Past Mathematical and Computational Finance Seminar

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.

 In the first part of the talk, we present some recent and new developments in the theory of control and optimal stopping with nonlinear expectations. We first introduce an optimal stopping game with nonlinear expectations (Generalized Dynkin Game) in a non-Markovian framework and study its links with nonlinear doubly reflected BSDEs. We then present some new results (which are part of an ongoing work) on mixed stochastic stochastic control/optimal stopping problems (as well as stochastic control/optimal stopping game problems) in a non-Markovian framework and their relation with constrained reflected BSDEs with lower obstacle (resp. upper obstacle). These results are obtained using some technical tools of stochastic analysis. In the second part of the talk, we discuss applications to the $\cal{E}^g$ pricing of American options and Game options in complete and incomplete markets (based on joint works with M.C.Quenez and Agnès Sulem).
 

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.

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.

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.

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.

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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.

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.)
 

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

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.

Cover’s celebrated theorem states that the long run yield of a properly chosen “universal” portfolio is as good as the long run yield of the best retrospectively chosen constant rebalanced portfolio. We formulate an abstract principle behind such a universality phenomenon valid for general optimization problems in the long run. This allows to obtain new results on modelfree portfolio optimization, in particular in continuous time, involving larger classes of investment strategies. These modelfree results are complemented by a comparison with the log-optimal numeraire portfolio when fixing a stochastic model for the asset prices. The talk is based on joint work with Walter Schachermayer and Leonard Wong.

Thomas Piketty's influential book “Capital in the Twenty-First Century” documents the marked and unequivocal rise of income and wealth inequality observed across the developed world 
in the last three decades. His extrapolations into the distant future are much more controversial and has 
has been subject to various criticisms from both mainstreams and heterodox economists. This motivates the search for an alternative standpoint incorporating 
heterodox insights such as endogenous money and the lessons from the Cambridge capital controversies. We argue that the Goodwin-Keen approach paves the road towards such an alternative.
We first consider a modified Goodwin-Keen model driven by consumption by households, instead of investment by firms, leading to the same qualitative features 
of the original Keen 1995 model, namely the existence of an undesirable equilibrium characterized by infinite private debt ratio and zero employment, 
in addition to a desirable one with finite debt and non-zero employment. By further subdividing the household sector into workers and investors, we are able to investigate their relative 
income and wealth ratios for in the context of these two long-run equilibria, providing a testable link between asymptotic inequality and private debt accumulation.

We consider impatient decision makers when their assets' prices are in undesirable low regions for a significant amount of time, and they are risk averse to negative price jumps. We wish to study the unusual reactions of investors under such adverse market conditions. In mathematical terms, we study the optimal exercising of an American call option in a random time-horizon under spectrally negative Lévy models. The random time-horizon is modeled by an alarm of the so-called Omega default clock in insurance, which goes off when the cumulative amount of time spent by the asset price in an undesirable low region exceeds an independent exponential random time. We show that the optimal exercise strategies vary both quantitatively and qualitatively with the levels of impatience and nervousness of the investors, and we give a complete characterization of all optimal exercising thresholds. 

The share of market making conducted by high-frequency trading (HFT) firms has been rising steadily. A distinguishing feature of HFTs is that they trade intraday, ending the day flat. To shed light on the economics of HFTs, and in a departure from existing market making theories, we model an HFT that has access to unlimited leverage intraday but must fund any end-of-day inventory at an exogenously determined cost. Even though the inventory costs only occur at the end of the day, they impact intraday price and liquidity dynamics. This gives rise to an intraday endogenous price impact mechanism. As time approaches the end of the trading day, the sensitivity of prices to inventory levels intensifies, making price impact stronger and widening bid-ask spreads. Moreover, imbalances of buy and sell orders may catalyze hikes and drops of prices, even under fixed supply and demand functions. Empirically, we show that these predictions are borne out in the U.S. Treasury market, where bid-ask spreads and price impact tend to rise towards the end of the day. Furthermore, price movements are negatively correlated with changes in inventory levels as measured by the cumulative net trading volume.
 

(based on joint work with Tobias Adrian, Erik Vogt, and Hongzhong Zhang)

We propose a new flexible unified framework for studying the time consistency property suited for a large class of maps defined on the set of all cash flows and that are postulated to satisfy only two properties -- monotonicity and locality. This framework integrates the existing forms of time consistency for dynamic risk measures and dynamic performance measures (also known as acceptability indices). The time consistency is defined in terms of an update rule, a novel notion that would be discussed into details and illustrated through various examples. Finally, we will present some connections between existing popular forms of time consistency. 
This is a joint work with Tomasz R. Bielecki and Marcin Pitera.

We develop a Bayesian methodology for systemic risk assessment in financial networks such as the interbank market. Nodes represent participants in the network and weighted directed edges represent liabilities. Often, for every participant, only the total liabilities and total assets within this network are observable. However, systemic risk assessment needs the individual liabilities. We propose a model for the individual liabilities, which, following a Bayesian approach, we then condition on the observed total liabilities and assets and, potentially, on certain observed individual liabilities. We construct a Gibbs sampler to generate samples from this conditional distribution. These samples can be used in stress testing, giving probabilities for the outcomes of interest. As one application we derive default probabilities of individual banks and discuss their sensitivity with respect to prior information included to model the network. An R-package implementing the methodology is provided. (This is joint work with Axel Gandy (Imperial College London).)

We propose a randomised version of the Heston model--a widely used stochastic volatility model in mathematical finance--assuming that the starting point of the variance process is a random variable. In such a system, we study the small- and large-time behaviours of the implied volatility, and show that the proposed randomisation generates a short-maturity smile much steeper (`with explosion') than in the standard Heston model, thereby palliating the deficiency of classical stochastic volatility models in short time. We precisely quantify the speed of explosion of the smile for short maturities in terms of the right tail of the initial distribution, and in particular show that an explosion rate of $t^\gamma$ (gamma in [0,1/2]) for the squared implied volatility--as observed on market data--can be obtained by a suitable choice of randomisation. The proofs are based on large deviations techniques and the theory of regular variations. Joint work with Fangwei Shi (Imperial College London)

We introduce sufficient conditions for the solution of a multi-dimensional, Markovian BSDE to have a density. We show that a system of BSDEs possesses a density if its corresponding semilinear PDE exhibits certain regularity properties, which we verify in the case of several examples.

Backward SDEs have proven to be a useful tool in mathematical finance. Their applications include the solution to various pricing and equilibrium problems in complete and incomplete markets, the estimation of value adjustments in the presence of funding costs, and the solution to many utility/risk optimisation type of problems.
In this work, we prove an explicit error expansion for the approximation of BSDEs. We focus our work on studying the cubature  method of solution. To profit fully from these expansions in this case, e.g. to design high order approximation methods, we need in addition to control the complexity growth of the base algorithm. In our work, this is achieved by using a sparse grid representation. We present several numerical results that confirm the efficiency of our new method. Based on joint work with J.F. Chassagneux.