Past Mathematical and Computational Finance Internal Seminar

We pursue robust approach to pricing and hedging in mathematical
finance. We develop a general discrete time setting in which some
underlying assets and options are available for dynamic trading and a
further set of European options, possibly with varying maturities, is
available for static trading. We include in our setup modelling beliefs by
allowing to specify a set of paths to be considered, e.g.
super-replication of a contingent claim is required only for paths falling
in the given set. Our framework thus interpolates between
model-independent and model-specific settings and allows to quantify the
impact of making assumptions. We establish suitable FTAP and
Pricing-Hedging duality results which include as special cases previous
results of Acciaio et al. (2013), Burzoni et al. (2016) as well the
Dalang-Morton-Willinger theorem. Finally, we explain how to treat further
problems, such as insider trading (information quantification) or American
options pricing.
Based on joint works with Burzoni, Frittelli, Hou, Maggis; Aksamit, Deng and Tan.
 

In this talk, we present and analyse a class of “filtered” numerical schemes for second order Hamilton-Jacobi-Bellman (HJB) equations, with a focus on examples arising from stochastic control problems in financial engineering. We start by discussing more widely the difficulty in constructing compact and accurate approximations. The key obstacle is the requirement in the established convergence analysis of certain monotonicity properties of the schemes. We follow ideas in Oberman and Froese (2010) to introduce a suitable local modification of high order schemes, which are necessarily non-monotone, by “filtering” them with a monotone scheme. Thus, they can be proven to converge and still show an overall high order behaviour for smooth enough value functions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests. 

This talk is based on joint work with Olivier Bokanowski and Athena Picarelli.

Zhenru Wang
Title: Multi-Index Monte Carlo Estimators for a Class of Zakai SPDEs
Abstract:   
We first propose a space-time Multi-Index Monte Carlo (MIMC) estimator for a one-dimensional parabolic SPDE of Zakai type. We compare the computational cost required for a prescribed accuracy with the Multilevel Monte Carlo (MLMC) method of Giles and Reisinger (2012). Then we extend the estimator to a two-dimensional variant of SPDE. The theoretical analysis shows the benefit of using MIMC in high dimensional problems over MLMC methods. Numerical tests confirm these finding empirically.

Vadim Kaushansky
Title: An extended structural default model with jump risk
Abstact:
We consider a structural default model in an interconnected banking network as in Itkin and Lipton (2015), where there are mutual obligations between each pair of banks. We analyse the model numerically for the case of two banks with jumps in their asset value processes. Specifically, we develop a finite difference method for the resulting two-dimensional partial integro-differential equation, and study its stability and consistency. By applying this method, we compute joint and marginal survival probabilities, as well as prices of credit default swaps (CDS) and first-to-default swaps (FTD), Credit and Debt Value Adjustments (CVA and DVA).

 

We will introduce variants of the optimal transport problem, namely martingale optimal transport problem and subharmonic martingale transport problem. Their motivation is partly from mathematical finance. We will see that in dimension greater than one, the additional constraints imply interesting and deep mathematical subtlety on the attainment of dual problem, and it also affects heavily on the geometry of optimal solutions. If time permits, we will introduce still another variant of the martingale transport problem, called the multi-martingale optimal transport problem.

In practice, stochastic decision problems are often based on statistical estimates of probabilities. We all know that statistical error may be significant, but it is often not so clear how to incorporate it into our decision making. In this informal talk, we will look at one approach to this problem, based on the theory of nonlinear expectations. We will consider the large-sample theory of these estimators, and also connections to `robust statistics' in the sense of Huber.

Mining massive amounts of sequentially ordered data and inferring structural properties is nowadays a standard task (in finance, etc). I will present some results that combine and extend ideas from rough paths and machine learning that allow to give a general non-parametric approach with strong theoretical guarantees. Joint works with F. Kiraly and T. Lyons.

