# Minisymposia

There will be minisymposia on:

Systemic Risk.
Speakers:
Rama Cont (Imperial College London and CNRS) – Systemic risk in heterogeneous financial networks: the case for targeted capital requirements
Tom Hurd (McMaster University) – Illiquidity and Insolvency Cascades in the Interbank Network
Garence Staraci (Yale University) – Systemic Risk: About its Nature, Regulation and some Modeling Caveats.
Stochastic control, optimization.
Speakers:
Bruno Bouchard (University Paris Dauphine) – Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions
Marcel Nutz (Columbia University) – Nonlinear Lévy Processes and their Characteristics
Huyen Pham (University Paris Diderot) – Randomization approach and backward SDE representation for optimal control of non-Markovian SDEs
Numerical probability.
Speakers:
Thomas Gerstner (Frankfurt University) – Parallel adaptive multilevel Monte Carlo simulation
Ahmed Kebaier (University Paris 13) – Importance Sampling for the Multilevel Monte Carlo method
John Schoenmakers (Weierstrass Institute Berlin) – Multilevel dual valuation and multilevel policy iteration for pricing American options
Asymptotic methods.
Speakers:
Stefano De Marco (Ecole Polytechnique) – Density asymptotics for diffusions, old and new
Antoine Jacquier (Imperial College London) – Asymptotics of forward implied volatility
Johannes Muhle-Karbe (ETH Zürich) – Optimal liquidity provision in limit order markets
Order book modeling and microstructure.
Speakers:
Jim Gatheral (Baruch College) – Minimizing execution costs
Charles-Albert Lehalle (Capital Fund Management) – Modeling and Understanding Market Microstructure: the case of Limit Orderbook Dynamics
Fabrizio Lillo (Scuola Normale Superiore di Pisa and Santa Fe Institute) – Modeling the coupled return-spread high frequency dynamics of large tick assets
Model-free hedging.
Speakers:
Mathias Beiglbock (University of Vienna) – A “Variational Principle” for Model-Independent Finance
Alexander Cox (University of Bath) – Optimal robust bounds for variance options and asymptotically extreme models
Gilles Pagès (University Pierre et Marie Curie) – Convex order for path-dependent derivatives: a dynamic programming approach
Numerical methods for non-linear equations in finance (CVA, BSDE…).
Speakers:
Jean-Francois Chassagneux (Imperial College London) – Stability analysis of approximation schemes for BSDEs
Samuel Cohen (Oxford University) – EBSDEs and risk averse networks
Stéphane Crépey (University Evry) – Wrong-Way and Gap Risk Modeling
Pierre Henry-Labordère (Société Générale) – Branching diffusion for non-linear equations
Practitioner sessions
Speakers:
P. Balland (UBS) – Normal Expansion of SABR
P. Delanoe (HSBC) – Local Correlation With Local Vol and Stochastic Vol
L. De Leo (CFM) – Smile in the low moments (How to trade options, and survive to tell the tale)
J. Guyon (Bloomberg) – Calibration of Local Correlation Models to Basket Smiles
B. Huge (Danske Bank) – Wots me Delda
N. Kahale (ESCP Europe) – Super-Replication of Financial Derivatives Via Convex Programing
C. Martini (Zeliade Systems) – Calibration of the SSVI model and applications
J.J. Rabeyrin (BNP Paribas) – How to survive in a non linear world?
A. Reghai (Natixis) – Financial Mathematics + Scientific Computation : the winning combination
P. Balland: Normal Expansion of SABR
Abstract: The SABR dynamic is very popular in interest rates because it provides an intuitive parametrisation of volatility smiles. This dynamic assumes a CEV backbone, and consequently, the rate is eventually absorbed at zero. The classic approximation for call prices under SABR is based on an asymptotic expansion of the implied lognormal volatility known as SABR formula. Since the rate in a lognormal model is not absorbed at zero, the asymptotic expansion loses accuracy for low strikes when the CEV exponent is small. This difference in asymptotes between SABR and the lognormal model can significantly affect the accuracy of the SABR formula. In this paper, we propose a more accurate approximation by expanding the volatility implied from the Normal SABR model instead of the lognormal volatility.

M. Beiglbock: A “Variational Principle” for Model-Independent Finance
Abstract: Model-independent pricing has grown into an independent field in Mathematical Finance during the last 15 years. A driving inspiration in this area has been the fruitful connection to the Skorokhod embedding problem. We discuss a more recent approach to model-independent pricing, based on a link to Monge-Kantorovich optimal transport. Based on a similar technique in optimal transport we derive a “variational principle” that is applicable to model-independent pricing. This transport-viewpoint also sheds new light on Skorokhod’s classical problem.

