FX Smile in the Heston Model

The Heston model stands out from the class of stochastic volatility (SV) models mainly for two reasons. Firstly, the process for the volatility is non-negative and mean-reverting, which is what we observe in the markets. Secondly, there exists a fast and easily implemented semi-analytical solution for European options. In this article we adapt the original work of Heston (1993) to a foreign exchange (FX) setting. We discuss the computational aspects of using the semi-analytical formulas, performing Monte Carlo simulations, checking the Feller condition, and option pricing with FFT. In an empirical study we show that the smile of vanilla options can be reproduced by suitably calibrating three out of five model parameters.Comment: Chapter prepared for the 2nd edition of Statistical Tools for Finance and Insurance, P.Cizek, W.Haerdle, R.Weron (eds.), Springer-Verlag, forthcoming in 201

FX Smile in the Heston Model

The Heston model stands out from the class of stochastic volatility (SV) models mainly for two reasons. Firstly, the process for the volatility is nonnegative and mean-reverting, which is what we observe in the markets. Secondly, there exists a fast and easily implemented semi-analytical solution for European options. In this article we adapt the original work of Heston (1993) to a foreign exchange (FX) setting. We discuss the computational aspects of using the semi-analytical formulas, performing Monte Carlo simulations, checking the Feller condition, and option pricing with FFT. In an empirical study we show that the smile of vanilla options can be reproduced by suitably calibrating three out of five model parameters.Heston model; vanilla option; stochastic volatility; Monte Carlo simulation; Feller condition; option pricing with FFT;

FPGA-based Anomalous trajectory detection using SOFM

A system for automatically classifying the trajectory of a moving object in a scene as usual or suspicious is presented. The system uses an unsupervised neural network (Self Organising Feature Map) fully implemented on a reconfigurable hardware architecture (Field Programmable Gate Array) to cluster trajectories acquired over a period, in order to detect novel ones. First order motion information, including first order moving average smoothing, is generated from the 2D image coordinates (trajectories). The classification is dynamic and achieved in real-time. The dynamic classifier is achieved using a SOFM and a probabilistic model. Experimental results show less than 15\% classification error, showing the robustness of our approach over others in literature and the speed-up over the use of conventional microprocessor as compared to the use of an off-the-shelf FPGA prototyping board

Pricing derivatives in stochastic volatility models using the finite difference method

The Heston stochastic volatility model is one extension of the Black-Scholes model which describes the money markets more accurately so that more realistic prices for derivative products are obtained. From the stochastic differential equation of the underlying financial product a partial differential equation (p.d.e.) for the value function of an option can be derived. This p.d.e. can be solved with the finite difference method (f.d.m.). The stability and consistency of the method is examined. Furthermore a boundary condition is proposed to reduce the numerical error. Finally a non uniform structured grid is derived which is fairly optimal for the numerical result in the most interesting point.Das stochastische Volatilitaetsmodell von Heston ist eines der Erweiterungen des Black-Scholes-Modells. Von der stochastischen Differentialgleichung fuer den unterliegenden Prozess kann eine partielle Differentialgleichung fuer die Wertfunktion einer Option abgeleitet werden. Es wird die Loesung mittels Finiter Differenzenmethode untersucht (Konsistenz, Stabilitaet). Weiterhin wird eine Randbedingung und ein spezielles nicht-uniformes Netz vorgeschlagen, was zu einer starken Reduzierung des numerischen Fehlers der Wertfunktion in einem ganz bestimmten Punkt fuehrt

Illustration of stochastic processes and the finite difference method in finance

The presentation shows sample paths of stochastic processes in form of animations. Those stochastic procsses are usually used to model financial quantities like exchange rates, interest rates and stock prices. In the second part the solution of the Black-Scholes PDE using the finite difference method is illustrated.Der Vortrag zeigt Animationen von Realisierungen stochstischer Prozesse, die zur Modellierung von Groessen im Finanzbereich haeufig verwendet werden (z.B. Wechselkurse, Zinskurse, Aktienkurse). Im zweiten Teil wird die Loesung der Black-Scholes Partiellen Differentialgleichung mittels Finitem Differenzenverfahren graphisch veranschaulicht

Pricing derivatives in stochastic volatility models using the finite difference method

The Heston stochastic volatility model is one extension of the Black-Scholes model which describes the money markets more accurately so that more realistic prices for derivative products are obtained. From the stochastic differential equation of the underlying financial product a partial differential equation (p.d.e.) for the value function of an option can be derived. This p.d.e. can be solved with the finite difference method (f.d.m.). The stability and consistency of the method is examined. Furthermore a boundary condition is proposed to reduce the numerical error. Finally a non uniform structured grid is derived which is fairly optimal for the numerical result in the most interesting point.Das stochastische Volatilitaetsmodell von Heston ist eines der Erweiterungen des Black-Scholes-Modells. Von der stochastischen Differentialgleichung fuer den unterliegenden Prozess kann eine partielle Differentialgleichung fuer die Wertfunktion einer Option abgeleitet werden. Es wird die Loesung mittels Finiter Differenzenmethode untersucht (Konsistenz, Stabilitaet). Weiterhin wird eine Randbedingung und ein spezielles nicht-uniformes Netz vorgeschlagen, was zu einer starken Reduzierung des numerischen Fehlers der Wertfunktion in einem ganz bestimmten Punkt fuehrt

Modelling spikes and pricing swing options in electricity markets

Most electricity markets exhibit high volatilities and occasional distinctive price spikes, which result in demand for derivative products which protect the holder against high prices. In this paper we examine a simple spot price model that is the exponential of the sum of an Ornstein-Uhlenbeck and an independent mean-reverting pure jump process. We derive the moment generating function as well as various approximations to the probability density function of the logarithm of the spot price process at maturity T. Hence we are able to calibrate the model to the observed forward curve and present semi-analytic formulae for premia of path-independent options as well as approximations to call and put options on forward contracts with and without a delivery period. In order to price path-dependent options with multiple exercise rights like swing contracts a grid method is utilized which in turn uses approximations to the conditional density of the spot process.Energy derivatives, Financial mathematics, Stochastic jumps, Numerical methods for option pricing, Continuous time models, Derivative pricing models,

GW4: An FPGA-driven image segmentation algorithm

We describe “GW4,” an efficient video segmentation algorithm designed for FPGA implementation. The algorithm detects moving foreground objects against a multimodal background; it is motivated by two well-known adaptive background differencing algorithms, Grimson's algorithm and W4. GW4 is designed specifically for implementation on reconfigurable FPGA hardware, avoiding the use of floating point numbers and transcendental operations, and operates at real-time frame rates on 640x480 video streams. We present experimental results indicating processing speeds, and superior segmentation performance to Grimson's algorithm

Is there an informationally passive benchmark for option pricing incorporating maturity?

Figlewski proposed testing the incremental contribution of the Black-Scholes model by comparing its performance against an “informationally passive” benchmark, which was defined to be an option pricing formula satisfying static no-arbitrage constraints. In this paper we extend Figlewski's analysis to include options of more than one maturity. Once maturity has been included in the model, any “informationally passive” call pricing function is consistent with some “active” model. In this sense, the notion of a passive model cannot be extended to pricing formulas incorporating option maturity. We derive the index dynamics of the active model implicit in Figlewski's implied G example. These dynamics are far more complicated than the dynamics of the Samuelson-Black-Scholes or Bachelier models. The main implication of our analysis is that an appropriate benchmark for assessing option pricing models should in fact have simple dynamics, such as those of Bachelier or the Black-Scholes models. This is despite the fact that the maturity extension of Figlewski's model gives as good a fit as the Black-Scholes model.