Wei Title: Adaptive timestep Methods for non-globally Lipschitz SDEs

Wei Abstract: Explicit Euler and Milstein methods are two common ways to simulate the numerical solutions of
SDEs for its computability and implementability, but they require global Lipschitz continuity on both
drift and diffusion coefficients. By assuming the boundedness of the p-th moments of exact solution
and numerical solution, strong convergence of the Euler-type schemes for locally Lipschitz drift has been
proved in [HMS02], including the implicit Euler method and the semi-implicit Euler method. However,
except for some special cases, implicit-type Euler method requires additional computational cost, which
is very inefficient in practice. Explicit Euler method then is shown to be divergent in [HJK11] for non-
Lipschitz drift. Explicit tamed Euler method proposed in [HJK + 12], shows the strong convergence for the
one-sided Lipschitz condition with at most polynomial growth and it is also extended to tamed Milstein
method in [WG13]. In this paper, we propose a new adaptive timestep Euler method, which shows the
strong convergence under locally Lipschitz drift and gains the standard convergence order under one-sided
Lipschitz condition with at most polynomial growth. Numerical experiments also demonstrate a better
performance of our scheme, especially for large initial value and high dimensions, by comparing the mean
square error with respect to the runtime. In addition, we extend this adaptive scheme to Milstein method
and get a higher order strong convergence with commutative noise.

Alexander Title: Functionally-generated portfolios and optimal transport

Alexander Abstract: I will showcase some ongoing research, in which I try to make links between the class of functionally-generated portfolios from Stochastic Portfolio Theory, and certain optimal transport problems.

In the first part of the talk I will review Bob Fernholz' theory of functionally generated portfolios. In the second part I will discuss questions related to the existence of short-term arbitrage opportunities.
This is joint work with Bob Fernholz and Ioannis Karatzas

We give necessary and sufficient conditions for the variance of the partial sums of stationary processes to be regularly varying in terms of the spectral measure associated with the shift operator. In the case of reversible Markov chains, or with normal transition operator we also give necessary and sufficient conditions in terms of the spectral measure of the transition operator.  

The two spectral measures are then linked through the use of harmonic measure.



This is joint work with S. Utev(University of Leicester, UK) and M. Peligrad (University of Cincinnati, USA).

This talk will discuss work-in-progress on the numerical approximation
of reflected diffusions arising from applications in engineering, finance
and network queueing models.  Standard numerical treatments with
uniform timesteps lead to 1/2 order strong convergence, and hence
sub-optimal behaviour when using multilevel Monte Carlo (MLMC).

In simple applications, the MLMC variance can be improved by through
a reflection "trick".  In more general multi-dimensional applications with
oblique reflections an alternative method uses adaptive timesteps, with
smaller timesteps when near the boundary.  In both cases, numerical
results indicate that we obtain the optimal MLMC complexity.

This is based on joint research with Eike Muller, Rob Scheichl and Tony
Shardlow (Bath) and Kavita Ramanan (Brown).

When estimated volatilities are not in perfect agreement with reality, delta hedged option portfolios will incur a non-zero profit-and-loss over time. There is, however, a surprisingly simple formula for the resulting hedge error, which has been known since the late 90s. We call this The Fundamental Theorem of Derivative Trading. This is a survey with twists of that result. We prove a more general version and discuss various extensions (including jumps) and applications (including deriving the Dupire-Gyo ̈ngy-Derman-Kani formula). We also consider its practical consequences both in simulation experiments and on empirical data thus demonstrating the benefits of hedging with implied volatility.

We examine the Foreign Exchange (FX) spot price spreads with and without Last Look on the transaction. We assume that brokers are risk-neutral and they quote spreads so that losses to latency arbitrageurs (LAs) are recovered from other traders in the FX market. These losses are reduced if the broker can reject, ex-post, loss-making trades by enforcing the Last Look option which is a feature of some trading venues in FX markets. For a given rejection threshold the risk-neutral broker quotes a spread to the market so that her expected profits are zero. When there is only one venue, we find that the Last Look option reduces quoted spreads. If there are two venues we show that the market reaches an equilibrium where traders have no incentive to migrate. The equilibrium can be reached with both venues coexisting, or with only one venue surviving. Moreover, when one venue enforces Last Look and the other one does not, counterintuitively, it may be the case that the Last Look venue quotes larger spreads.


a working version of the paper may be found here

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2630662

We provide a characterization in terms of Fatou property for weakly closed monotone sets in the space of P-quasisure bounded random variables, where P is a (eventually non-dominated) class of probability measures. Our results can be applied to obtain a topological deduction of the First Fundamental Theorem of Asset Pricing for discrete time processes, the dual representation of the superhedging price and more in general the robust dual representation for (quasi)convex increasing functionals.
This is a joint paper with T. Meyer-Brandis and G. Svindland.
 