B. Bouchard: Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions
Abstract: We study a class of stochastic target games where one player tries to find a strategy such that the state process almost-surely reaches a given target, no matter which action is chosen by the opponent. Our main result is a geometric dynamic programming principle which allows us to characterize the value function as the viscosity solution of a non-linear partial differential equation. Because abstract measurable selection arguments cannot be used in this context, the main obstacle is the construction of measurable almost-optimal strategies. We propose a novel approach where smooth supersolutions are used to define almost-optimal strategies of Markovian type, similarly as in ver- ification arguments for classical solutions of Hamilton–Jacobi–Bellman equations. The smooth supersolutions are constructed by an extension of Krylov’s method of shaken coefficients. We apply our results to a problem of option pricing under model uncertainty with different interest rates for borrowing and lending. Joint work with Marcel Nutz.

J.F. Chassagneux: Stability analysis of approximation schemes for BSDEs
Abstract: In this talk, I will focus on the stability issues of numerical approximations of BSDEs. By extending the ‘classical’ stability analysis for the Euler Scheme, I will first show that it is possible to prove convergence results for a large class of multi-stage/multi-step schemes. Under some suitable smoothness assumptions, these schemes have high order of convergence. I will then introduce the notion of ‘numerical stability’ for approximations of BSDEs and discuss it in details for one-step scheme. It turns out that this notion is very important in practice. For the Euler scheme, I will exhibit a sufficient and necessary condition for numerical stability.

S. Cohen: EBSDEs and risk averse networks
Abstract: When studying a financial network, one is often interested in the importance of a particular node. This can be measured in various ways, for example, by the ergodic probabilities of an associated Markov chain. We consider ergodic BSDEs based on countable state Markov chains, and use these to derive nonlinear, risk-averse versions of these probabilities and similar quantities. With this machinery, one can also consider various problems in ergodic stochastic control, and can incorporate model or statistical uncertainties into the assessment of the importance of different nodes and groups of nodes.

A. Cox: Optimal robust bounds for variance options and asymptotically extreme models
Abstract: Robust, or model-independent properties of the variance swap are well-known, and date back to Dupire and Neuberger, who showed that, given the price of co-terminal call options, the price of a variance swap was exactly specified under the assumption that the price process is continuous. In Cox and Wang we showed that a lower bound on the price of a variance call could be established using a solution to the Skorokhod embedding problem due to Root. In this talk, I will show that the Heston-Nandi model is ‘asymptotically extreme’ in the sense that, for large maturities, the Heston-Nandi model gives prices for variance call options which are approximately the lowest values consistent with the same call price data. (Joint work with Jiajie Wang).

S. Crepey: Wrong-Way and Gap Risk Modeling
Abstract: We develop a reduced-form approach for a class of nonstandard BSDEs with random terminal time, random but not necessarily endowed with an intensity (possibly predictable). This is done under a relaxation of the classical immersion hypothesis, stated in terms of a suitably changed probability measure. These results are then applied to wrong-way and gap risks modeling in counterparty risk applications. (Joint work with S. Song).

P. Delanoe: Local Correlation With Local Vol and Stochastic Vol
Abstract: It is well known that a model with constant correlation is unable to fit the index smile. We therefore introduce a local correlation model to solve this issue and present our calibration procedure in the case where our underlyings follow a local vol model or a stochastic vol model. We also introduce new local formulae – which are equivalent to Dupire formulae – for Rainbow options.

L. De Leo: Smile in the low moments (How to trade options, and survive to tell the tale)
Abstract: We take the viewpoint of an option trader who is willing to trade an option and hold it until expiry while re-hedging his position on a regular basis. A fair price estimation is needed in order to decide whether the quoted option is to be sell or bought. One way is to estimate the implied volatility by looking at the the underlying price dynamics. This procedure usually involves high moments of the return distribution, which makes it a numerically unstable task. We present an alternative formula based on low order moments. We show how the coefficients of this formula can be computed as the payoff of exotic options, and how the calculation of these payoffs can be made more precise using the Hedged Monte Carlo method. In the second part of the talk, we show how the formalism can be applied to compute the skew-stickiness ratio in a generic non-linear model.

