Inverse Reinforcement Learning for Marketing

Learning customer preferences from an observed behaviour is an important topic in the marketing literature. Structural models typically model forward-looking customers or firms as utility-maximizing agents whose utility is estimated using methods of Stochastic Optimal Control. We suggest an alternative approach to study dynamic consumer demand, based on Inverse Reinforcement Learning (IRL). We develop a version of the Maximum Entropy IRL that leads to a highly tractable model formulation that amounts to low-dimensional convex optimization in the search for optimal model parameters. Using simulations of consumer demand, we show that observational noise for identical customers can be easily confused with an apparent consumer heterogeneity.Comment: 18 pages, 5 figure

Quantum KAM Technique and Yang-Mills Quantum Mechanics

We study a quantum analogue of the iterative perturbation theory by Kolmogorov used in the proof of the Kolmogorov-Arnold-Moser (KAM) theorem. The method is based on sequent canonical transformations with a "running" coupling constant \lm,\lm^{2},\lm^{4} etc. The proposed scheme, as its classical predecessor, is "superconvergent" in the sense that after the n-th step, a theory is solved to the accuracy of order \lm^{2^{n-1}} . It is shown that the Kolmogorov technique corresponds to an infinite resummation of the usual perturbative series. The corresponding expansion is convergent for the quantum anharmonic oscillator due to the fact that it turns out to be identical to the Pade series. The method is easily generalizable to many-dimensional cases. The Kolmogorov technique is further applied to a non-perturbative treatment of Yang-Mills quantum mechanics. A controllable expansion for the wave function near the origin is constructed. For large fields, we build an asymptotic adiabatic expansion in inverse powers of the field. This asymptotic solution contains arbitrary constants which are not fixed by the boundary conditions at infinity. To find them, we approximately match the two expansions in an intermediate region. We also discuss some analogies between this problem and the method of QCD sum rules.Comment: 26 pages, latex, no figure

QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds

This paper presents a discrete-time option pricing model that is rooted in Reinforcement Learning (RL), and more specifically in the famous Q-Learning method of RL. We construct a risk-adjusted Markov Decision Process for a discrete-time version of the classical Black-Scholes-Merton (BSM) model, where the option price is an optimal Q-function, while the optimal hedge is a second argument of this optimal Q-function, so that both the price and hedge are parts of the same formula. Pricing is done by learning to dynamically optimize risk-adjusted returns for an option replicating portfolio, as in the Markowitz portfolio theory. Using Q-Learning and related methods, once created in a parametric setting, the model is able to go model-free and learn to price and hedge an option directly from data, and without an explicit model of the world. This suggests that RL may provide efficient data-driven and model-free methods for optimal pricing and hedging of options, once we depart from the academic continuous-time limit, and vice versa, option pricing methods developed in Mathematical Finance may be viewed as special cases of model-based Reinforcement Learning. Further, due to simplicity and tractability of our model which only needs basic linear algebra (plus Monte Carlo simulation, if we work with synthetic data), and its close relation to the original BSM model, we suggest that our model could be used for benchmarking of different RL algorithms for financial trading applicationsComment: 30 pages (minor changes in the presentation, updated references

Implied Multi-Factor Model for Bespoke CDO Tranches and other Portfolio Credit Derivatives

This paper introduces a new semi-parametric approach to the pricing and risk management of bespoke CDO tranches, with a particular attention to bespokes that need to be mapped onto more than one reference portfolio. The only user input in our framework is a multi-factor model (a "prior" model hereafter) for index portfolios, such as CDX.NA.IG or iTraxx Europe, that are chosen as benchmark securities for the pricing of a given bespoke CDO. Parameters of the prior model are fixed, and not tuned to match prices of benchmark index tranches. Instead, our calibration procedure amounts to a proper reweightening of the prior measure using the Minimum Cross Entropy method. As the latter problem reduces to convex optimization in a low dimensional space, our model is computationally efficient. Both the static (one-period) and dynamic versions of the model are presented. The latter can be used for pricing and risk management of more exotic instruments referencing bespoke portfolios, such as forward-starting tranches or tranche options, and for calculation of credit valuation adjustment (CVA) for bespoke tranches.Comment: 40 pages, 10 figure

Keep It Real: Tail Probabilities of Compound Heavy-Tailed Distributions

We propose an analytical approach to the computation of tail probabilities of compound distributions whose individual components have heavy tails. Our approach is based on the contour integration method, and gives rise to a representation of the tail probability of a compound distribution in the form of a rapidly convergent one-dimensional integral involving a discontinuity of the imaginary part of its moment generating function across a branch cut. The latter integral can be evaluated in quadratures, or alternatively represented as an asymptotic expansion. Our approach thus offers a viable (especially at high percentile levels) alternative to more standard methods such as Monte Carlo or the Fast Fourier Transform, traditionally used for such problems. As a practical application, we use our method to compute the operational Value at Risk (VAR) of a financial institution, where individual losses are modeled as spliced distributions whose large loss components are given by power-law or lognormal distributions. Finally, we briefly discuss extensions of the present formalism for calculation of tail probabilities of compound distributions made of compound distributions with heavy tails.Comment: 23 pages, 3 figure

