Quantifying uncertainty in finance is a major concern when one has to address properly risk management issues with uncertainty with respect to the model and to its inputs, for instance.
In this session, we have selected different up-to-date contributions: how to derive a metamodel in credit risk, where a direct sampling of the loss is quite time-consuming (because of large number of obligors)? how to use to cleverly interpolate financial quantities (interest rate curve, implied volatility surface) with kriging techniques accounting for arbitrage-free conditions? how to account for uncertain model (with a prior distribution on the copula dependency) to compute extreme losses in capital allocation problems? or to address asset management problems (portfolio optimisations)?
The tools cover Gaussian processes, MCMC, splitting, Polynomial Chaos Expansion, Stochastic Approximations.
08:30
Kriging for arbitrage-free construction of financial term-structures
Areski Cousin | Universite de Strasbourg | France
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Areski Cousin | Universite de Strasbourg | France
In some situations where market information is incomplete or not reliable, building financial term-structures (interest-rate curves, credit curves, volatility surfaces, ...) may be associated with a significant degree of uncertainty.
We propose a new arbitrage-free construction method that extends classical spline techniques by additionally allowing for quantification of uncertainty. The proposed method is based on a generalization of kriging regression models to linear equality constraints (market-fit conditions) and shape-preserving constraints (no-arbitrage conditions).
Prices of illiquid instruments can also be incorporated when considered as noisy observations.
We define the most likely response surface and the most-likely noise values and show how to build confidence bands.
The Gaussian process hyper-parameters under the construction constraints are estimated using maximum likelihood.
The method is illustrated on Euro Stoxx 50 option prices by building no-arbitrage volatility surfaces and their corresponding confidence bands.
09:00
Meta-model of a large credit risk portfolio in the Gaussian copula model
Clément Rey | Ecole Polytechnique | France
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Clément Rey | Ecole Polytechnique | France
We propose a meta-model for the loss distribution of a large credit
portofolio in the Gaussian copula model. In the family of Gaussian copula models, we find two sources of randomness which are the systemic/common risk shared by every member of the portofolio and the idiosyncratic risk which is related to one member and is independent from all other sources of noise. The meta-model is built from a truncation of the Wiener chaos
decomposition with respect to the systemic risk. It leads to a truncated Wiener chaos decomposition with random weigths depending only on the idiosyncratic risk. Moreover, the portofolio being large, we derive a Central Limit theorem for those random weights. We thus propose an extension of our meta-model using Gaussian approximation instead of random weights. This method significantly reduce the computational time needed to simulate the credit Loss and then to estimate risk measures for instance.
09:30
Uncertain Quantification Stochastic Algorithm as a Decision-Making tool for Portfolio Optimisation
Linda Chamakh | BNP Paribas | France
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Linda Chamakh | BNP Paribas | France
Investment decisions are always based on some part of uncertainty. These uncertainties stem from either the statistical error inherent to any empirical estimation procedure, or to the model error.
Portfolio optimization essentially relies on the estimation of the trends and covariance matrix of the assets returns on the asset pool considered. Indeed, the optimal portfolio is given by the allocation which maximizes the Sharpe ratio, e.g. the portfolio returns mean over its standard deviation. Adopting an SA point of view on this maximization problem, we can adapt the UQSA algorithm to derive a portfolio optimization scheme converging to the Sharpe ratio and portfolio distributions for a given model on the returns moments uncertainty. This approach provides an optimal range of portfolio allocation and can serve as a decision-making tool for investors.
10:00
Risk measures of a mixture model, an approach with mixture of MCMC and splitting
Cyril Benezet | Ecole Polytechnique | France
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Authors:
Emmanuel Gobet | CMAP - Ecole Polytechnique | France
Cyril Benezet | Ecole Polytechnique | France
Rodrigo Targino | FGV - Fundacao Getulio Vargas | Brazil
In this talk, we consider the problem of computing VaR or Expected Shortfall of a loss arising in capital allocation problems, where the business lines are dependent with uncertain copula dependency. A prior distribution is determined from the data and we are back to computing the VaR/ES of a mixture model. We design a splitting and MCMC scheme able to capture extremes far in the tails of mixture model. We establish convergence results (impact of the sampling size) and illustrate the experiments with realistic examples.