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)
4:16 PM (0 minutes ago)
Ladies and Gentlemen,
Although the round four sphere itself has totally integrable geodesic flow, generic hypersurfaces of the sphere have positive topological entropy and therefore ergodic geodesic flow. This is a relatively easy corollary of a theorem that says that for compact manifolds an open dense set of generic metrics have positive topological entropy and therefore ergodic geodesic flow, so some analysis shows that generic embedding metrics into a four sphere (which should be for radius 1/h) which can be characterized by additional Gauss-Codazzi relations also have ergodic geodesic flow and should therefore fall under the quantum ergodicity theorems. The attached paper has the generic metrics result.
The key issue is how to squeeze the solution of a constant coefficient Riccati equation. For the Heston model the key point for getting an explicit solution for the probability density is by solving a Riccati equation . For the time-fractional case you want to solve instead where is the waiting time of the fractional Poisson process. The trick is to replace the time variable with . So you have to insert this expression into the explicit formula for the Heston probability density in DragulescuYakovenko2002 eq (18) say.
Einstein never liked quantum mechanics for very good reason. It’s an insane description of the world. I guess I am chasing a ‘hidden variable’ theory which I am sure is right because otherwise I would not be able to explain the redshift so accurately. The physical three-dimensional universe is a hypersurface (including the electromagnetic field) of a single fixed-radius four-sphere whose fixed radius gives rise to all quantization. I am confused about why Kaluza-Klein’s extra dimensions should be small. It makes little sense to me because quantization in the scale of would require a large circle of radius and not small circles. Quantum mechanics is some sort of mathematical gimmick with a Hilbert space and noncommuting operator observables that capture the small scale by localizing and linearizing at the tangent space of the physical universe in the small and then going berserk with eigenfunction expansion basis. On the other hand, it is frustrating to work in a vacuum on my interpretation of what makes sense. Kaluza-Klein is a guide but there is more going on because there is no -invariance in an S4 universe because there is a normal bundle but no principal -bundle. This should lead to experimental predictions correcting MAXWELL’S EQUATIONS which I no longer believe could be describing the universe exactly. Maxwell’s equations would be true if the extra structure were a principal circle-bundle with fibers of length but I cannot believe that this can be right because it should be an embedding where there are circles of the right length along the normal vector but they can intersect. It is the curvature of the ambient 4-sphere that should produce a ‘cosmological constant’ and also produce the correct level for redshift (not expansion) which are both the right order of magnitude. This is frustrating, because I am sure I am right but I am also living in the wrong age for this because a century of physics went the wrong way and got itself so confused with so much technical work that recovery seems impossible almost.