S. De Marco: Density asymptotics for diffusions, old and new
Abstract: The tools of large deviations theory and classical heat kernel asymptotics have been widely applied in the quantitative finance literature. Very recently, revisiting and extending the work of classical authors (Varadhan, Bismut, Léandre, Ben Arous…) has lead to new interesting applications: checkable conditions for the validity of marginal density expansions (Deuschel et al. 2013), asymptotics for partially conditioned diffusions, yielding extrapolation tool for Dupire local volatilities (DM Friz 2013), asymptotics and special phenomena in multi-dimensional models. If the small-time regime has been often tackled in the previous literature, the more general small-noise regime, though requiring to overtake some non-trivial technical complications, lends itself to understand both small-time and in some cases spatial asymptotics. Recent work of different authors on linear combinations of multi-dimensional processes paves the road for the applications of these techniques to market models in higher dimension.

T. Gerstner: Parallel adaptive multilevel Monte Carlo simulation
Abstract: We use the multilevel Monte Carlo method to compute prices of financial derivatives and combine this method with two adaptive algorithms. In the first algorithm we consider time discretization and sample size as two separate dimensions and use dimension-adaptive refinement to optimize the error with respect to these dimensions in relation to the computational costs. The second algorithm uses locally adaptive timestepping and is constructed especially for non-Lipschitz payoff functions whose weak and strong order of convergence is reduced when the Euler-Maruyama method is used to discretize the underlying stochastic differential equation. The numerical results show that for barrier options the convergence order for smooth payoffs can be recovered in these cases. In order to implement these methods on parallel computers we apply parallelization techniques which are able to obtain a linear speed-up.

J. Guyon: Calibration of Local Correlation Models to Basket Smiles
Abstract: Allowing correlation to be local, i.e., state-dependent, in multi-asset models allows better hedging by incorporating correlation moves in the delta. When options on a basket, be it a stock index, a cross FX rate, or an interest rate spread, are liquidly traded, one may want to calibrate a local correlation to these option prices. Only two particular solutions have been suggested so far in the literature. Both solutions impose a particular dependency of the correlation matrix on the asset values that one has no reason to undergo. By combining the particle method (see Being Particular About Calibration, Guyon and Henry-Labordere, Risk magazine, January 2012) with a new simple idea, we build whole families of calibrated local correlation models, which include the two existing models as special cases. For the first time, one can now design a calibrated local correlation in order to fit a view on the correlation skew, or reproduce historical correlation, or match some exotic option prices, thus improving the pricing, hedging, and risk-management of multi-asset derivatives. We also show how to generalize this technique to calibrate (i) models that combine stochastic interest rates, stochastic dividend yield, local stochastic volatility, and local correlation; and (ii) single-asset path-dependent volatility models. Numerical results show the wide variety of calibrated local correlations and give insight on a difficult (still unsolved) problem: find lower bounds/upper bounds on general multi-asset option prices given the whole surfaces of implied volatilities of a basket and its constituents.

T. Hurd: Illiquidity and Insolvency Cascades in the Interbank Network
Abstract: This talk will focus on some issues that arise when trying to gain a mathematical understanding of financial systemic risk. What are the most important attributes of the interbank network that determine its resilience to the cascading of financial shocks that are the definition of systemic risk? One point of view is that “toy models” of systemic risk, especially if amenable to exact probabilistic analysis, are the appropriate starting point in understanding such complex adapted systems. This approach can be useful, for example, in mapping out how network stability depends on behavioural assumptions, key structural parameters, and other model variables, and can serve as a guide to applying slower, possibly surer, Monte Carlo methods. I will present a rather intricate solvable model of “double cascades” that result when normal banks can become either stressed (illiquid) or insolvent. How does this model work, how useful is it, and what are some of the conclusions that can be drawn from it?

A. Jacquier: Asymptotics of forward implied volatility
Abstract: We study here the asymptotic behaviour of the forward implied volatility (namely the implied volatility corresponding to forward-start European options). Our tools rely on (finite-dimensional) large deviations and saddlepoint analysis, albeit not always relying on standard convexity arguments. We shall also relate this to the Freidlin-Wentzell approach for sample paths. From a practical point of view, this sheds light on the dynamics of forward implied volatilities, which we highlight numerically in the Heston model.

N. Kahale: Super-Replication of Financial Derivatives Via Convex Programing
Abstract: We give a method based on convex programming to calculate the optimal super-replicating and sub-replicating prices and corresponding hedging strategies of a financial derivative in terms of other financial derivatives. Our method finds a model that matches the superreplicating (or sub-replicating) price within an arbitrary precision and is consistent with the other financial derivatives prices. Applications include robust replication in terms of call prices with various strikes and maturities of forward start options, volatility and variance swaps and derivatives, cliquets calls, barrier options, lookback and Asian options. Numerical examples show that, in some cases, the super-replicating and/or sub-replicating prices are within 10% of the Black-and-Scholes price but considerably differ from it in other cases. Our method can take into account bid-ask spreads, interest rates and dividends and various limitations to the diffusion model.

