Efficient pricing algorithms for exotic derivatives

Since the Nobel-prize winning papers of Black and Scholes and Merton in 1973, the derivatives market has evolved into a multi-trillion dollar market. Structures which were once considered as exotic are now commonplace, appearing in retail products such as mortgages and investment notes. At the same time, new and more complex structures are invented on a regular basis. To price and risk manage such products, a financial engineer will typically: (1) choose a model which is both economically plausible and analytically tractable, (2) calibrate the model to the prices of traded options, and (3) price the exotic option with the calibrated model, using appropriate numerical techniques. This thesis mainly deals with the second and third steps in this process. For the analytically tractable class of affine models, containing among others the Black-Scholes model and Heston’s stochastic volatility model, it deals with topics such as the robust pricing of European options via Fourier inversion, the pricing of Bermudan options using convolution based methods, the simulation of stochastic volatility models and the pricing of Asian options. A separate chapter deals with a completely different topic, the mathematical properties of the principal components of term structure data. Roger Lord (1977) holds cum laude Master’s degrees in both Applied Mathematics (Eindhoven University of Technology) and Econometrics (Tilburg University). After graduating he joined Cardano Risk Management in 2001 as a financial engineer. Deciding to pursue a PhD degree, he joined Erasmus University Rotterdam as a PhD candidate in 2003. Throughout his PhD he held a part-time position as a quantitative analyst at the Derivatives Research & Validation team of Rabobank International. He has published articles in Applied Mathematical Finance, the Journal of Computational Finance, Mathematical Finance, Quantitative Finance and SIAM Journal on Scientific Computing, and presented his research at several international conferences. Since October 2006 he joined Rabobank International’s Financial Engineering team in London as a quantitative analyst, developing front-office pricing models for interest rate derivatives

Optimal Fourier Inversion in Semi-analytical Option Pricing

At the time of writing this article, Fourier inversion is the computational method of choice for a fast and accurate calculation of plain vanilla option prices in models with an analytically available characteristic function. Shifting the contour of integration along the complex plane allows for different representations of the inverse Fourier integral. In this article, we present the optimal contour of the Fourier integral, taking into account numerical issues such as cancellation and explosion of the characteristic function. This allows for robust and fast option pricing for almost all levels of strikes and maturities

Level-Slope-Curvature - Fact or Artefact?

The first three factors resulting from a principal components analysis of term structure data are in the literature typically interpreted as driving the level, slope and curvature of the term structure. Using slight generalisations of theorems from total positivity, we present sufficient conditions under which level, slope and curvature are present. These conditions have the nice interpretation of restricting the level, slope and curvature of the correlation surface. It is proven that the Schoenmakers-Coffey correlation matrix also brings along such factors. Finally, we formulate and corroborate our conjecture that the order present in correlation matrices causes slope

Why the Rotation Count Algorithm Works

The characteristic functions of many affine jump-diffusion models, such as Heston’s stochastic volatility model and all of its extensions, involve multivalued functions such as the complex logarithm. If we restrict the logarithm to its principal branch, as is done in most software packages, the characteristic function can become discontinuous, leading to completely wrong option prices if options are priced by Fourier inversion. In this paper we prove under non-restrictive conditions on the parameters that the rotation count algorithm of Kahl and Jäckel chooses the correct branch of the complex logarithm. Under the same restrictions we prove that in an alternative formulation of the characteristic function the principal branch is the correct one. Seen as this formulation is easier to implement and numerically more stable than Heston’s formulation, it should be the preferred one. The remainder of this paper shows how complex discontinuities can be avoided in the Schöbel-Zhu model and the exact simulation algorithm of the Heston model, recently proposed by Broadie and Kaya. Finally, we show that Matytsin’s SVJJ model has a closed-form characteristic function, though the complex discontinuities that arise there due to the branch switching of the exponential integral cannot be avoided under all circumstances

A Comparison of Biased Simulation Schemes for Stochastic Volatility Models

When using an Euler discretisation to simulate a mean-reverting square root process, one runs into the problem that while the process itself is guaranteed to be nonnegative, the discretisation is not. Although an exact and efficient simulation algorithm exists for this process, at present this is not the case for the Heston stochastic volatility model, where the variance is modelled as a square root process. Consequently, when using an Euler discretisation, one must carefully think about how to fix negative variances. Our contribution is threefold. Firstly, we unify all Euler fixes into a single general framework. Secondly, we introduce the new full truncation scheme, tailored to minimise the upward bias found when pricing European options. Thirdly and finally, we numerically compare all Euler fixes to a recent quasi-second order scheme of Kahl and Jäckel and the exact scheme of Broadie and Kaya. The choice of fix is found to be extremely important. The full truncation scheme by far outperforms all biased schemes in terms of bias, root-mean-squared error, and hence should be the preferred discretisation method for simulation of the Heston model and extensions thereof

National Evaluation of the Partnerships for Older People Projects: Interim Report of Progress

