On the generation of arbitrage-free stock price models using Lie symmetry analysis

AbstractIn Bell and Stelljes (2009) a scheme for constructing explicitly solvable arbitrage-free models for stock prices is proposed. Under this scheme solutions of a second-order (1+1)-partial differential equation, containing a rational parameter p drawn from the interval [1/2,1], are used to generate arbitrage-free models of the stock price. In this paper Lie symmetry analysis is employed to propose candidate models for arbitrage-free stock prices. For all values of p, many solutions of the determining partial differential equation are constructed algorithmically using routines of Lie symmetry analysis. As such the present study significantly extends the work by Bell and Stelljes who found only two arbitrage-free models based on two simple solutions of the determining equation, corresponding to p=1/2 and p=1

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

Diagnosis and Prediction of Market Rebounds in Financial Markets

We introduce the concept of "negative bubbles" as the mirror image of standard financial bubbles, in which positive feedback mechanisms may lead to transient accelerating price falls. To model these negative bubbles, we adapt the Johansen-Ledoit-Sornette (JLS) model of rational expectation bubbles with a hazard rate describing the collective buying pressure of noise traders. The price fall occurring during a transient negative bubble can be interpreted as an effective random downpayment that rational agents accept to pay in the hope of profiting from the expected occurrence of a possible rally. We validate the model by showing that it has significant predictive power in identifying the times of major market rebounds. This result is obtained by using a general pattern recognition method which combines the information obtained at multiple times from a dynamical calibration of the JLS model. Error diagrams, Bayesian inference and trading strategies suggest that one can extract genuine information and obtain real skill from the calibration of negative bubbles with the JLS model. We conclude that negative bubbles are in general predictably associated with large rebounds or rallies, which are the mirror images of the crashes terminating standard bubbles.Comment: 49 pages, 14 figure

Lie symmetry analysis on pricing power options under the Heston dynamic and some fractional financial models

The rise of computational mathematics in financial markets has accelerated the bloom of various financial models. For instance, the Black-Scholes-Merton model, the Vasicek model, the Cox-Ingersoll-Ross model, the Heston model, etc. Each of these models often produces challenging partial differential equations. The Lie symmetry method appears to be a powerful tool to solve these types of equations. In this study, we apply Lie's method to the power options model under the Heston dynamic. The infinitesimal operators, the optimal systems, the invariant solutions, and the conservation laws are presented. Lie analysis is also an efficient tool to solve the fractional differential equations which involve the differentiation of a function with respect to its independent variable(s) to a non-integer order. Fractional differential equations are well known for their ability to describe the memory effect in various natural phenomena. We apply the Lie symmetry analysis to a time-fractional Black-Scholes-Merton model, as well as an arbitrage-free stock price model. The results of the analysis which include the infinitesimal operators or generators, the optimal systems, and the invariant solutions of the above models are presented. The visual representations of the invariant solutions are provided alongside discussions and comparisons with the solutions from their corresponding non-fractional models

Pricing American Options by Exercise Rate Optimization

We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black-Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black-Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model

Modelling FX smile : from stochastic volatility to skewness

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WARNING: Physics Envy May Be Hazardous To Your Wealth!

The quantitative aspirations of economists and financial analysts have for many years been based on the belief that it should be possible to build models of economic systems - and financial markets in particular - that are as predictive as those in physics. While this perspective has led to a number of important breakthroughs in economics, "physics envy" has also created a false sense of mathematical precision in some cases. We speculate on the origins of physics envy, and then describe an alternate perspective of economic behavior based on a new taxonomy of uncertainty. We illustrate the relevance of this taxonomy with two concrete examples: the classical harmonic oscillator with some new twists that make physics look more like economics, and a quantitative equity market-neutral strategy. We conclude by offering a new interpretation of tail events, proposing an "uncertainty checklist" with which our taxonomy can be implemented, and considering the role that quants played in the current financial crisis.Comment: v3 adds 2 reference