Quantum Mechanics, Path Integrals and Option Pricing: Reducing the Complexity of Finance

Quantum Finance represents the synthesis of the techniques of quantum theory (quantum mechanics and quantum field theory) to theoretical and applied finance. After a brief overview of the connection between these fields, we illustrate some of the methods of lattice simulations of path integrals for the pricing of options. The ideas are sketched out for simple models, such as the Black-Scholes model, where analytical and numerical results are compared. Application of the method to nonlinear systems is also briefly overviewed. More general models, for exotic or path-dependent options are discussed.Comment: 10 pages, 4 figures, presented by C.Coriano at the Intl. Workshop "Nonlinear Physics, THeory and Experiment II", Gallipoli, Lecce, June 28-July 6, 200

Price Dividend Ratio Factors : Proxies for Long Run Risk

We evaluate the empirical support for a broad class of long run risk models using information in factors extracted through principal component analysis of the covariance matrix of log price dividend ratios of twenty five equity portfolios formed on Size and Book-to-Market. We identify two price-dividend ratio factor proxies for economy wide long run risk, one tracking the volatility of the growth rate in economy wide aggregate consumption, and the other predicting the growth rates in the stock index portfolio dividends and aggregate consumption, consistent with the implications of these models. We show that that the long run risk factor driving expected consumption growth is not recoverable from the cross section of excess returns alone. The price dividend ratio factors perform better than the stock index price dividend ratio and the corporate yield spread, and has information in addition to what is in the slope of the term structure of interest rates, in forecasting the growth rate in real time consumption and stock index dividends. The covariance of excess returns with factor innovations explain the cross section of excess returns on size, book/market, earnings/price ratio, long term reversal, and short term reversal sorted portfolios in a manner robust to look-ahead and useless factor biases. Our findings suggest that the widely used Fama and French (1993) three factor model and the long run risk models studied in the literature are not necessarily inconsistent with each other. They may be representing the same underlying phenomenon, but emphasizing different aspects of reality.

Comparison of Field Theory Models of Interest Rates with Market Data

We calibrate and test various variants of field theory models of the interest rate with data from eurodollars futures. A model based on a simple psychological factor are seen to provide the best fit to the market. We make a model independent determination of the volatility function of the forward rates from market data.Comment: 9 figure

Hamiltonian and Potentials in Derivative Pricing Models: Exact Results and Lattice Simulations

The pricing of options, warrants and other derivative securities is one of the great success of financial economics. These financial products can be modeled and simulated using quantum mechanical instruments based on a Hamiltonian formulation. We show here some applications of these methods for various potentials, which we have simulated via lattice Langevin and Monte Carlo algorithms, to the pricing of options. We focus on barrier or path dependent options, showing in some detail the computational strategies involved.Comment: 27 pages, 11 figures 1 subsection added (4.1). Slightly longer appendi

Hedging in Field Theory Models of the Term Structure

We use path integrals to calculate hedge parameters and efficacy of hedging in a quantum field theory generalization of the Heath, Jarrow and Morton (HJM) term structure model which parsimoniously describes the evolution of imperfectly correlated forward rates. We also calculate, within the model specification, the effectiveness of hedging over finite periods of time. We use empirical estimates for the parameters of the model to show that a low dimensional hedge portfolio is quite effective.Comment: 18 figures, Invited Talk, International Econophysics Conference, Bali, 28-31 August 200

Comparison of Field Theory Models of Interest Rates with Market Data

We calibrate and test various variants of field theory models of the interest rate with data from eurodollars futures. A model based on a simple psychological factor are seen to provide the best fit to the market. We make a model independent determination of the volatility function of the forward rates from market data.