.094459        9.186975e-02
49  WGO  2002-02-12            0.109614        8.576695e-02
50  WGO  2002-02-13            0.110616        1.103984e-01
51  WGO  2002-02-14            0.116418        1.109057e-01
52  WAG  2002-02-08            0.059161        6.123514e-02
53  WAG  2002-02-11            0.067877        7.072952e-02
54  WAG  2002-02-12            0.073040        7.199622e-02
55  WAG  2002-02-13            0.060819        5.949971e-02

The situation is very similar to the great breakthrough of Black, Scholes and Merton whose model had a constant volatility assumption which everyone knew was not correct and did not hold in the sense that our alpha parameter is in theory constant per stock but the estimated alpha from last trade volatility surface is not constant.  A possibility that I might explore later is a third stochastic process for alpha but for now we have a clear model that is closed form, almost as fast as Heston model, and that improves the fit of Heston model overwhelmingly often (using last trades and not using bid-ask averages of option prices).
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Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON
MODEL IN MORE THAN 90% IN 56 CASES
From: “zulfikar.ahmed@gmail.com” <zulfikar.ahmed@gmail.com>
To: jfrenkel@fma.if.usp.br, jhansen@uchicago.edu, jharrington@uchicago.edu,
jharris@andrew.cmu.edu, jhp@uchicago.edu, jhricko_4@juno.com, jhs@math.ubc.ca,
jhutasoi@andrew.cmu.edu, jianjunp@andrew.cmu.edu, jinha@uchicago.edu,
jjbrehm@uchicago.edu, jlind@andrew.cmu.edu, jlondon@uchicago.edu, jlw@jhu.edu,
jmateo@uchicago.edu, jmerseth@uchicago.edu, jmetcalf@ucsc.edu,
jmg@uchicago.edu, jmogel@ucsc.edu, joel@scipp.ucsc.edu, joelms@andrew.cmu.edu,
john.aldrich@vch.ca, john.beatty@interchange.ubc.ca,
johncrawford53@hotmail.com, jose.oliveira@dfn.if.usp.br, josesoto@uem.mz,
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11:09 AM (1 hour ago)

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11:09 AM (1 hour ago)

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11:09 AM (1 hour ago)

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11:09 AM (1 hour ago)

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11:09 AM (1 hour ago)

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11:09 AM (1 hour ago)

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11:09 AM (1 hour ago)

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DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed;
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11:09 AM (1 hour ago)

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11:09 AM (1 hour ago)

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Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON
 MODEL IN MORE THAN 90% IN 56 CASES
From: "zulfikar.ahmed@gmail.com" <zulfikar.ahmed@gmail.com>
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Cc:
Date: Tue, 22 Mar 2016 11:09:16 -0700
Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON MODEL IN MORE THAN 90% IN 56 CASES

 

​Ladies and Gentlemen,

I am pleased to announce that our time-fractional Heston model fits the volatility surfaces determined by the LAST TRADE of call options statistically significantly better than the Heston model in 51/55 of the cases (stock/day) that I have checked last night.  The model differs from the Heston model by a single extra parameter 0<alpha<2.  The comparison is to calibrate both the Heston and Zulf models on a volatility surface and record the error of fit (sum of absolute difference of implied volatilities). The paired t-test of these errors give:

>>> ttest_rel(x.ix[:,2],x.ix[:,3])
(array(4.27139160907284), 7.7558339944524236e-05)

