Market memory and fat tail consequences in option pricing on the expOU stochastic volatility model

The expOU stochastic volatility model is capable of reproducing fairly well most important statistical properties of financial markets daily data. Among them, the presence of multiple time scales in the volatility autocorrelation is perhaps the most relevant which makes appear fat tails in the return distributions. This paper wants to go further on with the expOU model we have studied in Ref. 1 by exploring an aspect of practical interest. Having as a benchmark the parameters estimated from the Dow Jones daily data, we want to compute the price for the European option. This is actually done by Monte Carlo, running a large number of simulations. Our main interest is to "see" the effects of a long-range market memory from our expOU model in its subsequent European call option. We pay attention to the effects of the existence of a broad range of time scales in the volatility. We find that a richer set of time scales brings to a higher price of the option. This appears in clear contrast to the presence of memory in the price itself which makes the price of the option cheaper.Comment: 9 pages, 4 figures, APFA5 Torin

Diversity and Arbitrage in a Regulatory Breakup Model

In 1999 Robert Fernholz observed an inconsistency between the normative assumption of existence of an equivalent martingale measure (EMM) and the empirical reality of diversity in equity markets. We explore a method of imposing diversity on market models by a type of antitrust regulation that is compatible with EMMs. The regulatory procedure breaks up companies that become too large, while holding the total number of companies constant by imposing a simultaneous merge of other companies. The regulatory events are assumed to have no impact on portfolio values. As an example, regulation is imposed on a market model in which diversity is maintained via a log-pole in the drift of the largest company. The result is the removal of arbitrage opportunities from this market while maintaining the market's diversity.Comment: 21 page

A Fast Mean-Reverting Correction to Heston's Stochastic Volatility Model

We propose a multi-scale stochastic volatility model in which a fast mean-reverting factor of volatility is built on top of the Heston stochastic volatility model. A singular pertubative expansion is then used to obtain an approximation for European option prices. The resulting pricing formulas are semi-analytic, in the sense that they can be expressed as integrals. Difficulties associated with the numerical evaluation of these integrals are discussed, and techniques for avoiding these difficulties are provided. Overall, it is shown that computational complexity for our model is comparable to the case of a pure Heston model, but our correction brings significant flexibility in terms of fitting to the implied volatility surface. This is illustrated numerically and with option data

Perturbed Copula: Introducing the skew effect in the co-dependence

Gaussian copulas are widely used in the industry to correlate two random variables when there is no prior knowledge about the co-dependence between them. The perturbed Gaussian copula approach allows introducing the skew information of both random variables into the co-dependence structure. The analytical expression of this copula is derived through an asymptotic expansion under the assumption of a common fast mean reverting stochastic volatility factor. This paper applies this new perturbed copula to the valuation of derivative products; in particular FX quanto options to a third currency. A calibration procedure to fit the skew of both underlying securities is presented. The action of the perturbed copula is interpreted compared to the Gaussian copula. A real worked example is carried out comparing both copulas and a local volatility model with constant correlation for varying maturities, correlations and skew configurations.Comment: 34 pages, 6 figures and 3 table

Calibration of the Distance Scale from Cepheids

We have used the infrared surface brightness technique to obtain a new absolute calibration of the Cepheid PL relation in optical and near-infrared bands from improved data on Galactic stars. The infrared surface brightness distances to the Galactic variables are consistent with direct interferometric Cepheid distance measurements, and with the PL calibration coming from Hipparcos parallaxes of nearby Cepheids, but are more accurate than these determinations. We find that in all bands, the Galactic Cepheid PL relation appears to be slightly, but significantly steeper than the corresponding relation defined by the LMC Cepheids. Since the slope of our LMC Cepheid sample is clearly better defined than the one of the much smaller Galactic sample, we fit the LMC slopes to our Galactic calibrating Cepheid sample (which introduces only a small uncertainty) to obtain our final, adopted and improved absolute calibrations of the Cepheid PL relations in the VIWJHK bands. Comparing the absolute magnitudes of 10-day period Cepheids in both galaxies which are only slightly affected by the different Galactic and LMC slopes of the PL relation, we derive values for the LMC distance modulus in all these bands which can be made to agree extremely well under reasonable assumptions for both, the reddening law, and the adopted reddenings of the LMC Cepheids. This yields, as our current best estimate from Cepheid variables, a LMC distance modulus of 18.55 +- 0.06.Comment: to be published in: "Stellar Candles", Lecture Notes in Physics (http://link.springer.de/series/lnpp