Past Mathematical and Computational Finance Internal Seminar

This talk will discuss efficient numerical methods for estimating the
probability of a large portfolio loss, and associated risk measures such
as VaR and CVaR.  These involve nested expectations, and following
Bujok, Hambly & Reisinger (2015) we use the number of samples for the
inner conditional expectation as the key approximation parameter in the
Multilevel Monte Carlo formulation.  The main difference in this case is
the indicator function in the definition of the probability. Here we
build on previous work by Gordy & Juneja (2010) who analyse the use of a
fixed number of inner samples , and Broadie, Du & Moallemi (2011) who
develop and analyse an adaptive algorithm.  I will present the
algorithm, outline the main theoretical results and give the numerical
results for a representative model problem.  I will also discuss the
extension to real portfolios with a large number of options based on
multiple underlying assets.

Joint work with Abdul-Lateef Haji-Ali

This work deals with a class of stochastic optimal control problems in the presence of state constraints. It is well known that for such problems the value function is, in general, discontinuous, and its characterisation by a Hamilton-Jacobi equation requires additional assumptions involving an interplay between the boundary of the set of constraints and the dynamics
of the controlled system. Here, we give a characterization of the epigraph of the value function without assuming the usual controllability assumptions. To this end, the stochastic optimal control problem is first translated into a state-constrained stochastic target problem. Then a level-set approach is used to describe the backward reachable sets of the new target problem. It turns out that these backward reachable sets describe the value function. The main advantage of our approach is that it allows us to easily handle the state constraints by an exact penalisation. However, the target problem involves a new state variable and a new control variable that is unbounded.
 

In the 12 months from the middle of June 2016 to the middle of June 2017, a number of events occurred in a relatively short period of time, all of which either had, or had the potential to have,  a considerably volatile impact upon financial markets. The events referred to here are the Brexit  referendum (23 June 2016), the US election (8 November 2016), the 2017 French elections (23 April and 7 May 2017) and the surprise 2017 UK parliamentary election (8 June 2017). 
All of these events - the Brexit referendum and the Trump election in particular - were notable both for their impact upon financial markets after the event and the degree to which the markets failed to anticipate these events. A natural question to ask is whether these could have been predicted, given information freely available in the financial markets beforehand. In this talk, we focus on market expectations for price action around Brexit and the Trump election, based on information available in the traded foreign exchange options market. We also investigate the horizon date of 30 March 2019, when the two year time window that started with the Article 50 notification on 29 March 2017 will terminate.
Mathematically, we construct a mixture model corresponding to two scenarios for the GBPUSD exchange rate after the referendum vote, one scenario for “remain” and one for “leave”. Calibrating this model to four months of market data, from 24 February to 22 June 2016, we find that a “leave” vote was associated with a predicted devaluation of the British pound to approximately 1.37 USD per GBP, a 4.5% devaluation, and quite consistent with the observed post-referendum exchange rate move down from 1.4877 to 1.3622. We find similar predictive power for USDMXN in the case of the 2016 US presidential election. We argue that we can apply the same bimodal mixture model technique to construct two states of the world corresponding to soft Brexit (continued access to the single market) and hard Brexit (failure of negotiations in this regard).
 

In this paper, we consider the pricing and hedging of a financial derivative for an insider trader, in a model-independent setting. In particular, we suppose that the insider wants to act in a way which is independent of any modelling assumptions, but that she observes market information in the form of the prices of vanilla call options on the asset. We also assume that both the insider’s information, which takes the form of a set of impossible paths, and the payoff of the derivative are time-invariant. This setup allows us to adapt recent work of Beiglboeck, Cox, and Huesmann [BCH16] to prove duality results and a monotonicity principle, which enables us to determine geometric properties of the optimal models. Moreover, we show that this setup is powerful, in that we are able to find analytic and numerical solutions to certain pricing and hedging problems. (Joint with B. Acciaio and M. Huesmann)

We discuss pathwise pricing-hedging dualities in continuous time and on a frictionless market consisting of finitely many risky assets with continuous price trajectories.

Rita Maria del Rio Chanona:

Global financial contagion on a Multiplex Network

We explore the global financial system, in particular the risk of global financial contagion through network theory. Although there is extensive literature on contagion in networks, we argue that it is important to consider different channels of contagion. Therefore we deem into the multilayer framework, where nodes are countries and each layer represents a different type of financial obligation. The multiplex network is built using data provided by collaborators in the IMF. We study contagion with a percolation model and conclude that financial shocks can be amplified considerably when the multilayer structure is taken into account.


Johannes Wiesel:

Robust Superhedging vs Robust Statistics

In this talk I try to reconcile the different understanding of robustness in mathematical finance and statistics. Motivated by recent advances in the estimation of risk measures, I present estimators for the superhedging price of a claim given a history of observed prices. I discuss weak efficiency and convergence speed of these estimators. Besides I explain how to apply classical notions of sensitivity for the estimation procedure. This talk is based on ongoing work with Jan Obloj.