Here is the actual table of errors

    UPL  2003-01-02  0.0957002682349182  0.0773492111916326
0   UPL  2003-01-03            0.078657        6.652813e-02
1   UPL  2003-01-06            0.069270        5.893240e-02
2   KMI  2002-02-08            0.007780        6.266435e-03
3   KMI  2002-02-11            0.008966        5.994275e-03
4   KMI  2002-02-12            0.008309        5.424351e-03
5   KMI  2002-02-13            0.007713        5.128947e-03
6   KMI  2002-02-14            0.005948        5.272464e-03
7   PAC  2006-11-21            0.000290        1.745138e-07
8   PAC  2006-11-22            0.000558        3.908786e-07
9   NFG  2003-01-02            0.088293        8.326848e-02
10  NFG  2003-01-03            0.092200        1.840495e-02
11  NFG  2003-01-06            0.082446        6.947264e-02
12  NFG  2003-01-07            0.089481        1.946255e-02
13  NFG  2003-01-08            0.091463        8.470383e-02
14  OXY  2002-02-08            0.034763        3.187211e-02
15  OXY  2002-02-11            0.029676        3.690711e-02
16  OXY  2002-02-12            0.022510        1.772629e-02
17  OXY  2002-02-13            0.017657        2.116841e-02
18  OXY  2002-02-14            0.024343        3.202859e-02
19  AEM  2002-02-08            0.123033        1.217102e-01
20  AEM  2002-02-11            0.110734        6.377745e-02
21  AEM  2002-02-12            0.100758        9.943284e-02
22  AEM  2002-02-13            0.088986        8.803400e-02
23  AEM  2002-02-14            0.103508        9.958264e-02
24  NWY  2006-10-11            0.005319        4.450291e-08
25  NWY  2006-10-12            0.001743        1.144806e-07
26  NWY  2006-10-13            0.004078        2.571337e-06
27  NWY  2006-10-16            0.004726        3.099954e-03
28  TEF  2002-02-08            0.010447        5.958070e-03
29  TEF  2002-02-11            0.009764        8.375140e-03
30  CIB  2006-08-18            0.001899        6.670807e-07
31  CIB  2006-08-21            0.001459        8.779171e-09
32  CIB  2006-08-22            0.004307        1.318464e-08
33  CIB  2006-08-23            0.006848        4.271603e-07
34  SKO  2002-02-08            0.059888        4.294716e-02
35  SKO  2002-02-11            0.061167        4.417510e-02
36  SKO  2002-02-12            0.061014        4.901067e-02
37  SKO  2002-02-13            0.055221        4.730380e-02
38  SKO  2002-02-14            0.061628        4.659908e-02
39  PAG  2007-07-03            0.005586        3.715073e-08
40  PAG  2007-07-05            0.008058        1.047600e-06
41  PAG  2007-07-06            0.007521        5.312208e-07
42  PAG  2007-07-09            0.007519        4.822871e-07
43  PAG  2007-07-10            0.006269        7.045978e-07
44  RBK  2002-02-08            0.119141        1.042945e-01
45  RBK  2002-02-11            0.128607        1.035408e-01
46  RBK  2002-02-12            0.131435        1.201102e-01
47  WGO  2002-02-08            0.109837        9.434335e-02
48  WGO  2002-02-11            0.094459        9.186975e-02
49  WGO  2002-02-12            0.109614        8.576695e-02
50  WGO  2002-02-13            0.110616        1.103984e-01
51  WGO  2002-02-14            0.116418        1.109057e-01
52  WAG  2002-02-08            0.059161        6.123514e-02
53  WAG  2002-02-11            0.067877        7.072952e-02
54  WAG  2002-02-12            0.073040        7.199622e-02
55  WAG  2002-02-13            0.060819        5.949971e-02

The situation is very similar to the great breakthrough of Black, Scholes and Merton whose model had a constant volatility assumption which everyone knew was not correct and did not hold in the sense that our alpha parameter is in theory constant per stock but the estimated alpha from last trade volatility surface is not constant.  A possibility that I might explore later is a third stochastic process for alpha but for now we have a clear model that is closed form, almost as fast as Heston model, and that improves the fit of Heston model overwhelmingly often (using last trades and not using bid-ask averages of option prices).
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Date: Tue, 22 Mar 2016 11:09:16 -0700
Message-ID: <CAOKUxJBApka+wV4-R2nY6UkNdDF36=nFw7otMYBTodK=XB1+zQ@mail.gmail.com>
Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON
MODEL IN MORE THAN 90% IN 56 CASES
From: “zulfikar.ahmed@gmail.com” <zulfikar.ahmed@gmail.com>

slcooper@illinois.edu, james P <strivergreen@gmail.com>, bm@falconfone.com
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Cc:
Date: Tue, 22 Mar 2016 11:09:16 -0700
Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON MODEL IN MORE THAN 90% IN 56 CASES

 

​Ladies and Gentlemen,

I am pleased to announce that our time-fractional Heston model fits the volatility surfaces determined by the LAST TRADE of call options statistically significantly better than the Heston model in 51/55 of the cases (stock/day) that I have checked last night.  The model differs from the Heston model by a single extra parameter 0<alpha<2.  The comparison is to calibrate both the Heston and Zulf models on a volatility surface and record the error of fit (sum of absolute difference of implied volatilities). The paired t-test of these errors give:

>>> ttest_rel(x.ix[:,2],x.ix[:,3])
(array(4.27139160907284), 7.7558339944524236e-05)

Here is the actual table of errors

    UPL  2003-01-02  0.0957002682349182  0.0773492111916326
0   UPL  2003-01-03            0.078657        6.652813e-02
1   UPL  2003-01-06            0.069270        5.893240e-02
2   KMI  2002-02-08            0.007780        6.266435e-03
3   KMI  2002-02-11            0.008966        5.994275e-03
4   KMI  2002-02-12            0.008309        5.424351e-03
5   KMI  2002-02-13            0.007713        5.128947e-03
6   KMI  2002-02-14            0.005948        5.272464e-03
7   PAC  2006-11-21            0.000290        1.745138e-07
8   PAC  2006-11-22            0.000558        3.908786e-07
9   NFG  2003-01-02            0.088293        8.326848e-02
10  NFG  2003-01-03            0.092200        1.840495e-02
11  NFG  2003-01-06            0.082446        6.947264e-02
12  NFG  2003-01-07            0.089481        1.946255e-02
13  NFG  2003-01-08            0.091463        8.470383e-02
14  OXY  2002-02-08            0.034763        3.187211e-02
15  OXY  2002-02-11            0.029676        3.690711e-02
16  OXY  2002-02-12            0.022510        1.772629e-02
17  OXY  2002-02-13            0.017657        2.116841e-02
18  OXY  2002-02-14            0.024343        3.202859e-02
19  AEM  2002-02-08            0.123033        1.217102e-01
20  AEM  2002-02-11            0.110734        6.377745e-02
21  AEM  2002-02-12            0.100758        9.943284e-02
22  AEM  2002-02-13            0.088986        8.803400e-02
23  AEM  2002-02-14            0.103508        9.958264e-02
24  NWY  2006-10-11            0.005319        4.450291e-08
25  NWY  2006-10-12            0.001743        1.144806e-07
26  NWY  2006-10-13            0.004078        2.571337e-06
27  NWY  2006-10-16            0.004726        3.099954e-03
28  TEF  2002-02-08            0.010447        5.958070e-03
29  TEF  2002-02-11            0.009764        8.375140e-03
30  CIB  2006-08-18            0.001899        6.670807e-07
31  CIB  2006-08-21            0.001459        8.779171e-09
32  CIB  2006-08-22            0.004307        1.318464e-08
33  CIB  2006-08-23            0.006848        4.271603e-07
34  SKO  2002-02-08            0.059888        4.294716e-02
35  SKO  2002-02-11            0.061167        4.417510e-02
36  SKO  2002-02-12            0.061014        4.901067e-02
37  SKO  2002-02-13            0.055221        4.730380e-02
38  SKO  2002-02-14            0.061628        4.659908e-02
39  PAG  2007-07-03            0.005586        3.715073e-08
40  PAG  2007-07-05            0.008058        1.047600e-06
41  PAG  2007-07-06            0.007521        5.312208e-07
42  PAG  2007-07-09            0.007519        4.822871e-07
43  PAG  2007-07-10            0.006269        7.045978e-07
44  RBK  2002-02-08            0.119141        1.042945e-01
45  RBK  2002-02-11            0.128607        1.035408e-01
46  RBK  2002-02-12            0.131435        1.201102e-01
47  WGO  2002-02-08            0.109837        9.434335e-02
48  WGO  2002-02-11            0.094459        9.186975e-02
49  WGO  2002-02-12            0.109614        8.576695e-02
50  WGO  2002-02-13            0.110616        1.103984e-01
51  WGO  2002-02-14            0.116418        1.109057e-01
52  WAG  2002-02-08            0.059161        6.123514e-02
53  WAG  2002-02-11            0.067877        7.072952e-02
54  WAG  2002-02-12            0.073040        7.199622e-02
55  WAG  2002-02-13            0.060819        5.949971e-02

