Pricing of Spread Options on a Bivariate Jump Market and Stability to Model Risk

Abstract

We study the pricing of spread options and we obtain a Margrabe-type formula for a bivariate jump-diffusion model. Moreover, we study the robustness of the price to model risk, in the sense that we consider two types of bivariate jump-diffusion models: one allowing for infinite activity small jumps and one not. In the second model, an adequate continuous component describes the small variation of prices. We illustrate our computations by several examples.     

The Minimal Entropy Martingale Measure and Numerical Option Pricing for the Barndorff–Nielsen–Shephard Stochastic Volatility Model

Abstract

We develop and apply a numerical scheme for pricing options in the stochastic volatility model proposed by Barndorff–Nielsen and Shephard. This non-Gaussian Ornstein–Uhlenbeck type of volatility model gives rise to an incomplete market, and we consider the option prices under the minimal entropy martingale measure. To numerically price options with respect to this risk neutral measure, one needs to consider a Black and Scholes type of partial differential equation, with an integro-term arising from the volatility process. We suggest finite difference schemes to solve this parabolic integro-partial differential equation, and derive appropriate boundary conditions for the finite difference method. As an application of our algorithm, we consider price deviations from the Black and Scholes formula for call options, and the implications of the stochastic volatility on the shape of the volatility smile.     

Determination of fluorine in uranium oxyfluoride particles as an indicator of particle age

As swipe samples from enrichment activities typically contain uranium particles with a detectable amount of fluorine, the question was raised whether the analysis of fluorine in particles could complement the information on the uranium isotope ratios.     

A PDE representation of the density of the minimal entropy martingale measure in stochastic volatility markets

Under general conditions stated in Rheinländer [An entropy approach to the stein/stein model with correlation. Preprint, 2003, ETH Zürich.], we prove that in a stochastic volatility market the Radon–Nikodym density of the minimal entropy martingale measure (MEMM) can be expressed in terms of the solution of a semilinear PDE. The semilinear PDE is suggested by the dynamic programming approach to the utility indifference pricing problem of contingent claims. One of our main results is the existence and uniqueness of a classical solution of the semilinear PDE in the case of a general stochastic volatility model with additive noise correlated with the asset price. Our results are applied to the Stein–Stein and Heston stochastic volatility models.     

Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations

We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton-Jacobi-Bellman equations by introducing a new notion of consistency. Our results are robust and general - they improve and extend earlier results by Krylov, Barles, and Jakobsen. We apply our general results to various schemes including Crank-Nicholson type finite difference schemes, splitting methods, and the classical approximation by piecewise constant controls. In the first two cases our results are new, and in the last two cases the results are obtained by a new method which we develop here.

     

Assessing market uncertainty by means of a time-varying intermittency parameter for asset price fluctuations

Maximum likelihood estimation techniques for multifractal processes are applied to high-frequency data in order to quantify intermittency in the fluctuations of asset prices. From time records as short as one month these methods permit extraction of a meaningful intermittency parameter λ$\lambda$ characterising the degree of volatility clustering. We can therefore study the time evolution of volatility clustering and test the statistical significance of this variability. By analysing data from the Oslo Stock Exchange, and comparing the results with the investment grade spread, we find that the estimates of λ$\lambda$ are lower at times of high market uncertainty.     

On Numerical Approximations of Fractional and Nonlocal Mean Field Games

We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.

     

Analytical strategies for successful enantioselective separation of atropisomeric polybrominated biphenyls 132 and 149 in environmental samples

Some of the polybrominated biphenyls (PBBs) found in the environment are axially chiral, due to hindered rotation about the interannular phenyl-phenyl bond. This applies for PBB congeners having two or more bromine substituents in ortho-position to this bond. In this study analytical methods were developed that allow determining the enantiomer fraction (EF) of axially chiral (atropisomeric) PBBs in environmental samples. A white-tailed sea eagle egg was used as test sample. The egg extract was purified and further fractionated by normal phase (NP) high performance liquid chromatography (HPLC), yielding enriched fractions of axially chiral PBB 132 and PBB 149. Gas chromatographic (GC) enantioseparation of the atropisomers of PBB 149 was achieved on one of nine tested modified cyclodextrin phases. Due to coelution with an unknown brominated compound, conventional GC/ECNI-MS, which is based on the detection of the bromide ion, did not allow for the establishment of the EF. However, by means of GC/EI-MS-MS it was possible to verify an EF of 0.42-0.43, i.e. a significant enantiomeric enrichment of the second eluting atropisomer of PBB 149 in the white-tailed sea eagle egg. This is the first proof of non-racemic proportions of a chiral PBB in environmental samples. Despite the testing of nine different chiral stationary phases, GC enantioseparation of PBB 132 or other atropisomeric PBB congeners failed. For this reason, an enantioselective reversed-phase HPLC method was developed. This method proved to be a powerful tool for the separation of PBB atropisomers. It was found that even a standard of the di-ortho substituted PBB 153 could be partially separated into atropisomers at 0 degrees C but already enantiomerized at 5 degrees C. For establishing the EF of PBB 132 in the bird egg sample a combination of enantioselective HPLC followed by non-chiral gas chromatography was employed. Using enantioselective HPLC, the atropisomers of PBB 132 were quantitatively targeted into two separate fractions at room temperature (20 degrees C). After addition of internal standards for volume adjustment the relative amounts of the atropisomers in the isolated fractions were quantified by using non-chiral GC/EI-MS analysis. A deviation from the racemic mixture of the atropisomers of PBB 132 in the egg extract could not be statistically proven.     

A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations

We formulate and prove a non-local “maximum principle for semicontinuous functions” in the setting of fully nonlinear and degenerate elliptic integro-partial differential equations with integro operators of second order. Similar results have been used implicitly by several researchers to obtain compare/uniqueness results for integro-partial differential equations, but proofs have so far been lacking.