 

Christoph Siebenbrunner:

Clearing Algorithms and Network Centrality

I show that the solution of a standard clearing model commonly used in contagion analyses for financial systems can be expressed as a specific form of a generalized Katz centrality measure under conditions that correspond to a system-wide shock. This result provides a formal explanation for earlier empirical results which showed that Katz-type centrality measures are closely related to contagiousness. It also allows assessing the assumptions that one is making when using such centrality measures as systemic risk indicators. I conclude that these assumptions should be considered too strong and that, from a theoretical perspective, clearing models should be given preference over centrality measures in systemic risk analyses.


Andreas Sojmark:

An SPDE Model for Systemic Risk with Default Contagion

In this talk, I will present a structural model for systemic risk, phrased as an interacting particle system for $N$ financial institutions, where each institution is removed upon default and this has a contagious effect on the rest of the system. Moreover, the financial instituions display herding behavior and they are exposed to correlated noise, which turns out to be an important driver of the contagion mechanism. Ultimately, the motivation is to provide a clearer connection between the insights from dynamic mean field models and the detailed study of contagion in the (mostly static) network-based literature. Mathematically, we prove a propagation of chaos type result for the large population limit, where the limiting object is characterized as the unique solution to a nonlinear SPDE on the positive half-line with Dirichlet boundary. This is based on joint work with Ben Hambly and I will also point out some interesting future directions, which are part of ongoing work with Sean Ledger.

Martingale optimal transport is a variant of the classical optimal transport problem where a martingale constraint is imposed on the coupling. In a recent paper, Beiglböck, Nutz and Touzi show that in dimension one there is no duality gap and that the dual problem admits an optimizer. A key step towards this achievement is the characterization of the polar sets of the family of all martingale couplings. Here we aim to extend this characterization to arbitrary finite dimension through a deeper study of the convex order

 

I spent a number of years trading government bonds and interest-rate derivatives for Barclays Capital. This included the period of the financial crisis, and I was a colleague of some of the Barclays traders charged with fraud related to LIBOR rate manipulation. I will present a some examples of common trading scenarios, and some of the ethical issues these might raise for quants.
 

The efficient pricing and hedging of vanilla and exotic foreign exchange options requires an adequate model that takes into account both the local and the stochastic features of the volatility dynamics. In this joint work, we put forward a four-factor hybrid local-stochastic volatility (LSV) model that combines state-of-the-art dynamics for the exchange rate with stochastic dynamics for the domestic and foreign short rates, and provide a consistent and self-contained calibration and pricing framework.
For the calibration, we propose a novel and generic algorithm that builds on the particle method of Guyon and Labordere. We combine it with new variance reduction techniques to accelerate convergence and use control variates derived from a pure local volatility model, the stochastic short rates and the two-factor Heston-type LSV model. Our numerical experiments show a dramatic variance reduction that allows us to calibrate the four-factor model at almost no extra computational cost. The method can be applied to a large class of hybrid LSV models and is not restricted to our particular choice of the diffusion.
For the pricing, we propose a Monte Carlo simulation scheme that combines the full truncation Euler (FTE) scheme for the stochastic volatility and the stochastic short rates with the log-Euler scheme for the exchange rate. We find a lower bound on the explosion time of exponential moments of FTE approximations, and prove the strong convergence of the exchange rate approximations and the convergence of Monte Carlo estimators for a number of vanilla and exotic options. We then carry out numerical experiments to justify our choice of model and demonstrate convergence.
 

Speaker: Yixuan Wang
Titile: Minimum resting time with market orders
Abstract:  Regulators have been discussing possible rules to control high frequency trading and decrease market speed, and minimum resting time is one of them. We develop a simple mathematical model, and derive an asymptotic expression of the expected PnL, which is also the performance criteria that a market maker would like to maximize by choosing the optimal depth at which she posts the limit order. We investigate the comparative statistics of the optimal depth with each parameters, an in particular the comparative statistics show that the minimum resting time will decrease the market liquidity, forcing the market makers to post limit orders of volume 1.


Speaker: Marco Pangallo
Title: Does learning converge in generic games?
Abstract: In game theory, learning has often been proposed as a convincing method to achieve coordination on an equilibrium. But does learning converge, and to what? We start investigating the drivers of instability in the simplest possible non-trivial setting, that is generic 2-person, 2-strategy normal form games. In payoff matrices with a unique mixed strategy equilibrium the players may follow the best-reply cycle and fail to converge to the Nash Equilibrium (NE): we rather observe limit cycles or low-dimensional chaos. We then characterize the cyclic structure of games with many moves as a combinatorial problem: we quantify exactly how many best-reply configurations give rise to cycles or to NE, and show that acyclic (e.g. coordination, potential, supermodular) games become more and more rare as the number of moves increases (a fortiori if the payoffs are negatively correlated and with more than two players).  In most games the learning dynamics ends up in limit cycles or high-dimensional chaotic attractors, preventing the players to coordinate. Strategic interactions would then be governed by learning in an ever-changing environment, rather than by rational and fully-informed equilibrium thinking.
Collaborators: J. D. Farmer, T. Galla, T. Heinrich, J. Sanders

The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this talk, we consider an extension of the weak formulation of the optimal Skorokhod embedding problem. Using the classical convex duality approach together with the optimal stopping theory, we establish some duality. Moreover, based on the duality, we provide an alternative proof of the monotonicity principle proved by Beiglbock, Cox and Huesmann.