From Bachelier to Dupire via optimal transport

Famously, mathematical finance was started by Bachelier in his 1900 PhD thesis where – among many other achievements – he also provided a formal derivation of the Kolmogorov forward equation. This also forms the basis for Dupire’s (again formal) solution to the problem of finding an arbitrage-free model calibrated to a given volatility surface. The latter result has rigorous counterparts in the theorems of Kellerer and Lowther. In this survey article, we revisit these hallmarks of stochastic finance, highlighting the role played by some optimal transport results in this context.

Approximation of SDE solutions using local asymptotic expansions

We develop an asymptotic expansion for small time of the density of the solution of a non-degenerate system of stochastic differential equations with smooth coefficients, and apply this to the stepwise approximation of solutions. The asymptotic expansion, which takes the form of a multivariate Edgeworth-type expansion, is obtained from the Kolmogorov forward equation using some standard PDE results. To generate one step of the approximation to the solution, we use a Cornish–Fisher type expansion derived from the Edgeworth expansion. To interpret the approximation generated in this way as a strong approximation we use couplings between the (normal) random variables used and the Brownian path driving the SDE. These couplings are constructed using techniques from optimal transport and Vaserstein metrics. The type of approximation so obtained may be regarded as intermediate between a conventional strong approximation and a weak approximation. In principle approximations of any order can be obtained, though for higher orders the algebra becomes very heavy. In order 1/2 the method gives the usual Euler approximation; in order 1 it gives a variant of the Milstein method, but which needs only normal variables to be generated. However the method is somewhat limited by the non-degeneracy requirement.

An Incompressible Polymer Fluid Interacting with a Koiter Shell

We study a mutually coupled mesoscopic-macroscopic-shell system of equations modeling a dilute incompressible polymer fluid which is evolving and interacting with a flexible shell of Koiter type. The polymer constitutes a solvent-solute mixture where the solvent is modelled on the macroscopic scale by the incompressible Navier–Stokes equation and the solute is modelled on the mesoscopic scale by a Fokker–Planck equation (Kolmogorov forward equation) for the probability density function of the bead-spring polymer chain configuration. This mixture interacts with a nonlinear elastic shell which serves as a moving boundary of the physical spatial domain of the polymer fluid. We use the classical model by Koiter to describe the shell movement which yields a fully nonlinear fourth order hyperbolic equation. Our main result is the existence of a weak solution to the underlying system which exists until the Koiter energy degenerates or the flexible shell approaches a self-intersection.

Estimation of population genetic parameters using an EM algorithm and sequence data from experimental evolution populations

Motivation Evolve and resequence (E&R) experiments show promise in capturing real-time evolution at genome-wide scales, enabling the assessment of allele frequency changes SNPs in evolving populations and thus the estimation of population genetic parameters in the Wright–Fisher model (WF) that quantify the selection on SNPs. Currently, these analyses face two key difficulties: the numerous SNPs in E&R data and the frequent unreliability of estimates. Hence, a methodology for efficiently estimating WF parameters is needed to understand the evolutionary processes that shape genomes. Results We developed a novel method for estimating WF parameters (EMWER), by applying an expectation maximization algorithm to the Kolmogorov forward equation associated with the WF model diffusion approximation. EMWER was used to infer the effective population size, selection coefficients and dominance parameters from E&R data. Of the methods examined, EMWER was the most efficient method for selection strength estimation in multi-core computing environments, estimating both selection and dominance with accurate confidence intervals. We applied EMWER to E&R data from experimental Drosophila populations adapting to thermally fluctuating environments and found a common selection affecting allele frequency of many SNPs within the cosmopolitan In(3R)P inversion. Furthermore, this application indicated that many of beneficial alleles in this experiment are dominant. Availability and implementation Our C++ implementation of ‘EMWER’ is available at https://github.com/kojikoji/EMWER. Supplementary information Supplementary data are available at Bioinformatics online.

A FORWARD EQUATION FOR COMPUTING DERIVATIVES EXPOSURE

We derive a forward equation for computing the expected exposure of financial derivatives. Under general assumptions about the underlying diffusion process, we give an explicit decomposition of the exposure into an intrinsic value which can be directly deduced from the term structure of the forward mark-to-market, and a time value which expresses the variability of the future mark-to-market. Our approach is inspired by Dupire’s equation for local volatility and leads to an ordinary differential equation qualifying the evolution of the expected exposure with respect to the observation dates. We show how this approach can be linked with local times theory in dimension one and to the co-area formula in a higher dimension. As for numerical considerations, we show how this approach leads to an efficient numerical method in the case of one or two risk factors. The accuracy and time-efficiency of this forward representation in small dimension are of special interest in benchmarking XVA valuation adjustments at trade level.