A. Kebaier : Importance Sampling for the Multilevel Monte Carlo method
Abstract : (joint work with M. Ben Alaya and K. Hajji University Paris 13) The efficiency of Monte Carlo simulations is significantly improved when implemented with variance reduction methods. Among these methods we focus on the popular importance sampling technique based on producing a parametric transformation through a shift parameter $\theta$. The optimal choice of $\theta$ is approximated using Robbins-Monro procedures, provided that a non explosion condition is satisfied. In the First part of the talk, we introduce a new algorithm based on a combination of the Multilevel Monte Carlo method and the importance sampling technique. In the setting of discritized diffusions, the Multilevel Monte Carlo method is known for reducing efficiently the complexity compared to the classical Monte Carlo one. We prove the almost sure convergence of both constrained and unconstrained versions of the Robbins-Monro routine, towards the optimal shift $\theta^*$ that minimizes the variance associated to the Multilevel Monte Carlo. Then, we prove a central limit theorem for this new algorithm. In the second part of the talk we extend the above procedure in the case when no discretizing scheme is used. More precisely, we introduce a new algorithm reducing both variance and computational effort associated to the effective computation of option prices when the underlying asset process follows an exponential pure jump Lévy model.

C.-A. Lehalle. Modeling and Understanding Market Microstructure: the case of Limit Orderbook Dynamics
Abstract: Since the financial crisis and recent regulation adjustements, market microstructure became a key element of the functioning of financial markets. It is where the price formation process takes place, and its study allows to understand how liquidity states influence the outcome of the well known “offer and demand balance” that one hope to make the price converges towards the “fair price”. I will start from practical observations of market microstructure recent trends (following the book “Market Microstructure in Practice” by Lehalle, Laruelle, Burgot, Pelin and Lasnier) and then I will present the main features of a “Mean Field Game” modeling of limit orderbook dynamics (following the paper “Efficiency of the Price Formation Process in Presence of High Frequency Participants: a Mean Field Game analysis” by Lachapelle, Lasry, Lehalle and Lions).

F. Lillo: Modeling the coupled return-spread high frequency dynamics of large tick assets
Abstract: Large tick assets, i.e. assets where one tick movement is a significant fraction of the price and bid-ask spread is almost always equal to one tick, display a dynamics in which price changes and spread are strongly coupled. Examples of large tick assets are found among futures and equities. We introduce a Markov-switching modelling approach that describes this coupling and the dynamics of spread and return in transaction time. The latent Markov process is the transition process between spreads. Returns and spread are represented by discrete variables. A Markov-switching model with regressors can account for stochastic volatility, anomalous scaling of
kurtosis, and possible inefficiencies of the mid-price process. This autoregressive model is a finite Markov mixture of logit regressions. We show that it is possible to reduce this model to a double chain Markov model. We calibrate our models on Nasdaq stocks and we find that this models reproduce remarkably well the statistical properties of real data. Joint work with Gianbiagio Curato.

C. Martini: Calibration of the SSVI model and applications
Abstract: Gatheral and Jacquier achieved in 2012 a consistent (arbitrage-free) extension of the parametric SVI model in the maturity dimension. This Surface SVI (SSVI) model is parameterized by a correlation coefficient and a function which corresponds to the curvature of the smile at each maturity. We go through a re-parameterization of the SSVI model that lends itself to a nice 2-stages calibration procedure. Calibration examples on CBOE SPX delayed quotes are provided. Since the SSVI model does calibrate very well, we eventually get an explicit arbitrage-free parameterization of the market implied volatility surface. We compute the model-free Beiglböck-Juillet-Touzi-Henry Labordère optimal transport bounds of an exotic option in this setting. An executable version of this work is available on the Zanadu platform (joint work with I.Laachir, ENSTA and Zeliade Systems).

J. Muhle-Karbe: Optimal liquidity provision in limit order markets
Abstract: A small investor provides liquidity at the best bid and ask prices of a limit order market. For small spreads and frequent orders of other market participants, we explicitly determine the investor’s optimal policy and welfare. In doing so, we allow for general dynamics of the mid price, the spread, and the order flow, as well as for arbitrary preferences of the liquidity provider under consideration. (Joint work with Christoph Kühn).

