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

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

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

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

WHOI Hawaii Ocean Timeseries Station (WHOTS) : WHOTS-2 mooring turnaround cruise report

The Woods Hole Oceanographic Institution (WHOI) Hawaii Ocean Timeseries (HOT) Site (WHOTS), 100 km north of Oahu, Hawaii, is intended to provide long-term, high-quality air-sea fluxes as a coordinated part of the HOT program and contribute to the goals of observing heat, fresh water and chemical fluxes at a site representative of the oligotrophic North Pacific Ocean. This report documents recovery of the WHOTS-1 mooring, deployed in August 2004 near 22.75°N, 158°W, and deployment of the WHOTS-2 mooring at the same site. Both moorings were outfitted with Air-Sea Interaction Meteorology (ASIMET) systems to measure, record, and transmit the surface meteorological variables necessary to compute air-sea fluxes of heat, moisture and momentum. In cooperation with R. Lukas of the University of Hawaii, the upper 155 m of the moorings were outfitted with oceanographic sensors for the measurement of temperature, conductivity and velocity. The WHOTS mooring turnaround was done on the Scripps Institution of Oceanography Ship Melville, Cruise TUIM-10MV. The cruise took place between 23 and 30 July 2005.Funding was provided by the National Oceanic and Atmospheric Administration under Grant No. NA17RJ1223 and the Cooperative Institute for Climate and Ocean Research (CICOR)

WHOI Hawaii Ocean Timeseries Station (WHOTS) : WHOTS-4 2007 mooring turnaround cruise report

The Woods Hole Oceanographic Institution (WHOI) Hawaii Ocean Timeseries (HOT) Site (WHOTS), 100 km north of Oahu, Hawaii, is intended to provide long-term, high-quality air-sea fluxes as a part of the NOAA Climate Observation Program. The WHOTS mooring also serves as a coordinated part of the HOT program, contributing to the goals of observing heat, fresh water and chemical fluxes at a site representative of the oligotrophic North Pacific Ocean. The approach is to maintain a surface mooring outfitted for meteorological and oceanographic measurements at a site near 22.75°N, 158°W by successive mooring turnarounds. These observations will be used to investigate air–sea interaction processes related to climate variability. The first three WHOTS moorings (WHOTS-1 through 3) were deployed in August 2004, July 2005 and June 2006, respectively. This report documents recovery of the WHOTS-3 mooring and deployment of the fourth mooring (WHOTS-4). Both moorings used Surlyn foam buoys as the surface element and were outfitted with two Air–Sea Interaction Meteorology (ASIMET) systems. Each ASIMET system measures, records, and transmits via Argos satellite the surface meteorological variables necessary to compute air–sea fluxes of heat, moisture and momentum. The upper 155 m of the moorings were outfitted with oceanographic sensors for the measurement of temperature, conductivity and velocity in a cooperative effort with R. Lukas of the University of Hawaii. A pCO2 system was installed on the WHOT-3 buoy in a cooperative effort with Chris Sabine at the Pacific Marine Environmental Laboratory. The WHOTS mooring turnaround was done on the University of Hawaii research vessel Kilo Moana, Cruise KM-07-08, by the Upper Ocean Processes Group of the Woods Hole Oceanographic Institution. The cruise took place between 24 June and 1 July 2007. Operations began with deployment of the WHOTS-4 mooring on 25 June at approximately 22°40.2′N, 157°57.0′W in 4756 m of water. This was followed by meteorological intercomparisons and CTDs at the WHOTS-4 and WHOTS-3 sites. The WHOTS-3 mooring was recovered on June 28th followed by CTD operations at the HOT site and shipboard meteorological observations at several sites to the south of the mooring site. This report describes these cruise operations, as well as some of the in-port operations and pre-cruise buoy preparations.Funding was provided by the National Oceanic and Atmospheric Administration under Grant No. NA17RJ1223 for the Cooperative Institute for Climate and Ocean Research (CICOR)

WHOI Hawaii Ocean Timeseries Station (WHOTS) : WHOTS-5 2008 mooring turnaround cruise report

