My model for our actual universe is a hypersurface of a round four-sphere. In general, for generic metrics on closed spin three-manifolds, there are no harmonic spinors by anghel-genericvanishingtwisteddiracharmonicspinorsammann-dahl-humbert-surgeryharmonicspinors. Thus for generic hypersurfaces of the four-sphere, the kernel of the Dirac operator is zero, i.e. there are no massless fermions. So I think that the recent discovery of apparently massless Weyl fermions is probably a repetition of the error when neutrinos were considered massless on discovery. I don’t believe that there can be massless fermions in the actual universe.
Here is Chen-Kumagai two sided bound for jump diffusion with some assumption on the jump kernel similar to that of -stable jump . We use to denote two-sided bound. Suppose is the jump kernel where and . Then
For a Feller process with a pseudodifferential generator with symbol satisfying conditions and we can bound all moments of the exit time from a ball of radius using the Lemma 4.1 of Schilling’s 1998 paper schilling1998-growthholderregularityfellerprocesses. Let note that and then use the bound
Then recall that the norm can be expressed as and integrate the above bound.
This can be proved by Chern classes using the fact that the complexified tangent bundle of all spheres are trivial and the normal bundle of an oriented hypersurface is trivial. Thankfully the details are worked out here: The Triviality of the Complexified Tangent Bundles of Some Manifolds. Of course the nontrivial claim is that the actual universe is such a hypersurface but this is my hypothesis based on such issues as a thermal equilibrium for the cosmic background which defied Gaussian bounds for heat kernels and also the fact that the redshift slope is perfectly explained by an S4 geometry (predicted by it) and the fact that quasicrystals are parsimoniously explained as four-dimensional crystals.
Volatility is modeled as a mean-reverting process employing the argument that if not and there is a law that it had followed for the past century then it would be much larger. This is the same argument used to justify interest rates as mean-reverting. The ‘classical’ models of volatility and interest rates had been the Cox-Ingersoll-Ross (1985)/Feller (1951) process and the Ornstein-Uhlenbeck process, i.e. the models and . The former has become much more popular for volatility models because of the closed-form option pricing by Heston (1993) model. Eraker (2003) determined that jumps in volatility are necessary in addition to jumps in the price process, so volatility models must have jumps in addition to the mean reverting diffusion but this had not become standardized and in fact I was totally confused about this aspect simply because Heston and Bates models had been well-known. An affine class closed form model with jumps in volatility was published by Duffie-Pan-Singleton in 2000 that I have used to test which has excellent fits to the volatility surface and whose implementation is a minor variation of the Heston/Bates models. I had shown recently that in addition to jumps time-change by the inverse stable subordinator are necessary to fit volatility. Now we take the next step, which is to consider the problem of multivariate volatility models with jumps and inverse stable time change. The method I had used to fit the univariate volatility models is by minimizing the error of fit of the characteristic function of the theoretical model with the empirical characteristic function. Recall that for a series the empirical characteristic function is simply . This definition is valid for being vector quantities so we have to work with the theoretical multivariate characteristic function. The multidimensional analogue of the square root process is the Wishart process. It is the process governed by
A formula for the Laplace transform (and therefore the characteristic function) was derived for this matrix process by Bru (1991). More details for this process can be found here: AlessandroGnoatto and an option pricing model where the volatility follows the Wishart process can be found here: FonsecaWishartVolatilityModel. Under time change by an absolutely continuous time change of volatility the process satisfies
Now the Bru (1991) Laplace transform formula is given in laplacewishart
Now for the time-change we just insert the stopping time and integrate against the density of the inverse stable subordinator.
The Black-Scholes model was an intellectual breakthrough because for the first time the idea was established that rational prices exist for options independent of investor preferences but it was never a scientific model in the sense of quantitative mathematical models fitting actual data. Since 1973 there have been amazing improvements of the model by extensions of which the most promising seemed to be the stochastic volatility type models of which the most prominent for their reasonable fit to the implied volatility surfaces were the Heston (1993) model and the Bates (1997) and there are others as well. What we do here is turn the issues on their head and ask for an exact model for stochastic volatility and then return to the Black-Scholes argument for a partial differential equation. Heston had chosen to model stochastic volatility by a Feller (1951)/Cox-Ingersoll-Ross (1975) square root process and Bates had followed Merton (1976) by adding normally distributed jumps at Poisson random times. We show that an almost exact fit to volatility proxied by result from the model of a square root process with normally distributed jumps subordinated to inverse stable waiting times. Regardless of option pticing theory, this can be considered then as an exact scientific model for stochastic volatility and serve as the basis for a partial differential equation that must be satisfied for option prices. In Ito form time-change by inverse stable subordinator can be considered by change of density of the terms of the Ito equation for a square root process with jumps:
where and is the density of the inverse stable subordinator.
The diffusion component Ito formula can be used in a straightforward manner. The jump component is more complicated. There is an Ito formula with jumps for example Theorem 4.31 of menaldi-book-06-2013, but jumps were already a nontrivial problem when Robert Merton considered them in Merton1976 one way to deal with option pricing with jumps he considered is summing over jumps for log-normal jump distribution:
We used the jump term in the characteristic probability distribution of David Bates modified to account for the time-change. Bates’ jump term is described here (Bates_Scott):
The Bates PDE (non time-changed case) is standard and here is a useful version of how to deal with the jump term in the PDE.
Recall that a stable subordinator is an increasing Levy process whose density with respect to Lebesgue measure has Laplace transform ; the inverse stable subordinator and more details can be found in Meerschaert et. al. We fit a number of reasonable models of volatility that consider jumps and time changes from the basic square root diffusion and find overwhelming evidence that the combination of normally distributed jumps and inverse stable time change produce tight fits to empirical volatility of stocks. Next, the Ito formula above allows us to repeat the Black-Scholes-Heston standard argument for the partial differential equation that must be solved by assets: let be the asset price, consider the portfolio which we assume is riskless so . Now use Ito formula on and eliminate the stochasticity of by setting and and we obtain
The form of this partial differential equation is such that one can use the Heston method exactly. Recall that Heston solves for the characteristic functions of probabilities such that the plain vanilla call has price and finds these in the form where . The partial differential equation we obtain can be solved by replacing Heston’s solution with . In particular, for our claimed exact models for stochastic volatility, the closed form solution of Heston extends simply and thus we do not lose tractability. Now for the empirical results that provide evidence for our model:
The characteristic function of the square root process as well as the density is available in closed form and can be found for example in Mendoza-Arriega and Lintesky (2014). We derive the jump version using Fourier tranforms conditioned by number of jumps. We use Carr-Wu (2004) CarrWu2004-TCLP Theorem 1 to evaluate the characteristic function of time-change by evaluation of the characteristic function at the Laplace transform of the density of inverse stable subordinator. We fit these theoretical characteristic functions to empirical characteristic functions of using the weight , which is a standard technique (see Jiang-Knight-ECF and references for history of the method) The following are errors of fits to the models using 1000 days of daily return data per stock and shows that the square root with jumps and time-change fit the data extremely well.
The full code to generate these fits is the following:
def mittag_leffler_autocorrelation(lag, a, b, c, d, e):
s1 = b*t
s2 = c*t
numerator = a*t**(2.*a)*beta(a,a+1,s/t)
numerator = – abs(t/s)**a + a*abs(s)**(2.*a)*beta(a,a+1.)
denominator = abs(s*t)**a
denominator *= (2.*gamma(1+a)**2)/gamma(2*a+1.) – 1)
return d*exact_corr_func(a,s1,t) + e*exact_corr_func(a,s2,t)