Transition probability of Brownian motion in the octant and its application to default modeling

We derive a semi-analytic formula for the transition probability of three-dimensional Brownian motion in the positive octant with absorption at the boundaries. Separation of variables in spherical coordinates leads to an eigenvalue problem for the resulting boundary value problem in the two angular components. The main theoretical result is a solution to the original problem expressed as an expansion into special functions and an eigenvalue which has to be chosen to allow a matching of the boundary condition. We discuss and test several computational methods to solve a finite-dimensional approximation to this nonlinear eigenvalue problem. Finally, we apply our results to the computation of default probabilities and credit valuation adjustments in a structural credit model with mutual liabilities

High dimensional American options

Pricing single asset American options is a hard problem in mathematical finance. There are no closed form solutions available (apart from in the case of the perpetual option), so many approximations and numerical techniques have been developed. Pricing multi–asset (high dimensional) American options is still more difficult. We extend the method proposed theoretically by Glasserman and Yu (2004) by employing regression basis functions that are martingales under geometric Brownian motion. This results in more accurate Monte Carlo simulations, and computationally cheap lower and upper bounds to the American option price. We have implemented these models in QuantLib, the open–source derivatives pricing library. The code for many of the models discussed in this thesis can be downloaded from quantlib.org as part of a practical pricing and risk management library. We propose a new type of multi–asset option, the “Radial Barrier Option” for which we find analytic solutions. This is a barrier style option that pays out when a barrier, which is a function of the assets and their correlations, is hit. This is a useful benchmark test case for Monte Carlo simulations and may be of use in approximating multi–asset American options. We use Laplace transforms in this analysis which can be applied to give analytic results for the hitting times of Bessel processes. We investigate the asymptotic solution of the single asset Black–Scholes–Merton equation in the case of low volatility. This analysis explains the success of some American option approximations, and has the potential to be extended to basket options

Meshless Methods for Option Pricing and Risks Computation

In this thesis we price several financial derivatives by means of radial basis functions. Our main contribution consists in extending the usage of said numerical methods to the pricing of more complex derivatives - such as American and basket options with barriers - and in computing the associated risks. First, we derive the mathematical expressions for the prices and the Greeks of given options; next, we implement the corresponding numerical algorithm in MATLAB and calculate the results. We compare our results to the most common techniques applied in practice such as Finite Differences and Monte Carlo methods. We mostly use real data as input for our examples. We conclude radial basis functions offer a valid alternative to current pricing methods, especially because of the efficiency deriving from the free, direct calculation of risks during the pricing process. Eventually, we provide suggestions for future research by applying radial basis function for an implied volatility surface reconstruction

Computational Studies of Microscopic Superfluidity in the 4He Clusters

The physics that result in the decoupling of a molecule from a bosonic solvent at 0 K are studied. Fixed-node diffusion Monte Carlo (FNDMC) coupled with a Genetic Algorithm is used to perform simulations of the bosonic droplets doped with various molecules. The efficacy and accuracy of this approach is tested on a strongly coupled 2-dimensional quartic oscillator with excellent results. This algorithm is then applied to 4He-CO and 4He-HCN clusters respectively in an effort to determine the factors that result in the onset of microscopic superfluidity. The decoupling of the doped molecule from the bosonic solvent is found to be, primarily, a result of the combined effect of the repulsive interaction between the helium atoms and bose symmetry. The effects of rotor size versus molecular anisotropy in a NH3 molecule seeded into a 4He droplet is studied as well. Simulations are done using the accurate rotational constants (B0=9.945 cm-1, C0=6.229 cm-1) and using fudged versions of the rotational constants (Bfudged=0.9945 cm-1, Cfudged=0.6229 cm-1) for the |0011〉state. The simulations done with the fudged rotational constants experience a slightly smaller reduction than those done using the accurate rotational constants. This is attributed to the importance of molecular anisotropy versus the size of larger rotational constants in molecules whose rotational constants fall in an intermediate regime