The QLBS Q-Learner Goes NuQLear: Fitted Q Iteration, Inverse RL, and Option Portfolios

The QLBS model is a discrete-time option hedging and pricing model that is based on Dynamic Programming (DP) and Reinforcement Learning (RL). It combines the famous Q-Learning method for RL with the Black-Scholes (-Merton) model's idea of reducing the problem of option pricing and hedging to the problem of optimal rebalancing of a dynamic replicating portfolio for the option, which is made of a stock and cash. Here we expand on several NuQLear (Numerical Q-Learning) topics with the QLBS model. First, we investigate the performance of Fitted Q Iteration for a RL (data-driven) solution to the model, and benchmark it versus a DP (model-based) solution, as well as versus the BSM model. Second, we develop an Inverse Reinforcement Learning (IRL) setting for the model, where we only observe prices and actions (re-hedges) taken by a trader, but not rewards. Third, we outline how the QLBS model can be used for pricing portfolios of options, rather than a single option in isolation, thus providing its own, data-driven and model independent solution to the (in)famous volatility smile problem of the Black-Scholes model.Comment: 18 pages, 5 figure

Bayesian Entropic Inverse Theory Approach to Implied Option Pricing with Noisy Data

A popular approach to nonparametric option pricing is the Minimum Cross Entropy (MCE) method based on minimization of the relative Kullback-Leibler entropy of the price density distribution and a given reference density, with observable option prices serving as constraints. When market prices are noisy, the MCE method tends to overfit the data and often becomes unstable. We propose a non-parametric option pricing method whose input are noisy market prices of arbitrary number of European options with arbitrary maturities. Implied transition densities are calculated using the Bayesian inverse theory with entropic priors, with a reference density which may be estimated by the algorithm itself. In the limit of zero noise, our approach is shown to reduce to the canonical MCE method generalized to a multi-period case. The method can be used for a non-parametric pricing of American/Bermudan options with a possible weak path dependence.Comment: 23 pages, 6 figure

"Integrating in" and Effective Lagrangian for Non-Supersymmetric Yang-Mills Theory

Recently a non-supersymmetric analog of Veneziano-Yankielowicz (VY) effective Lagrangian has been proposed and applied for the analysis of the theta dependence in pure Yang-Mills theory. This effective Lagrangian is similar in many respects to the VY construction and, in particular, exhibits a kind of low energy holomorphy which is absent in the full YM theory. Here we incorporate a heavy fermion into this effective theory by using the "integrating in" technique. We find that, in terms of this extended theory, holomorphy of the effective Lagrangian for pure YM theory naturally implies a holomorphic dependence on the heavy fermion mass. It is shown that this analysis fixes, under certain assumptions, a dimensionless parameter which enters the effective Lagrangian and determines the number of nondegenerate vacuum sectors in pure YM theory. We also compare our results for the vacuum structure and theta dependence to those obtained recently by Witten on the basis of AdS/CFT correspondence.Comment: Latex 17 pages, no figures. Discussion is extended and new references are added. Final version to appear in Nucl. Phys.

BSLP: Markovian Bivariate Spread-Loss Model for Portfolio Credit Derivatives

BSLP is a two-dimensional dynamic model of interacting portfolio-level loss and spread (more exactly, loss intensity) processes. The model is similar to the top-down HJM-like frameworks developed by Schonbucher (2005) and Sidenius-Peterbarg-Andersen (SPA) (2005), however is constructed as a Markovian, short-rate intensity model. This property of the model enables fast lattice methods for pricing various portfolio credit derivatives such as tranche options, forward-starting tranches, leveraged super-senior tranches etc. A non-parametric model specification is used to achieve nearly perfect calibration to liquid tranche quotes across strikes and maturities. A non-dynamic version of the model obtained in the zero volatility limit of stochastic intensity is useful on its own as an arbitrage-free interpolation model to price non-standard index tranches off the standard ones.Comment: 42 pages, 9 figure

Climbing Down from the Top: Single Name Dynamics in Credit Top Down Models

In the top-down approach to multi-name credit modeling, calculation of singe name sensitivities appears possible, at least in principle, within the so-called random thinning (RT) procedure which dissects the portfolio risk into individual contributions. We make an attempt to construct a practical RT framework that enables efficient calculation of single name sensitivities in a top-down framework, and can be extended to valuation and risk management of bespoke tranches. Furthermore, we propose a dynamic extension of the RT method that enables modeling of both idiosyncratic and default-contingent individual spread dynamics within a Monte Carlo setting in a way that preserves the portfolio "top"-level dynamics. This results in a model that is not only calibrated to tranche and single name spreads, but can also be tuned to approximately match given levels of spread volatilities and correlations of names in the portfolio.Comment: 34 pages, 9 figure