The situation is very similar to the great breakthrough of Black, Scholes and Merton whose model had a constant volatility assumption which everyone knew was not correct and did not hold in the sense that our alpha parameter is in theory constant per stock but the estimated alpha from last trade volatility surface is not constant.  A possibility that I might explore later is a third stochastic process for alpha but for now we have a clear model that is closed form, almost as fast as Heston model, and that improves the fit of Heston model overwhelmingly often (using last trades and not using bid-ask averages of option prices).
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Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON MODEL IN MORE THAN 90% IN 56 CASES
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Message-ID: <CAOKUxJBApka+wV4-R2nY6UkNdDF36=nFw7otMYBTodK=XB1+zQ@mail.gmail.com>
Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON
 MODEL IN MORE THAN 90% IN 56 CASES
From: "zulfikar.ahmed@gmail.com" <zulfikar.ahmed@gmail.com>
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Cc:
Date: Tue, 22 Mar 2016 11:09:16 -0700
Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON MODEL IN MORE THAN 90% IN 56 CASES

 

​Ladies and Gentlemen,

I am pleased to announce that our time-fractional Heston model fits the volatility surfaces determined by the LAST TRADE of call options statistically significantly better than the Heston model in 51/55 of the cases (stock/day) that I have checked last night.  The model differs from the Heston model by a single extra parameter 0<alpha<2.  The comparison is to calibrate both the Heston and Zulf models on a volatility surface and record the error of fit (sum of absolute difference of implied volatilities). The paired t-test of these errors give:

>>> ttest_rel(x.ix[:,2],x.ix[:,3])
(array(4.27139160907284), 7.7558339944524236e-05)

Here is the actual table of errors

    UPL  2003-01-02  0.0957002682349182  0.0773492111916326
0   UPL  2003-01-03            0.078657        6.652813e-02
1   UPL  2003-01-06            0.069270        5.893240e-02
2   KMI  2002-02-08            0.007780        6.266435e-03
3   KMI  2002-02-11            0.008966        5.994275e-03
4   KMI  2002-02-12            0.008309        5.424351e-03
5   KMI  2002-02-13            0.007713        5.128947e-03
6   KMI  2002-02-14            0.005948        5.272464e-03
7   PAC  2006-11-21            0.000290        1.745138e-07
8   PAC  2006-11-22            0.000558        3.908786e-07
9   NFG  2003-01-02            0.088293        8.326848e-02
10  NFG  2003-01-03            0.092200        1.840495e-02
11  NFG  2003-01-06            0.082446        6.947264e-02
12  NFG  2003-01-07            0.089481        1.946255e-02
13  NFG  2003-01-08            0.091463        8.470383e-02
14  OXY  2002-02-08            0.034763        3.187211e-02
15  OXY  2002-02-11            0.029676        3.690711e-02
16  OXY  2002-02-12            0.022510        1.772629e-02
17  OXY  2002-02-13            0.017657        2.116841e-02
18  OXY  2002-02-14            0.024343        3.202859e-02
19  AEM  2002-02-08            0.123033        1.217102e-01
20  AEM  2002-02-11            0.110734        6.377745e-02
21  AEM  2002-02-12            0.100758        9.943284e-02
22  AEM  2002-02-13            0.088986        8.803400e-02
23  AEM  2002-02-14            0.103508        9.958264e-02
24  NWY  2006-10-11            0.005319        4.450291e-08
25  NWY  2006-10-12            0.001743        1.144806e-07
26  NWY  2006-10-13            0.004078        2.571337e-06
27  NWY  2006-10-16            0.004726        3.099954e-03
28  TEF  2002-02-08            0.010447        5.958070e-03
29  TEF  2002-02-11            0.009764        8.375140e-03
30  CIB  2006-08-18            0.001899        6.670807e-07
31  CIB  2006-08-21            0.001459        8.779171e-09
32  CIB  2006-08-22            0.004307        1.318464e-08
33  CIB  2006-08-23            0.006848        4.271603e-07
34  SKO  2002-02-08            0.059888        4.294716e-02
35  SKO  2002-02-11            0.061167        4.417510e-02
36  SKO  2002-02-12            0.061014        4.901067e-02
37  SKO  2002-02-13            0.055221        4.730380e-02
38  SKO  2002-02-14            0.061628        4.659908e-02
39  PAG  2007-07-03            0.005586        3.715073e-08
40  PAG  2007-07-05            0.008058        1.047600e-06
41  PAG  2007-07-06            0.007521        5.312208e-07
42  PAG  2007-07-09            0.007519        4.822871e-07
43  PAG  2007-07-10            0.006269        7.045978e-07
44  RBK  2002-02-08            0.119141        1.042945e-01
45  RBK  2002-02-11            0.128607        1.035408e-01
46  RBK  2002-02-12            0.131435        1.201102e-01
47  WGO  2002-02-08            0.109837        9.434335e-02
48  WGO  2002-02-11            0.094459        9.186975e-02
49  WGO  2002-02-12            0.109614        8.576695e-02
50  WGO  2002-02-13            0.110616        1.103984e-01
51  WGO  2002-02-14            0.116418        1.109057e-01
52  WAG  2002-02-08            0.059161        6.123514e-02
53  WAG  2002-02-11            0.067877        7.072952e-02
54  WAG  2002-02-12            0.073040        7.199622e-02
55  WAG  2002-02-13            0.060819        5.949971e-02