Evolution of oil droplets in a chemorobotic platform

Evolution, once the preserve of biology, has been widely emulated in software, while physically embodied systems that can evolve have been limited to electronic and robotic devices and have never been artificially implemented in populations of physically interacting chemical entities. Herein we present a liquid-handling robot built with the aim of investigating the properties of oil droplets as a function of composition via an automated evolutionary process. The robot makes the droplets by mixing four different compounds in different ratios and placing them in a Petri dish after which they are recorded using a camera and the behaviour of the droplets analysed using image recognition software to give a fitness value. In separate experiments, the fitness function discriminates based on movement, division and vibration over 21 cycles, giving successive fitness increases. Analysis and theoretical modelling of the data yields fitness landscapes analogous to the genotype–phenotype correlations found in biological evolution. , Trevor Hinkley, James Ward Taylor Kliment Yane

A Survey on Quantum Computational Finance for Derivatives Pricing and VaR

[Abstract]: We review the state of the art and recent advances in quantum computing applied to derivative pricing and the computation of risk estimators like Value at Risk. After a brief description of the financial derivatives, we first review the main models and numerical techniques employed to assess their value and risk on classical computers. We then describe some of the most popular quantum algorithms for pricing and VaR. Finally, we discuss the main remaining challenges for the quantum algorithms to achieve their potential advantages.Xunta de Galicia; ED431G 2019/01All authors acknowledge the European Project NExt ApplicationS of Quantum Computing (NEASQC), funded by Horizon 2020 Program inside the call H2020-FETFLAG-2020-01 (Grant Agreement 951821). Á. Leitao, A. Manzano and C. Vázquez wish to acknowledge the support received from the Centro de Investigación de Galicia “CITIC”, funded by Xunta de Galicia and the European Union (European Regional Development Fund- Galicia 2014-2020 Program), by Grant ED431G 2019/01

RBF approximation by partition of unity for valuation of options under exponential Lévy processes

International audienceThe prices of some European and American-style contracts on assets driven by a class of Markov processes containing, in particular, L\'{e}vy processes of pure jump type with infinite jump activity, are obtained numerically, as solutions of the partial integro-differential equations (PIDEs) they satisfy. This paper overcomes the ill-conditioning inherent in global meshfree methods by using localized RBF approximations known as the RBF partition of unity (RBF-PU) method for (PIDEs) arising in option pricing problems in L\'{e}vy driven assets. Then, Crank-Nicolson, LeapFrog (CNLF) is applied for time discretization. We treat the local term using an implicit step, and the nonlocal term using an explicit step, to avoid the inversion of the nonsparse matrix. For dealing with early exercise feature of American option and solving free boundary problem we use the implicit-explicit method combined with a penalty method. Efficiency and practical performance are demonstrated by numerical experiments for pricing European and American contracts. Suggested reviewers Mohan Kadalbajoo, Mehdi Dehghan, simon hubbert Submission Files Included in this PDF File Name [File Type] cover letter.pdf [Cover Letter] Research Highlights.pdf [Highlights] Levy_Fereshtian.pdf [Manuscript File] Submission Files Not Included in this PDF File Name [File Type] Paper_Levy.zip [LaTeX Source File] To view all the submission files, including those not included in the PDF, click on the manuscript title on your EVISE Homepage, then click 'Download zip file'

One-Class Classification: Taxonomy of Study and Review of Techniques

One-class classification (OCC) algorithms aim to build classification models when the negative class is either absent, poorly sampled or not well defined. This unique situation constrains the learning of efficient classifiers by defining class boundary just with the knowledge of positive class. The OCC problem has been considered and applied under many research themes, such as outlier/novelty detection and concept learning. In this paper we present a unified view of the general problem of OCC by presenting a taxonomy of study for OCC problems, which is based on the availability of training data, algorithms used and the application domains applied. We further delve into each of the categories of the proposed taxonomy and present a comprehensive literature review of the OCC algorithms, techniques and methodologies with a focus on their significance, limitations and applications. We conclude our paper by discussing some open research problems in the field of OCC and present our vision for future research.Comment: 24 pages + 11 pages of references, 8 figure