The Woods Hole Oceanographic Institution (WHOI) Hawaii Ocean Timeseries (HOT) Site (WHOTS), 100 km north of Oahu, Hawaii, is intended to provide long-term, high-quality air-sea fluxes as a part of the NOAA Climate Observation Program. The WHOTS mooring also serves as a coordinated part of the HOT program, contributing to the goals of observing heat, fresh water and chemical fluxes at a site representative of the oligotrophic North Pacific Ocean. The approach is to maintain a surface mooring outfitted for meteorological and oceanographic measurements at a site near 22.75°N, 158°W by successive mooring turnarounds. These observations will be used to investigate air–sea interaction processes related to climate variability. The first four WHOTS moorings (WHOTS-1 through 4) were deployed in August 2004, July 2005, June 2006, and June 2007, respectively. This report documents recovery of the WHOTS-4 mooring and deployment of the fifth mooring (WHOTS-5). Both moorings used Surlyn foam buoys as the surface element and were outfitted with two Air–Sea Interaction Meteorology (ASIMET) systems. Each ASIMET system measures, records, and transmits via Argos satellite the surface meteorological variables necessary to compute air–sea fluxes of heat, moisture and momentum. The upper 155 m of the moorings were outfitted with oceanographic sensors for the measurement of temperature, conductivity and velocity in a cooperative effort with R. Lukas of the University of Hawaii. A pCO2 system was installed on the WHOTS-5 buoy in a cooperative effort with Chris Sabine at the Pacific Marine Environmental Laboratory. The WHOTS mooring turnaround was done on the University of Hawaii research vessel Kilo Moana, Cruise KM-08-08, by the Upper Ocean Processes Group of the Woods Hole Oceanographic Institution. The cruise took place between 3 and 11 June 2008. Operations began with deployment of the WHOTS-5 mooring on 5 June at approximately 22°46.1'N, 157°54.1'W in 4702 m of water. This was followed by meteorological intercomparisons and CTDs at the WHOTS-4 site. A period of calmer weather was taken advantage of to recover WHOTS-4 on 6 June 2008. The Kilo Moana then returned to the WHOTS-5 mooring for CTD operations and meteorological intercomparisons. This report describes these cruise operations, as well as some of the in-port operations and pre-cruise buoy preparations.Funding was provided by the National Oceanic and Atmospheric Administration under Grant No. NA17RJ1223 for the Cooperative Institute for Climate and Ocean Research (CICOR)

WHOI Hawaii Ocean Timeseries Station (WHOTS) : WHOTS-6 2009 mooring turnaround cruise report

The Woods Hole Oceanographic Institution (WHOI) Hawaii Ocean Timeseries Site (WHOTS), 100 km north of Oahu, Hawaii, is intended to provide long-term, high-quality air-sea fluxes as a part of the NOAA Climate Observation Program. The WHOTS mooring also serves as a coordinated part of the Hawaiian Ocean Timeseries (HOT) program, contributing to the goals of observing heat, fresh water and chemical fluxes at a site representative of the oligotrophic North Pacific Ocean. The approach is to maintain a surface mooring outfitted for meteorological and oceanographic measurements at a site near 22.75°N, 158°W by successive mooring turnarounds. These observations will be used to investigate air–sea interaction processes related to climate variability. The first WHOTS mooring (WHOTS-1) was deployed in August 2004. Turnaround cruises for successive moorings (WHOTS-2 through WHOTS-5) have typically been in either June or July. This report documents recovery of the WHOTS-5 mooring and deployment of the sixth mooring (WHOTS-6). The moorings utilize Surlyn foam buoys as the surface element and are outfitted with two Air–Sea Interaction Meteorology (ASIMET) systems. Each ASIMET system measures, records, and transmits via Argos satellite the surface meteorological variables necessary to compute air–sea fluxes of heat, moisture and momentum. The upper 155 m of the mooring is outfitted with oceanographic sensors for the measurement of temperature, conductivity and velocity in a cooperative effort with R. Lukas of the University of Hawaii (UH). A pCO2 system is installed on the buoy in a cooperative effort with Chris Sabine at the Pacific Marine Environmental Laboratory. Dr. Frank Bradley, CSIRO, Australia, assisted with meteorological sensor comparisons. A NOAA “Teacher at Sea” and a NOAA “Teacher in the Lab” participated in the cruise. The WHOTS mooring turnaround was done on the University of Hawaii research vessel Kilo Moana, Cruise KM-09-16, by the Upper Ocean Processes Group of the Woods Hole Oceanographic Institution in cooperation with UH and NOAA’s Earth System Research Laboratory, Physical Sciences Division (ESRL/PSD). The cruise took place between 9 and 17 July 2009. Operations began with deployment of the WHOTS-6 mooring on 10 July at approximately 22°40.0'N, 157°57.0'W in 4758 m of water. This was followed by meteorological intercomparisons and CTDs at the WHOTS-6 and WHOTS-5 sites. The WHOTS-5 mooring was recovered on 15 July 2009. The Kilo Moana then moved to the HOT central site (22°45.0'N, 158°00.0'W) for CTD casts. This report describes the cruise operations in more detail, as well as some of the in-port operations and pre-cruise buoy preparations.Funding was provided by the National Oceanic and Atmospheric Administration under Grant No. NA17RJ1223 for the Cooperative Institute for Climate and Ocean Research (CICOR)

Photoswitching Mechanism of Cyanine Dyes

Photoswitchable fluorescent probes have been used in recent years to enable super-resolution fluorescence microscopy by single-molecule imaging.1-6 Among these probes are red carbocyanine dyes, which can be reversibly photoconverted between a fluorescent state and a dark state for hundreds of cycles, yielding several thousand detected photons per switching cycle, before permanent photobleaching occurs.7,8 While these properties make them excel-lent probes for super-resolution imaging, the mechanism by which cyanine dyes are photoconverted has yet to be determined. Such an understanding could prove useful for creating new photoswit-chable probes with improved properties. The photoconversion of red cyanine dyes into their dark states occurs upon illumination by red light and is facilitated by a primary thiol in solution,7,9 whereas agents with a secondary thiol do not support photoswitching. These observations suggest that the reactiv