The situation is very similar to the great breakthrough of Black, Scholes and Merton whose model had a constant volatility assumption which everyone knew was not correct and did not hold in the sense that our alpha parameter is in theory constant per stock but the estimated alpha from last trade volatility surface is not constant.  A possibility that I might explore later is a third stochastic process for alpha but for now we have a clear model that is closed form, almost as fast as Heston model, and that improves the fit of Heston model overwhelmingly often (using last trades and not using bid-ask averages of option prices).
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Message-ID: <CAOKUxJBApka+wV4-R2nY6UkNdDF36=nFw7otMYBTodK=XB1+zQ@mail.gmail.com>
Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON
MODEL IN MORE THAN 90% IN 56 CASES
From: “zulfikar.ahmed@gmail.com” <zulfikar.ahmed@gmail.com>

slcooper@illinois.edu, james P <strivergreen@gmail.com>, bm@falconfone.com
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Date: Tue, 22 Mar 2016 11:09:16 -0700
Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON MODEL IN MORE THAN 90% IN 56 CASES

 

​Ladies and Gentlemen,

I am pleased to announce that our time-fractional Heston model fits the volatility surfaces determined by the LAST TRADE of call options statistically significantly better than the Heston model in 51/55 of the cases (stock/day) that I have checked last night.  The model differs from the Heston model by a single extra parameter 0<alpha<2.  The comparison is to calibrate both the Heston and Zulf models on a volatility surface and record the error of fit (sum of absolute difference of implied volatilities). The paired t-test of these errors give:

>>> ttest_rel(x.ix[:,2],x.ix[:,3])
(array(4.27139160907284), 7.7558339944524236e-05)

