Spectral methods for volatility derivatives

In the first quarter of 2006 Chicago Board Options Exchange (CBOE) introduced, as one of the listed products, options on its implied volatility index (VIX). This created the challenge of developing a pricing framework that can simultaneously handle European options, forward-starts, options on the realized variance and options on the VIX. In this paper we propose a new approach to this problem using spectral methods. We use a regime switching model with jumps and local volatility defined in \cite{FXrev} and calibrate it to the European options on the S&P 500 for a broad range of strikes and maturities. The main idea of this paper is to "lift" (i.e. extend) the generator of the underlying process to keep track of the relevant path information, namely the realized variance. The lifted generator is too large a matrix to be diagonalized numerically. We overcome this difficulty by applying a new semi-analytic algorithm for block-diagonalization. This method enables us to evaluate numerically the joint distribution between the underlying stock price and the realized variance, which in turn gives us a way of pricing consistently European options, general accrued variance payoffs and forward-starting and VIX options.Comment: to appear in Quantitative Financ

Markov chains and the pricing of derivatives

A numerical method for pricing financial derivatives based on continuous-time Markov chains is proposed. It approximates the underlying stochastic process by a continuous-time Markov chain. We show how to construct a multi-dimensional continuous-time Markov chain such that it converges in distribution to a multi-dimensional diffusion process. The method is flexible enough to be applied to a model where the underlying process contains local volatility, stochastic volatility and jumps. Furthermore, we introduce a method to approximate the dynamics of the realized variance of a Markov chain and an algorithm to reduce the complexity of computing the joint probability distribution between the realized variance and the underlying

A Numerical Method for Pricing Electricity Derivatives for Jump-Diffusion Processes Based on Continuous Time Lattices

We present a numerical method for pricing derivatives on electricity prices. The method is based on approximating the generator of the underlying process and can be applied for stochastic processes that are combinations of diusions and jump processes. The method is accurate even in the case of processes with fast mean-reversion and jumps of large magnitude. We illustrate the speed and accuracy of the method by pricing European and Bermudan options and calculating the hedge ratios of European options for the Geman-Roncoroni model for electricity prices.Electricity derivatives; operator methods

Foundations of Technical Analysis: Computational Algorithms, Statistical Inference, and Empirical Implementation

Technical analysis, also known as "charting", has been a part of financial practice for many decades, yet little academic research has been devoted to a systematic evaluation of this discipline. One of the main obstacles is the highly subjective nature of technical analysis---the presence of geometric shapes in historical price charts is often in the eyes of the beholder. In this paper, we propose a systematic and automatic approach to technical pattern recognition using nonparametric kernel regression, and apply this method to a large number of US stocks from 1962 to 1996 to evaluate the effectiveness of technical analysis. By comparing the unconditional empirical distribution of daily stock returns to the conditional distribution---conditioned on specific technical indicators such as head-and-shoulders or double-bottoms---we find that over the 31-year sample period, several technical indicators do provide incremental information and may have some practical value.

Asset Prices and Trading Volume Under Fixed Transactions Costs

We propose a dynamic equilibrium model of asset prices and trading volume with heterogeneous agents facing fixed transactions costs. We show that even small fixed costs can give rise to large 'no-trade' regions for each agent's optimal trading policy and a significant illiquidity discount in asset prices. We perform a calibration exercise to illustrate the empirical relevance of our model for aggregate data. Our model also has implications for the dynamics of order flow, bid/ask spreads, market depth, the allocation of trading costs between buyers and sellers, and other aspects of market microstructure, including a square-root power law between trading volume and fixed costs which we confirm using historical US stock market data from 1993 to 1997.

A Numerical Method for Pricing Electricity Derivatives for Jump-Diffusion Processes Based on Continuous Time Lattices

We present a numerical method for pricing derivatives on electricity prices. The method is based on approximating the generator of the underlying process and can be applied for stochastic processes that are combinations of diusions and jump processes. The method is accurate even in the case of processes with fast mean-reversion and jumps of large magnitude. We illustrate the speed and accuracy of the method by pricing European and Bermudan options and calculating the hedge ratios of European options for the Geman-Roncoroni model for electricity prices

A Numerical Method for Pricing Electricity Derivatives for Jump-Diffusion Processes Based on Continuous Time Lattices

We present a numerical method for pricing derivatives on electricity prices. The method is based on approximating the generator of the underlying process and can be applied for stochastic processes that are combinations of diusions and jump processes. The method is accurate even in the case of processes with fast mean-reversion and jumps of large magnitude. We illustrate the speed and accuracy of the method by pricing European and Bermudan options and calculating the hedge ratios of European options for the Geman-Roncoroni model for electricity prices

Dynamic Phase Transitions in Cell Spreading

We monitored isotropic spreading of mouse embryonic fibroblasts on fibronectin-coated substrates. Cell adhesion area versus time was measured via total internal reflection fluorescence microscopy. Spreading proceeds in well-defined phases. We found a power-law area growth with distinct exponents a_i in three sequential phases, which we denote basal (a_1=0.4+-0.2), continous (a_2=1.6+-0.9) and contractile (a_3=0.3+-0.2) spreading. High resolution differential interference contrast microscopy was used to characterize local membrane dynamics at the spreading front. Fourier power spectra of membrane velocity reveal the sudden development of periodic membrane retractions at the transition from continous to contractile spreading. We propose that the classification of cell spreading into phases with distinct functional characteristics and protein activity patterns serves as a paradigm for a general program of a phase classification of cellular phenotype. Biological variability is drastically reduced when only the corresponding phases are used for comparison across species/different cell lines.Comment: 4 pages, 5 figure

Security of quantum bit string commitment depends on the information measure

Unconditionally secure non-relativistic bit commitment is known to be impossible in both the classical and the quantum world. However, when committing to a string of n bits at once, how far can we stretch the quantum limits? In this letter, we introduce a framework of quantum schemes where Alice commits a string of n bits to Bob, in such a way that she can only cheat on a bits and Bob can learn at most b bits of information before the reveal phase. Our results are two-fold: we show by an explicit construction that in the traditional approach, where the reveal and guess probabilities form the security criteria, no good schemes can exist: a+b is at least n. If, however, we use a more liberal criterion of security, the accessible information, we construct schemes where a=4 log n+O(1) and b=4, which is impossible classically. Our findings significantly extend known no-go results for quantum bit commitment.Comment: To appear in PRL. Short version of quant-ph/0504078, long version to appear separately. Improved security definition and result, one new lemma that may be of independent interest. v2: added funding reference, no other change

Possibility, Impossibility and Cheat-Sensitivity of Quantum Bit String Commitment

Unconditionally secure non-relativistic bit commitment is known to be impossible in both the classical and the quantum worlds. But when committing to a string of n bits at once, how far can we stretch the quantum limits? In this paper, we introduce a framework for quantum schemes where Alice commits a string of n bits to Bob in such a way that she can only cheat on a bits and Bob can learn at most b bits of information before the reveal phase. Our results are two-fold: we show by an explicit construction that in the traditional approach, where the reveal and guess probabilities form the security criteria, no good schemes can exist: a+b is at least n. If, however, we use a more liberal criterion of security, the accessible information, we construct schemes where a=4log n+O(1) and b=4, which is impossible classically. We furthermore present a cheat-sensitive quantum bit string commitment protocol for which we give an explicit tradeoff between Bob's ability to gain information about the committed string, and the probability of him being detected cheating.Comment: 10 pages, RevTex, 2 figure. v2: title change, cheat-sensitivity adde