Past Mathematical and Computational Finance Seminar

9 November 2017
16:00
to
17:30
Abstract

This paper formulates an utility indifference pricing model for investors trading in a discrete time financial market under non-dominated model uncertainty.
The investors preferences are described by strictly increasing concave random functions defined on the positive axis. We prove that under suitable
conditions the multiple-priors utility indifference prices of a contingent claim converge to its multiple-priors superreplication price. We also
revisit the notion of certainty equivalent for random utility functions and establish its relation with the absolute risk aversion.

  • Mathematical and Computational Finance Seminar
19 October 2017
16:00
to
17:30
Julien Guyon
Abstract

We derive sharp bounds for the prices of VIX futures using the full information of S&P 500 smiles. To that end, we formulate the model-free sub/superreplication of the VIX by trading in the S&P 500 and its vanilla options as well as the forward-starting log-contracts. A dual problem of minimizing/maximizing certain risk-neutral expectations is introduced and shown to yield the same value. The classical bounds for VIX futures given the smiles only use a calendar spread of log-contracts on the S&P 500. We analyze for which smiles the classical bounds are sharp and how they can be improved when they are not. In particular, we introduce a tractable family of functionally generated portfolios which often improves the classical spread while still being tractable, more precisely, determined by a single concave/convex function on the line. Numerical experiments on market data and SABR smiles show that the classical lower bound can be improved dramatically, whereas the upper bound is often close to optimal.

  • Mathematical and Computational Finance Seminar
12 October 2017
16:00
to
17:30
Charles-Albert Lehalle
Abstract

This talk explains how to formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a " background noise " (or " mean field "). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. Here the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of " extended MFG ", we hence provide generic results to address these " MFG of controls ", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of " heterogenous preferences " (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can " learn " it day after day, observing others' behaviors.

  • Mathematical and Computational Finance Seminar
15 June 2017
16:00
to
17:30
Claudio Fontana
Abstract

We consider the problem of modelling the term structure of defaultable bonds, under minimal assumptions on the default time. In particular, we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. It turns out that this requires the introduction of an additional term in the forward rate approach by Heath, Jarrow and Morton (1992). This term is driven by a random measure encoding information about those times where default can happen with positive probability.  In this framework, we  derive necessary and sufficient conditions for a reference probability measure to be a local martingale measure for the large financial market of credit risky bonds, also considering general recovery schemes. This is based on joint work with Thorsten Schmidt.

  • Mathematical and Computational Finance Seminar
8 June 2017
16:00
to
17:30
Antoine Savine
Abstract

This document reviews the so called least square methodology (LSM) and its application for the valuation and risk of callable exotics and regulatory value adjustments (xVA). We derive valuation algorithms for xVA, both with or without collateral, that are particularly accurate, efficient and practical. These algorithms are based on a reformulation of xVA, designed by Jesper Andreasen and implemented in Danske Bank's award winning systems, that hasn't been previously published in full. We then investigate the matter of risk sensitivities, in the context of Algorithmic Automated Differentiation (AAD). A rather recent addition to the financial mathematics toolbox, AAD is presently generally acknowledged as a vastly superior alternative to the classical estimation of risk sensitivities through finite differences, and the only practical means for the calculation of the large number of sensitivities in the context of xVA. The theory and implementation of AAD, the related check-pointing techniques, and their application to Monte-Carlo simulations are explained in numerous textbooks and articles, including Giles and Glasserman's pioneering Smoking Adjoints. We expose an extension to LSM, and, in particular, we derive an original algorithm that resolves the matters of memory consumption and efficiency in differentiating simulations together with the LSM step.

  • Mathematical and Computational Finance Seminar

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