Here is the actual table of errors

    UPL  2003-01-02  0.0957002682349182  0.0773492111916326
0   UPL  2003-01-03            0.078657        6.652813e-02
1   UPL  2003-01-06            0.069270        5.893240e-02
2   KMI  2002-02-08            0.007780        6.266435e-03
3   KMI  2002-02-11            0.008966        5.994275e-03
4   KMI  2002-02-12            0.008309        5.424351e-03
5   KMI  2002-02-13            0.007713        5.128947e-03
6   KMI  2002-02-14            0.005948        5.272464e-03
7   PAC  2006-11-21            0.000290        1.745138e-07
8   PAC  2006-11-22            0.000558        3.908786e-07
9   NFG  2003-01-02            0.088293        8.326848e-02
10  NFG  2003-01-03            0.092200        1.840495e-02
11  NFG  2003-01-06            0.082446        6.947264e-02
12  NFG  2003-01-07            0.089481        1.946255e-02
13  NFG  2003-01-08            0.091463        8.470383e-02
14  OXY  2002-02-08            0.034763        3.187211e-02
15  OXY  2002-02-11            0.029676        3.690711e-02
16  OXY  2002-02-12            0.022510        1.772629e-02
17  OXY  2002-02-13            0.017657        2.116841e-02
18  OXY  2002-02-14            0.024343        3.202859e-02
19  AEM  2002-02-08            0.123033        1.217102e-01
20  AEM  2002-02-11            0.110734        6.377745e-02
21  AEM  2002-02-12            0.100758        9.943284e-02
22  AEM  2002-02-13            0.088986        8.803400e-02
23  AEM  2002-02-14            0.103508        9.958264e-02
24  NWY  2006-10-11            0.005319        4.450291e-08
25  NWY  2006-10-12            0.001743        1.144806e-07
26  NWY  2006-10-13            0.004078        2.571337e-06
27  NWY  2006-10-16            0.004726        3.099954e-03
28  TEF  2002-02-08            0.010447        5.958070e-03
29  TEF  2002-02-11            0.009764        8.375140e-03
30  CIB  2006-08-18            0.001899        6.670807e-07
31  CIB  2006-08-21            0.001459        8.779171e-09
32  CIB  2006-08-22            0.004307        1.318464e-08
33  CIB  2006-08-23            0.006848        4.271603e-07
34  SKO  2002-02-08            0.059888        4.294716e-02
35  SKO  2002-02-11            0.061167        4.417510e-02
36  SKO  2002-02-12            0.061014        4.901067e-02
37  SKO  2002-02-13            0.055221        4.730380e-02
38  SKO  2002-02-14            0.061628        4.659908e-02
39  PAG  2007-07-03            0.005586        3.715073e-08
40  PAG  2007-07-05            0.008058        1.047600e-06
41  PAG  2007-07-06            0.007521        5.312208e-07
42  PAG  2007-07-09            0.007519        4.822871e-07
43  PAG  2007-07-10            0.006269        7.045978e-07
44  RBK  2002-02-08            0.119141        1.042945e-01
45  RBK  2002-02-11            0.128607        1.035408e-01
46  RBK  2002-02-12            0.131435        1.201102e-01
47  WGO  2002-02-08            0.109837        9.434335e-02
48  WGO  2002-02-11            0.094459        9.186975e-02
49  WGO  2002-02-12            0.109614        8.576695e-02
50  WGO  2002-02-13            0.110616        1.103984e-01
51  WGO  2002-02-14            0.116418        1.109057e-01
52  WAG  2002-02-08            0.059161        6.123514e-02
53  WAG  2002-02-11            0.067877        7.072952e-02
54  WAG  2002-02-12            0.073040        7.199622e-02
55  WAG  2002-02-13            0.060819        5.949971e-02

The situation is very similar to the great breakthrough of Black, Scholes and Merton whose model had a constant volatility assumption which everyone knew was not correct and did not hold in the sense that our alpha parameter is in theory constant per stock but the estimated alpha from last trade volatility surface is not constant.  A possibility that I might explore later is a third stochastic process for alpha but for now we have a clear model that is closed form, almost as fast as Heston model, and that improves the fit of Heston model overwhelmingly often (using last trades and not using bid-ask averages of option prices).
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Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON
 MODEL IN MORE THAN 90% IN 56 CASES
From: "zulfikar.ahmed@gmail.com" <zulfikar.ahmed@gmail.com>
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Cc:
Date: Tue, 22 Mar 2016 11:09:16 -0700
Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON MODEL IN MORE THAN 90% IN 56 CASES

 

​Ladies and Gentlemen,

I am pleased to announce that our time-fractional Heston model fits the volatility surfaces determined by the LAST TRADE of call options statistically significantly better than the Heston model in 51/55 of the cases (stock/day) that I have checked last night.  The model differs from the Heston model by a single extra parameter 0<alpha<2.  The comparison is to calibrate both the Heston and Zulf models on a volatility surface and record the error of fit (sum of absolute difference of implied volatilities). The paired t-test of these errors give:

>>> ttest_rel(x.ix[:,2],x.ix[:,3])
(array(4.27139160907284), 7.7558339944524236e-05)

