Past Nomura Seminar

We focus on the mathematical structure of systemic risk measures as proposed by Chen, Iyengar, and Moallemi (2013). In order to clarify the interplay between local and global risk assessment, we study the local specification of a systemic risk measure by a consistent family of conditional risk measures for smaller subsystems, and we discuss the appearance of phase transitions at the global level. This extends the analysis of spatial risk measures in Föllmer and Klϋppelberg (2015).

CANCELLED

In the context of surplus models of insurance risk theory, 
some rather surprising and simple identities are presented. This 
includes an
identity relating level crossing probabilities of continuous-time models 
under (randomized) discrete and continuous observations, as well as
reflection identities relating dividend payments and capital injections. 
Applications as well as extensions to more general underlying processes are
discussed.

 

The limit order book is the at the core of every modern, electronic financial market. In this talk, I will present some results pertaining to their statistical properties, mathematical modelling and numerical simulation. Questions such as ergodicity, dependencies, relation betwen time scales... will be addressed and sometimes answered to. Some on-going research projects, with applications to optimal trading and market making, will be evoked.

This seminar is run jointly with OMI.

Throughout the Western world, defined benefit pension plans are disappearing, replaced by defined contribution (DC) plans. Retail investors are thus faced with managing investments over a thirty year accumulation period followed by a twenty year decumulation phase. Holders of DC plans are thus truly long term investors. We consider dynamic mean variance asset allocation strategies for long term investors. We derive the "embedding result" which converts the mean variance objective into a form suitable for dynamic programming using an intuitive approach. We then discuss a semi-Lagrangian technique for numerical solution of the optimal control problem via a Hamilton-Jacob-Bellman PDE. Parameters for the inflation adjusted return of a stock index and a risk free bond are determined by examining 89 years of US data. Extensive synthetic market tests, and resampled backtests of historical data, indicate that the multi-period mean variance strategy achieves approximately the same expected terminal wealth as a constant weight strategy, while reducing the probability of shortfall by a factor of two to three.

In this talk, we will establish existence and uniqueness for a wide class of Markovian systems of backward stochastic differential equations (BSDE) with quadratic nonlinearities. This class is characterized by an abstract structural assumption on the generator, an a-priori local-boundedness property, and a locally-H\"older-continuous terminal condition. We present easily verifiable sufficient conditions for these assumptions and treat several applications, including stochastic equilibria in incomplete financial markets, stochastic differential games, and martingales on Riemannian manifolds. This is a joint work with Gordan Zitkovic.

We provide a representation result of parabolic semi-linear PDEs, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod (1964), Watanabe (1965) and McKean (1975), by allowing for polynomial nonlinearity in the pair (u,Du), where u is the solution of the PDE with space gradient Du. Similar to the previous literature, our result requires a non-explosion condition which restrict to "small maturity" or "small nonlinearity" of the PDE. Our main ingredient is the automatic differentiation technique as in Henry Labordere, Tan and Touzi (2015), based on the Malliavin integration by parts, which allows to account for the nonlinearities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation.

This is joint work with Richard A. Davis (Columbia Statistics) and Johannes Heiny (Copenhagen). In recent years the sample covariance matrix of high-dimensional vectors with iid entries has attracted a lot of attention. A deep theory exists if the entries of the vectors are iid light-tailed; the Tracy-Widom distribution typically appears as weak limit of the largest eigenvalue of the sample covariance matrix. In the heavy-tailed case (assuming infinite 4th moments) the situation changes dramatically. Work by Soshnikov, Auffinger, Ben Arous and Peche shows that the largest eigenvalues are approximated by the points of a suitable nonhomogeneous Poisson process. We follows this line of research. First, we consider a p-dimensional time series with iid heavy-tailed entries where p is any power of the sample size n. The point process of the scaled eigenvalues of the sample covariance matrix converges weakly to a Poisson process. Next, we consider p-dimensional heavy-tailed time series with dependence through time and across the rows. In particular, we consider entries with a linear dependence or a stochastic volatility structure. In this case, the limiting point process is typically a Poisson cluster process. We discuss the suitability of the aforementioned models for large portfolios of return series. 