This second interim report provides a summary of key findings from the National Evaluation of the Department of Health’s POPP Programme. These summary findings are based on data collected and analysed over the last two years of the POPP programme (April 2006 to March 2008) and are made available to support emerging learning around prevention and early intervention. As the majority of the pilot sites still have one year to run, these findings, outcomes and subsequent discussion may be subject to change. All the issues and evidence on which these findings are based will be made available in the Final Report of the National Evaluation to be published in Autumn 2009

Why the rotation count algorithm works

ABSTRACT The characteristic functions of many affine jump-diffusion models, such as Heston's stochastic volatility model and all of its extensions, involve multivalued functions such as the complex logarithm. If we restrict the logarithm to its principal branch, as is done in most software packages, the characteristic function can become discontinuous, leading to completely wrong option prices if options are priced by Fourier inversion. In this paper we prove under non-restrictive conditions on the parameters that the rotation count algorithm of Kahl and Jäckel chooses the correct branch of the complex logarithm. Under the same restrictions we prove that in an alternative formulation of the characteristic function the principal branch is the correct one. Seen as this formulation is easier to implement and numerically more stable than Heston's formulation, it should be the preferred one. The remainder of this paper shows how complex discontinuities can be avoided in the Schöbel-Zhu model and the exact simulation algorithm of the Heston model, recently proposed by Broadie and Kaya. Finally, we show that Matytsin's SVJJ model has a closed-form characteristic function, though the complex discontinuities that arise there due to the branch switching of the exponential integral cannot be avoided under all circumstances

Training of Instrumentalists and Development of New Technologies on SOFIA

This white paper is submitted to the Astronomy and Astrophysics 2010 Decadal Survey (Astro2010)1 Committee on the State of the Profession to emphasize the potential of the Stratospheric Observatory for Infrared Astronomy (SOFIA) to contribute to the training of instrumentalists and observers, and to related technology developments. This potential goes beyond the primary mission of SOFIA, which is to carry out unique, high priority astronomical research. SOFIA is a Boeing 747SP aircraft with a 2.5 meter telescope. It will enable astronomical observations anywhere, any time, and at most wavelengths between 0.3 microns and 1.6 mm not accessible from ground-based observatories. These attributes, accruing from the mobility and flight altitude of SOFIA, guarantee a wealth of scientific return. Its instrument teams (nine in the first generation) and guest investigators will do suborbital astronomy in a shirt-sleeve environment. The project will invest $10M per year in science instrument development over a lifetime of 20 years. This, frequent flight opportunities, and operation that enables rapid changes of science instruments and hands-on in-flight access to the instruments, assure a unique and extensive potential - both for training young instrumentalists and for encouraging and deploying nascent technologies. Novel instruments covering optical, infrared, and submillimeter bands can be developed for and tested on SOFIA by their developers (including apprentices) for their own observations and for those of guest observers, to validate technologies and maximize observational effectiveness.Comment: 10 pages, no figures, White Paper for Astro 2010 Survey Committee on State of the Professio

Characterization of the genomic features and expressed fusion genes in micropapillary carcinomas of the breast

Micropapillary carcinoma ( MPC ) is a rare histological special type of breast cancer, characterized by an aggressive clinical behaviour and a pattern of copy number aberrations ( CNAs ) distinct from that of grade‐ and oestrogen receptor ( ER )‐matched invasive carcinomas of no special type ( IC‐NSTs ). The aims of this study were to determine whether MPCs are underpinned by a recurrent fusion gene(s) or mutations in 273 genes recurrently mutated in breast cancer. Sixteen MPCs were subjected to microarray‐based comparative genomic hybridization ( aCGH ) analysis and Sequenom OncoCarta mutation analysis. Eight and five MPCs were subjected to targeted capture and RNA sequencing, respectively. aCGH analysis confirmed our previous observations about the repertoire of CNAs of MPCs . Sequencing analysis revealed a spectrum of mutations similar to those of luminal B IC‐NSTs , and recurrent mutations affecting mitogen‐activated protein kinase family genes and NBPF10 . RNA ‐sequencing analysis identified 17 high‐confidence fusion genes, eight of which were validated and two of which were in‐frame. No recurrent fusions were identified in an independent series of MPCs and IC‐NSTs . Forced expression of in‐frame fusion genes ( SLC2A1–FAF1 and BCAS4–AURKA ) resulted in increased viability of breast cancer cells. In addition, genomic disruption of CDK12 caused by out‐of‐frame rearrangements was found in one MPC and in 13% of HER2 ‐positive breast cancers, identified through a re‐analysis of publicly available massively parallel sequencing data. In vitro analyses revealed that CDK12 gene disruption results in sensitivity to PARP inhibition, and forced expression of wild‐type CDK12 in a CDK12 ‐null cell line model resulted in relative resistance to PARP inhibition. Our findings demonstrate that MPCs are neither defined by highly recurrent mutations in the 273 genes tested, nor underpinned by a recurrent fusion gene. Although seemingly private genetic events, some of the fusion transcripts found in MPCs may play a role in maintenance of a malignant phenotype and potentially offer therapeutic opportunities. © 2014 The Authors. The Journal of Pathology published by John Wiley & Sons Ltd on behalf of Pathological Society of Great Britain and Ireland.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106752/1/path4325.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/106752/2/path4325-sup-0001-AppendixS1.pd