Here is the actual table of errors

    UPL  2003-01-02  0.0957002682349182  0.0773492111916326
0   UPL  2003-01-03            0.078657        6.652813e-02
1   UPL  2003-01-06            0.069270        5.893240e-02
2   KMI  2002-02-08            0.007780        6.266435e-03
3   KMI  2002-02-11            0.008966        5.994275e-03
4   KMI  2002-02-12            0.008309        5.424351e-03
5   KMI  2002-02-13            0.007713        5.128947e-03
6   KMI  2002-02-14            0.005948        5.272464e-03
7   PAC  2006-11-21            0.000290        1.745138e-07
8   PAC  2006-11-22            0.000558        3.908786e-07
9   NFG  2003-01-02            0.088293        8.326848e-02
10  NFG  2003-01-03            0.092200        1.840495e-02
11  NFG  2003-01-06            0.082446        6.947264e-02
12  NFG  2003-01-07            0.089481        1.946255e-02
13  NFG  2003-01-08            0.091463        8.470383e-02
14  OXY  2002-02-08            0.034763        3.187211e-02
15  OXY  2002-02-11            0.029676        3.690711e-02
16  OXY  2002-02-12            0.022510        1.772629e-02
17  OXY  2002-02-13            0.017657        2.116841e-02
18  OXY  2002-02-14            0.024343        3.202859e-02
19  AEM  2002-02-08            0.123033        1.217102e-01
20  AEM  2002-02-11            0.110734        6.377745e-02
21  AEM  2002-02-12            0.100758        9.943284e-02
22  AEM  2002-02-13            0.088986        8.803400e-02
23  AEM  2002-02-14            0.103508        9.958264e-02
24  NWY  2006-10-11            0.005319        4.450291e-08
25  NWY  2006-10-12            0.001743        1.144806e-07
26  NWY  2006-10-13            0.004078        2.571337e-06
27  NWY  2006-10-16            0.004726        3.099954e-03
28  TEF  2002-02-08            0.010447        5.958070e-03
29  TEF  2002-02-11            0.009764        8.375140e-03
30  CIB  2006-08-18            0.001899        6.670807e-07
31  CIB  2006-08-21            0.001459        8.779171e-09
32  CIB  2006-08-22            0.004307        1.318464e-08
33  CIB  2006-08-23            0.006848        4.271603e-07
34  SKO  2002-02-08            0.059888        4.294716e-02
35  SKO  2002-02-11            0.061167        4.417510e-02
36  SKO  2002-02-12            0.061014        4.901067e-02
37  SKO  2002-02-13            0.055221        4.730380e-02
38  SKO  2002-02-14            0.061628        4.659908e-02
39  PAG  2007-07-03            0.005586        3.715073e-08
40  PAG  2007-07-05            0.008058        1.047600e-06
41  PAG  2007-07-06            0.007521        5.312208e-07
42  PAG  2007-07-09            0.007519        4.822871e-07
43  PAG  2007-07-10            0.006269        7.045978e-07
44  RBK  2002-02-08            0.119141        1.042945e-01
45  RBK  2002-02-11            0.128607        1.035408e-01
46  RBK  2002-02-12            0.131435        1.201102e-01
47  WGO  2002-02-08            0.109837        9.434335e-02
48  WGO  2002-02-11            0.094459        9.186975e-02
49  WGO  2002-02-12            0.109614        8.576695e-02
50  WGO  2002-02-13            0.110616        1.103984e-01
51  WGO  2002-02-14            0.116418        1.109057e-01
52  WAG  2002-02-08            0.059161        6.123514e-02
53  WAG  2002-02-11            0.067877        7.072952e-02
54  WAG  2002-02-12            0.073040        7.199622e-02
55  WAG  2002-02-13            0.060819        5.949971e-02

The situation is very similar to the great breakthrough of Black, Scholes and Merton whose model had a constant volatility assumption which everyone knew was not correct and did not hold in the sense that our alpha parameter is in theory constant per stock but the estimated alpha from last trade volatility surface is not constant.  A possibility that I might explore later is a third stochastic process for alpha but for now we have a clear model that is closed form, almost as fast as Heston model, and that improves the fit of Heston model overwhelmingly often (using last trades and not using bid-ask averages of option prices).
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Cc:
Date: Tue, 22 Mar 2016 11:09:16 -0700
Subject: ZULF MODEL FITS LAST TRADE VOLATILITY SURFACE BETTER THAN HESTON MODEL IN MORE THAN 90% IN 56 CASES

 

​Ladies and Gentlemen,

I am pleased to announce that our time-fractional Heston model fits the volatility surfaces determined by the LAST TRADE of call options statistically significantly better than the Heston model in 51/55 of the cases (stock/day) that I have checked last night.  The model differs from the Heston model by a single extra parameter 0<alpha<2.  The comparison is to calibrate both the Heston and Zulf models on a volatility surface and record the error of fit (sum of absolute difference of implied volatilities). The paired t-test of these errors give:

>>> ttest_rel(x.ix[:,2],x.ix[:,3])
(array(4.27139160907284), 7.7558339944524236e-05)

