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Benjamin Virrion
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We present a generic path-dependent importance sampling algorithm where the Girsanov induced change of probability on the path space is represented by a sequence of neural networks taking the past of the trajectory as an input. At each learning step, the neural networks parameters are trained so as to reduce the variance of the Monte Carlo estimator induced by this change of measure. This allows for a generic path dependent change of measure which can be used to reduce the variance of any path-dependent financial payoff. We show in our numerical experiments that for payoffs consisting of either a call, an asymmetric combination of calls and puts, a symmetric combination of calls and puts, a multi coupon autocall or a single coupon autocall, we are able to reduce the variance of the Monte Carlo estimators by factors between 2 and 9. The numerical experiments also show that the method is very robust to changes in the parameter values, which means that in practice, the training can be done offline and only updated on a weekly basis.
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Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are two risk measures which are widely used in the practice of risk management. This paper deals with the problem of computing both VaR and CVaR using stochastic approximation (with decreasing steps): we propose a first Robbins-Monro procedure based on Rockaffelar-Uryasevs identity for the CVaR. The convergence rate of this algorithm to its target satisfies a Gaussian Central Limit Theorem. As a second step, in order to speed up the initial procedure, we propose a recursive importance sampling (I.S.) procedure which induces a significant variance reduction of both VaR and CVaR procedures. This idea, which goes back to the seminal paper of B. Arouna, follows a new approach introduced by V. Lemaire and G. Pag`es. Finally, we consider a deterministic moving risk level to speed up the initialization phase of the algorithm. We prove that the convergence rate of the resulting procedure is ruled by a Central Limit Theorem with minimal variance and its efficiency is illustrated by considering several typical energy portfolios.
Annealed importance sampling is a means to assign equilibrium weights to a nonequilibrium sample that was generated by a simulated annealing protocol. The weights may then be used to calculate equilibrium averages, and also serve as an ``adiabatic signature of the chosen cooling schedule. In this paper we demonstrate the method on the 50-atom dileucine peptide, showing that equilibrium distributions are attained for manageable cooling schedules. For this system, as naively implemented here, the method is modestly more efficient than constant temperature simulation. However, the method is worth considering whenever any simulated heating or cooling is performed (as is often done at the beginning of a simulation project, or during an NMR structure calculation), as it is simple to implement and requires minimal additional CPU expense. Furthermore, the naive implementation presented here can be improved.
The dynamics of financial markets are driven by the interactions between participants, as well as the trading mechanisms and regulatory frameworks that govern these interactions. Decision-makers would rather not ignore the impact of other participants on these dynamics and should employ tools and models that take this into account. To this end, we demonstrate the efficacy of applying opponent-modeling in a number of simulated market settings. While our simulations are simplified representations of actual market dynamics, they provide an idealized playground in which our techniques can be demonstrated and tested. We present this work with the aim that our techniques could be refined and, with some effort, scaled up to the full complexity of real-world market scenarios. We hope that the results presented encourage practitioners to adopt opponent-modeling methods and apply them online systems, in order to enable not only reactive but also proactive decisions to be made.
We present an importance sampling algorithm that can produce realisations of Markovian epidemic models that exactly match observations, taken to be the number of a single event type over a period of time. The importance sampling can be used to construct an efficient particle filter that targets the states of a system and hence estimate the likelihood to perform Bayesian parameter inference. When used in a particle marginal Metropolis Hastings scheme, the importance sampling provides a large speed-up in terms of the effective sample size per unit of computational time, compared to simple bootstrap sampling. The algorithm is general, with minimal restrictions, and we show how it can be applied to any discrete-state continuous-time Markov chain where we wish to exactly match the number of a single event type over a period of time.
In mathematical finance and other applications of stochastic processes, it is frequently the case that the characteristic function may be known but explicit forms for density functions are not available. The simulation of any distribution is greatly facilitated by a knowledge of the quantile function, by which uniformly distributed samples may be converted to samples of the given distribution. This article analyzes the calculation of a quantile function direct from the characteristic function of a probability distribution, without explicit knowledge of the density. We form a non-linear integro-differential equation that despite its complexity admits an iterative solution for the power series of the quantile about the median. We give some examples including tail models and show how to generate C-code for examples.
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