In finance, the default time of a counterparty is sometimes modeled as the
first passage time of a credit index process below a barrier. It is
therefore relevant to consider the following question:
   If we know the distribution of the default time, can we find a unique
barrier which gives this distribution? This is known as the Inverse
First Passage Time (IFPT) problem in the literature.
   We consider a more general `smoothed' version of the inverse first
passage time problem in which the first passage time is replaced by
the first instant that the time spent below the barrier exceeds an
independent exponential random variable. We show that any smooth
distribution results from some unique continuously differentiable
barrier. In current work with B. Ettinger and T. K. Wong, we use PDE
methods to show the uniqueness and existence of solutions to a
discontinuous version of the IFPT problem.

Systemic risk in financial markets occurs when activities that are beneficial to an agent in isolation (e.g. reducing microprudential risk) cause unintended consequences due to collective interactions (usually called macroprudential risk).  I will discuss three different mechanisms through which this occurs in financial markets.   Contagion can propagate due to the market impact of trading among agents with strongly overlapping portfolios, or due to cascading failures from chains of default caused by networks of interlinked counterparty exposures.  A proper understanding of these phenomena must take both dynamics and network effects into account.  I will discuss four different examples that illustrate these points.  The first is a simple model of the market dynamics induced by Basel-style risk management, which from extremely simple assumptions shows that excessive leverage can give rise to a slowly rising price bubble followed by an abrupt crash with a time period of 10 - 15 years.  The model gives rise to a chaotic attractor whose time series closely resembles the Great Moderation and subsequent crisis.   We show that alternatives to Basel can provide a better compromise between micro and macro prudential risk.   The second example is a model of leveraged value investors that yields clustered volatility and fat-tailed returns similar to those in financial markets.  The third example is the DebtRank algorithm, which uses a similar method to PageRank to correctly quantify the way risk propagates through networks of counterparty exposures and can be used as the basis of a systemic risk tax.  The fourth example will  be work in progress to provide an early warning system for financial stress caused by overlapping portfolios.  Finally I will discuss an often neglected source of financial risk due to imbalances in market ecologies.

tba

Pengyu Wei's title: Ranking ForexMaster Players

Abstract:

In this talk I will introduce ForexMaster, a simulated foreign exchange trading platform, and how I rank players on this platform. Different methods are compared. In particular, I use random forest and a carefully chosen feature set, which includes not only traditional performance measures like Sharp ratio, but also estimates from the Plackett-Luce ranking model, which has not been used in the financial modelling yet. I show players selected by this method have satisfactory out-of-sample performance, and the Plackett-Luce model plays an important role.

Alissa Kleinnijenhuis title: Stress Testing the European Banking System: Exposure Risk & Overlapping Portfolio Risk
Abstract:
Current regulatory stress testing, as for example done by the EBA, BoE and the FED, is microprudential, non-systemic. These stress tests do not take into account systemic risk, even though the official aim of the stress test is the "test the resilience of the financial system as a whole, and the individual banks therein, to another crisis".
 Two papers are being developed that look at the interconnections between banks. One paper investigates the systemic risk in the European banking system due to interbank exposures, using EBA data. The other paper, looks at the trade-off between individual and systemic risk with overlapping portfolios. The above two "channels of contagion" for systemic risk can be incorporated in stress tests to include systemic components to the traditional non-systemic stress tests.

We pursue robust approach to pricing and hedging in mathematical finance. We consider a continuous time setting in which some underlying assets and options, with continuous paths, are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. Motivated by the notion of prediction set in Mykland [03], we include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. super-replication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model--independent and model--specific settings and allows to quantify the impact of making assumptions or gaining information. We obtain a general pricing-hedging duality result: the infimum over superhedging prices is equal to supremum over calibrated martingale measures. In presence of non-trivial beliefs, the equality is between limiting values of perturbed problems. In particular, our results include the martingale optimal transport duality of Dolinsky and Soner [13] and extend it to multiple dimensions and multiple maturities.

This talk will describe methods for computing sharp upper bounds on the probability of a random vector falling outside of a convex set, or on the expected value of a convex loss function, for situations in which limited information is available about the probability distribution. Such bounds are of interest across many application areas in control theory, mathematical finance, machine learning and signal processing. If only the first two moments of the distribution are available, then Chebyshev-like worst-case bounds can be computed via solution of a single semidefinite program. However, the results can be very conservative since they are typically achieved by a discrete worst-case distribution. The talk will show that considerable improvement is possible if the probability distribution can be assumed unimodal, in which case less pessimistic Gauss-like bounds can be computed instead. Additionally, both the Chebyshev- and Gauss-like bounds for such problems can be derived as special cases of a bound based on a generalised definition of unmodality.

