Past 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
18 May 2017
16:00
to
17:30
Francesca Biagini
Abstract

We  study  the  concept  of   financial  bubble  under model uncertainty.
We suppose the agent to be endowed with a family Q of local martingale measures for  the  underlying  discounted  asset  price. The priors are allowed to be mutually singular to each other.
One fundamental issue is the definition of a well-posed concept of robust fundamental value of a given  financial asset.
Since in this setting we have no linear pricing system, we choose to describe robust fundamental values through superreplication prices.
To this purpose, we investigate a dynamic version of robust superreplication, which we use
to  introduce  the  notions  of  bubble  and  robust  fundamental  value  in  a  consistent way with the existing literature in the classical case of one prior.

This talk is based on the works [1] and [2].

[1] Biagini, F. , Föllmer, H. and Nedelcu, S. Shifting martingale measures
and the slow birth of a bubble as a submartingale, Finance and
Stochastics: Volume 18, Issue 2, Page 297-326, 2014.


[2] Biagini, F., Mancin, J.,
Financial Asset Price Bubbles under Model 
Uncertainty, Preprint, 2016.

  • Mathematical and Computational Finance Seminar
11 May 2017
16:00
to
17:30
Martin Herdegen
Abstract


We study risk-sharing equilibria with trading subject to small proportional transaction costs. We show that the frictionless equilibrium prices also form an "asymptotic equilibrium" in the small-cost limit. To wit, there exist asymptotically optimal policies for all agents and a split of the trading cost according to their risk aversions for which the frictionless equilibrium prices still clear the market. Starting from a frictionless equilibrium, this allows to study the interplay of volatility, liquidity, and trading volume.
(This is joint work with Johannes Muhle-Karbe, University of Michigan.)
 

  • Mathematical and Computational Finance Seminar
4 May 2017
16:00
to
17:30
Blanka Horvath
Abstract

We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the “rough” regime of Hurst pa- rameter H < 1/2. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation es- timates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approxi- mation formulae from CLT type log-moneyness deviations of order t1/2 (recent works of Alo`s, Le ́on & Vives and Fukasawa) to the wider moderate deviations regime.

This is work in collaboration with C. Bayer, P. Friz, A. Gulsashvili and B. Stemper

  • Mathematical and Computational Finance Seminar
27 April 2017
16:00
to
17:30
Arnulf Jentzen
Abstract

In this lecture I intend to review a few selected recent results on numerical approximations for high-dimensional nonlinear parabolic partial differential equations (PDEs), nonlinear stochastic ordinary differential equations (SDEs), and high-dimensional nonlinear forward-backward stochastic ordinary differential equations (FBSDEs). Such equations are key ingredients in a number of pricing models that are day after day used in the financial engineering industry to estimate prices of financial derivatives. The lecture includes content on lower and upper error bounds, on strong and weak convergence rates, on Cox-Ingersoll-Ross (CIR) processes, on the Heston model, as well as on nonlinear pricing models for financial derivatives. We illustrate our results by several numerical simulations and we also calibrate some of the considered derivative pricing models to real exchange market prices of financial derivatives on the stocks in the American Standard & Poor's 500 (S&P 500) stock market index.

  • Mathematical and Computational Finance Seminar

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