M. Nutz: Nonlinear Lévy Processes and their Characteristics
Abstract: A nonlinear Lévy process yields a model for Knightian uncertainty about the distribution of jumps. Such a model is tractable in that conditional expectations of Markovian functionals can be calculated by means of a partial integro-differential equation. In this talk, we develop a general construction for nonlinear Lévy processes with given characteristics. More precisely, given a set $U$ of Lévy triplets, we construct a sublinear expectation on Skorohod space under which the canonical process has stationary independent increments and a nonlinear generator corresponding to the supremum of all generators of classical Lévy processes with triplets in $U$. (Joint work with Ariel Neufeld.)

G. Pagès: Convex order for path-dependent derivatives: a dynamic programming approach
Abstract: Is the premium of an (American) option written on (a) convex path dependent payoff(s) monotonic as a functional of its volatility in a (possibly jumpy) local volatility model? We will she that the answer to this question is positive under natural assumptions . More generally, we will explore the functional convex order for martingale diffusions and stochastic integrals with respect to their diffusion coefficients in both Brownian and jump frameworks. We extend this result to the Snell envelope of functionals of these process i.e. to American options with (convex) pathwise dependent payoffs. Our approach relies on an extensive use of functional limit theorems to transfer the convex order property from discrete time Markov dynamics (like ARCH models representative of Euler schemes) to continuous time processes. Our assumptions are “bordered” by several counterexamples. We will also show that this purl probabilistic approach unifies and often extend former results which go back to Hajek, El Karoui-Jeanblanc-Shreve and more recently Rüschendorf.

H. Pham: Randomization approach and backward SDE representation for optimal control of non-Markovian SDEs
Abstract: We study optimal stochastic control problem for non-Markovian stochastic differential equations (SDEs) where the drift, diffusion coefficients, and gain functionals are path-dependent, and importantly we do not make any ellipticity assumption on the SDE. We develop a controls randomization approach, and prove that the value function can be reformulated under a family of dominated measures on an enlarged filtered probability space. This value function is then characterized by a backward SDE with nonpositive jumps under a single probability measure, which can be viewed as a path-dependent version of the Hamilton-Jacobi-Bellman (HJB) equation, and an extension to G-expectation. The derivation relies on original approximation results for continuous processes by pure jump processes, and circumvents delicate issues of dynamic programming. In the Markovian case, our BSDE representation provides a new probabilistic numerical scheme for solving fully nonlinear HJB equation, taking advantage of high dimensional features of Monte-Carlo methods.

J.J. Rabeyrin: How to survive in a non linear world?
Abstract: We’ll have a look at the different funding conditions that may appear in real life and will then write the corresponding equations to price derivatives. In a second step, we’ll then examine the different practical and organisational problems that may appear and discuss some practical solutions.

J. Schoenmakers: Multilevel dual valuation and multilevel policy iteration for pricing American options
Abstract: In this talk we propose two novel simulation based approaches for pricing American options. The first one (I) is in fact a multi level version of the well-known nested Monte Carlo algorithm of Andersen and Broadie (2004), whereas the second one (II) is a multi level version of simulation based policy iteration.
Add. I) We consider a sequence of martingales that converges to a given target martingale and decompose the original dual representation for American options into a sum of representations that correspond to different levels of approximation to the target martingale. By next replacing in each representation true conditional expectations with their Monte Carlo estimates, we arrive at a multilevel type dual Monte Carlo algorithm. The analysis of this algorithm reveals that the computational complexity of getting the corresponding target upper bound, due to the target martingale, can be significantly reduced. In particular, it turns out that using our new approach, we may obtain a multilevel version of the Andersen and Broadie algorithm that is, regarding complexity, virtually equivalent to a non-nested algorithm. (I) is joint work with Denis Belomestny and Fabian Dickmann.
Add. II) By constructing a sequence of Monte Carlo based policy iterations due different levels of accuracy we construct a multi level version of policy iteration with significantly improved complexity. In this context, we will present new convergence results regarding the bias and variance of simulation based Howard iteration (cf. Kolodko Sch. (2006)) and show that the multi level complexity is superior to the standard one. (II) is joint work with Denis Belomestny and Marcel Ladkau.

G. Staraci: Systemic Risk: About its Nature, Regulation and some Modeling Caveats.
Abstract: We first discuss, in light of historical events, the nature of systemic risk and the distinction between loss contagion and informational contagion. We then address the systemic risk regulation problem, and comment on the successes and failures of the US Dodd-Frank legislation for that purpose. We finally offer an additional insight of cascade dynamics and systemic risk within a graph theoretical approach.