Networks are a convenient way to represent systems of interacting entities. Many networks contain "communities" of nodes that are more densely connected to each other than to nodes in the rest of the network.

Most methods for detecting communities are designed for static networks. However, in many applications, entities and/or interactions between entities evolve in time.

We investigate "multilayer modularity maximization", a method for detecting communities in temporal networks. The main difference between this method and most previous methods for detecting communities in temporal networks is that communities identified in one temporal snapshot are not independent of connectivity patterns in other snapshots.  We show how the resulting partition reflects a trade-off between static community structure within snapshots and persistence of community structure between snapshots. As a focal example in our numerical experiments, we study time-dependent financial asset correlation networks.

Our study addresses the question of how an arbitrage-free semimartingale model is affected when the knowledge about a random time is added. Precisely, we focus on the No-Unbounded-Profit-with-Bounded-Risk condition, which is also known in the literature as the first kind of no arbitrage. In the general semimartingale setting, we provide a sufficient condition on the random time and price process for which the no arbitrage is preserved under filtration enlargement. Moreover we study the condition on the random time for which the no arbitrage is preserved for any process. This talk is based on a joint work with Tahir Choulli, Jun Deng and Monique Jeanblanc.

We consider a controlled stochastic system in presence of state-constraints. Under the assumption of exponential stabilizability of the system near a target set, we aim to characterize the set of points which can be asymptotically driven by an admissible control to the target with positive probability. We show that this set can be characterized as a level set of the optimal value function of a suitable unconstrained optimal control problem which in turn is the unique viscosity solution of a second order PDE which can thus be interpreted as a generalized Zubov equation.

In this talk we present a new type of Soblev norm defined in the space of functions of continuous paths. Under the Wiener probability measure the corresponding norm is suitable to prove the existence and uniqueness for a large type of system of path dependent quasi-linear parabolic partial differential equations (PPDE). We have establish 1-1 correspondence between this new type of PPDE and the classical backward SDE (BSDE). For fully nonlinear PPDEs, the corresponding Sobolev norm is under a sublinear expectation called G-expectation, in the place of Wiener expectation. The canonical process becomes a new type of nonlinear Brownian motion called G-Brownian motion. A similar 1-1 correspondence has been established. We can then apply the recent results of existence, uniqueness and principle of comparison for BSDE driven by G-Brownian motion to obtain the same result for the PPDE.

We study the problem of maximizing expected utility from terminal wealth in a semi-static market composed of derivative securities, which we assume can be traded only at time zero, and of stocks, which can be
traded continuously in time and are modeled as locally-bounded semi-martingales.

Using a general utility function defined on the positive real line, we first study existence and uniqueness of the solution, and then we consider the dependence of the outputs of the utility maximization problem on the price of the derivatives, investigating not only stability but also differentiability, monotonicity, convexity and limiting properties.

We analyze the portfolio choice problem of investors who maximize rank dependent utility in a single-period complete market. We propose a new
notion of less risk taking: choosing optimal terminal wealth that pays off more in bad states and less in good states of the economy. We prove that investors with a less risk averse preference relation in general choose more risky terminal wealth, receiving a risk premium in return for accepting conditional-zero-mean noise (more risk). Such general comparative static results do not hold for portfolio weights, which we demonstrate with a counter-example in a continuous-time model. This in turn suggests that our notion of less risk taking is more meaningful than the traditional notion based on holding less stocks.

This is a joint work with Xuedong He and Roy Kouwenberg.

I introduce Stochastic Portfolio Theory (SPT), which is an alternative approach to optimal investment, where the investor aims to beat an index instead of optimising a mean-variance or expected utility criterion. Portfolios which achieve this are called relative arbitrages, and simple and implementable types of such trading strategies have been shown to exist in very general classes of continuous semimartingale market models, with unspecified drift and volatility processes but realistic assumptions on the behaviour of stocks which come from empirical observation. I present some of my recent work on this, namely the so-called diversity-weighted portfolio with negative parameter. This portfolio outperforms the market quite significantly, for which I have found both theoretical and empirical evidence.