Here is the actual table of errors

    UPL  2003-01-02  0.0957002682349182  0.0773492111916326
0   UPL  2003-01-03            0.078657        6.652813e-02
1   UPL  2003-01-06            0.069270        5.893240e-02
2   KMI  2002-02-08            0.007780        6.266435e-03
3   KMI  2002-02-11            0.008966        5.994275e-03
4   KMI  2002-02-12            0.008309        5.424351e-03
5   KMI  2002-02-13            0.007713        5.128947e-03
6   KMI  2002-02-14            0.005948        5.272464e-03
7   PAC  2006-11-21            0.000290        1.745138e-07
8   PAC  2006-11-22            0.000558        3.908786e-07
9   NFG  2003-01-02            0.088293        8.326848e-02
10  NFG  2003-01-03            0.092200        1.840495e-02
11  NFG  2003-01-06            0.082446        6.947264e-02
12  NFG  2003-01-07            0.089481        1.946255e-02
13  NFG  2003-01-08            0.091463        8.470383e-02
14  OXY  2002-02-08            0.034763        3.187211e-02
15  OXY  2002-02-11            0.029676        3.690711e-02
16  OXY  2002-02-12            0.022510        1.772629e-02
17  OXY  2002-02-13            0.017657        2.116841e-02
18  OXY  2002-02-14            0.024343        3.202859e-02
19  AEM  2002-02-08            0.123033        1.217102e-01
20  AEM  2002-02-11            0.110734        6.377745e-02
21  AEM  2002-02-12            0.100758        9.943284e-02
22  AEM  2002-02-13            0.088986        8.803400e-02
23  AEM  2002-02-14            0.103508        9.958264e-02
24  NWY  2006-10-11            0.005319        4.450291e-08
25  NWY  2006-10-12            0.001743        1.144806e-07
26  NWY  2006-10-13            0.004078        2.571337e-06
27  NWY  2006-10-16            0.004726        3.099954e-03
28  TEF  2002-02-08            0.010447        5.958070e-03
29  TEF  2002-02-11            0.009764        8.375140e-03
30  CIB  2006-08-18            0.001899        6.670807e-07
31  CIB  2006-08-21            0.001459        8.779171e-09
32  CIB  2006-08-22            0.004307        1.318464e-08
33  CIB  2006-08-23            0.006848        4.271603e-07
34  SKO  2002-02-08            0.059888        4.294716e-02
35  SKO  2002-02-11            0.061167        4.417510e-02
36  SKO  2002-02-12            0.061014        4.901067e-02
37  SKO  2002-02-13            0.055221        4.730380e-02
38  SKO  2002-02-14            0.061628        4.659908e-02
39  PAG  2007-07-03            0.005586        3.715073e-08
40  PAG  2007-07-05            0.008058        1.047600e-06
41  PAG  2007-07-06            0.007521        5.312208e-07
42  PAG  2007-07-09            0.007519        4.822871e-07
43  PAG  2007-07-10            0.006269        7.045978e-07
44  RBK  2002-02-08            0.119141        1.042945e-01
45  RBK  2002-02-11            0.128607        1.035408e-01
46  RBK  2002-02-12            0.131435        1.201102e-01
47  WGO  2002-02-08            0.109837        9.434335e-02
48  WGO  2002-02-11            0.094459        9.186975e-02
49  WGO  2002-02-12            0.109614        8.576695e-02
50  WGO  2002-02-13            0.110616        1.103984e-01
51  WGO  2002-02-14            0.116418        1.109057e-01
52  WAG  2002-02-08            0.059161        6.123514e-02
53  WAG  2002-02-11            0.067877        7.072952e-02
54  WAG  2002-02-12            0.073040        7.199622e-02
55  WAG  2002-02-13            0.060819        5.949971e-02

The situation is very similar to the great breakthrough of Black, Scholes and Merton whose model had a constant volatility assumption which everyone knew was not correct and did not hold in the sense that our alpha parameter is in theory constant per stock but the estimated alpha from last trade volatility surface is not constant.  A possibility that I might explore later is a third stochastic process for alpha but for now we have a clear model that is closed form, almost as fast as Heston model, and that improves the fit of Heston model overwhelmingly often (using last trades and not using bid-ask averages of option prices).
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