1. Minimising Regret in Portfolio Optimisation (Simões)

When looking for an "optimal" portfolio the traditional approach is to either try to minimise risk or maximise profit. While this approach is probably correct for someone investing their own wealth, usually traders and fund managers have other concerns. They are often assessed taking into account others' performance, and so their decisions are molded by that. We will present a model for this decision making process and try to find our own "optimal" portfolio.

2. Systemic risk in financial networks (Murevics)

Abstract: In this paper I present a framework for studying systemic risk and financial contagion in interbank networks. The current financial health of institutions is expressed through an abstract measure of robustness, and the evolution of robustness in time is described through a system of stochastic differential equations. Using this model I then study how the structure of the interbank lending network affects the spread of financial contagion through different contagion channels and compare the results for different network structures. Finally I outline the future directions for developing this model.

1. Calibration and Pricing of Financial Derivatives using Forward PDEs (Mariapragassam)

Nowadays, various calibration techniques are in use in the financial industry and the exact re-pricing of call options is a must-have standard. However, practitioners are increasingly interested in taking into account the quotes of other derivatives as well.
We describe our approach to the calibration of a specific Local-Stochastic volatility model proposed by the FX group at BNP Paribas. We believe that forward PDEs are powerful tools as they allow to achieve stable and fast best-fit routines. We will expose our current results on this matter.

Joint work with Prof. Christoph Reisinger

2. Infinite discrete-time investment and consumption problem (Li)

We study the investment and consumption problem in infinite discrete-time framework. In our problem setting, we do not need the wealth process to be positive at any time point. We first analyze the time-consistent case and give the convergence of value function for infinite-horizon problem by value functions of finite-horizon problems.

Then we discuss the time-consistent case, and hope the value functions of finite-horizon problems will still converge to the infinite-horizon problem.

1. A Hybrid Monte-Carlo Partial Differential Solver for Stochastic  Volatility Models (Cozma)

In finance, Monte-Carlo and Finite Difference methods are the most popular approaches for pricing options. If the underlying asset is modeled by a multidimensional system of stochastic differential equations, an analytic solution is rarely available and working under a given computational budget comes at the cost of accuracy. The mixed Monte-Carlo partial differential solver introduced by Loeper and Pironneau (2009) is one way to overcome this issue and we investigate it thoroughly for a number of stochastic volatility models. Our main concern is to provide a rigorous mathematical proof of the convergence of the hybrid method under different frameworks, which in turn justifies the use of Monte-Carlo simulations to compute the expected discounted payoff of the financial derivative. Then, we carry out a quantitative assessment based on a European call option by comparison with alternative numerical methods.

2. tbc (Brackmann)

Peter Grindrod and Stephen Haben (UoOx)

We are dwelling in the Big Data age. The diversity of the uses of Big Data unleashes limitless possibilities. Many people are talking about ways to use Big Data to track the collective human behaviours, monitor electoral popularity, and predict financial fluctuations in stock markets, etc. Big Data reveals both challenges and opportunities, which are not only related to technology but also to human itself. This talk will cover various current topics and trends in Big Data research. The speaker will share his relevant experiences on how to use analytics tools to obtain key metrics on online social networks, as well as present the challenges of Big Data analytics.

Bio: Ning Wang (Ph.D) works as Researcher at the Oxford Internet Institute. His research is driven by a deep interest in analysing a wide range of sociotechnical problems by exploiting Big Data approaches, with the hope that this work could contribute to the intersection of social behavior and computational systems.

We are dwelling in the Big Data age. The diversity of the uses

of Big Data unleashes limitless possibilities. Many people are talking

about ways to use Big Data to track the collective human behaviours,

monitor electoral popularity, and predict financial fluctuations in

stock markets, etc. Big Data reveals both challenges and opportunities,

which are not only related to technology but also to human itself. This

talk will cover various current topics and trends in Big Data research.

The speaker will share his relevant experiences on how to use analytics

tools to obtain key metrics on online social networks, as well as

present the challenges of Big Data analytics.

\\

Bio: Ning Wang (Ph.D) works as Researcher at the Oxford Internet

Institute. His research is driven by a deep interest in analysing a wide

range of sociotechnical problems by exploiting Big Data approaches, with

the hope that this work could contribute to the intersection of social

behavior and computational systems.

This seminar has been cancelled