We describe the possible influence of stochastic 
dependence on the evaluation of
the risk of joint portfolios and establish relevant risk bounds.Some 
basic tools for this purpose are  the distributional transform,the 
rearrangement method and extensions of the classical Hoeffding -Frechet 
bounds based on duality theory.On the other hand these tools find also 
essential applications to various problems of optimal investments,to the 
construction of cost-efficient payoffs as well as to various optimal 
hedging problems.We
discuss in detail the case of optimal payoffs in Levy market models as 
well as utility optimal payoffs and hedgings
with state dependent utilities.

The problem of optimal stopping with finite horizon in discrete time
is considered in view of maximizing the expected gain. The algorithm
presented in this talk is completely nonparametric in the sense that it
uses observed data from the past of the process up to time -n+1 (n being
a natural number), not relying on any specific model assumption. Kernel
regression estimation of conditional expectations and prediction theory
of individual sequences are used as tools.
The main result is that the algorithm is universally consistent: the
achieved expected gain converges to the optimal value for n tending to
infinity, whenever the underlying process is stationary and ergodic.
An application to exercising American options is given.

In this talk, we present a pathwise method to construct confidence 
intervals on the value of some discrete time stochastic dynamic 
programming equations, which arise, e.g., in nonlinear option pricing 
problems such as credit value adjustment and pricing under model 
uncertainty. Our method generalizes the primal-dual approach, which is 
popular and well-studied for Bermudan option pricing problems. In a 
nutshell, the idea is to derive a maximization problem and a 
minimization problem such that the value processes of both problems 
coincide with the solution of the dynamic program and such that 
optimizers can be represented in terms of the solution of the dynamic 
program. Applying an approximate solution to the dynamic program, which 
can be precomputed by any algorithm, then leads to `close-to-optimal' 
controls for these optimization problems and to `tight' lower and upper 
bounds for the value of the dynamic program, provided that the algorithm 
for constructing the approximate solution was `successful'. We 
illustrate the method numerically in the context of credit value 
adjustment and pricing under uncertain volatility.
The talk is based on joint work with C. Gärtner, N. Schweizer, and J. 
Zhuo.

tba

With few exceptions, optimal stopping assumes that the underlying system is stopped immediately after the decision is made. 
In fact, most stoppings take time. This has been variously referred to as "time-to-build", "investment lag" and "gestation period", 
which is often non negligible. 
In this talk, we consider a class of optimal stopping/switching problems with delivery lags, or equivalently, delayed information, 
by using reflected BSDE method. As an example, we study American put option with delayed exercise, and show that it can be decomposed 
as a European put option and a premium, the latter of which involves a new optimal stopping problem where the investor decides when to stop
to collect the Greek theta of such a European option. We also give a complete characterization of the optimal exercise boundary by resorting to free boundary analysis.  

Joint work with Zhou Yang and Mihail Zervos. 

The large majority of risk-sharing transactions involve few agents, each of whom can heavily influence the structure and the prices of securities. This paper proposes a game where agents' strategic sets consist of all possible sharing securities and pricing kernels that are consistent with Arrow-Debreu sharing rules. First, it is shown that agents' best response problems have unique solutions, even when the underlying probability space is infinite. The risk-sharing Nash equilibrium admits a finite-dimensional characterisation and it is proved to exist for general number of agents and be unique in the two-agent game. In equilibrium, agents choose to declare beliefs on future random outcomes different from their actual probability assessments, and the risk-sharing securities are endogenously bounded, implying (amongst other things) loss of efficiency. In addition, an analysis regarding extremely risk tolerant agents indicates that they profit more from the Nash risk-sharing equilibrium as compared to the Arrow-Debreu one.
(Joint work with Michail Anthropelos)

Motivated by the European sovereign debt crisis, we propose a hybrid sovereign default model which combines an accessible part which takes into account the movement of the sovereign solvency and the impact of critical political events, and a totally inaccessible part for the idiosyncratic credit risk. We obtain closed-form formulas for the probability that the default occurs at political critical dates in a Markovian CEV process setting. Moreover, we introduce a generalized density framework for the hybrid default times and deduce the compensator process of default. Finally we apply the hybrid model and the generalized density to the valuation of sovereign bond and explain the significant jumps in the long-term government bond yield during the sovereign crisis.

In this talk, I will present a family of forward performance processes in
discrete time. These processes are predictable with regards to the market
information. Examples from a binomial setting will be given which include
the time-monotone exponential forward process and the completely monotonic
family.

Following a quick introduction to derivatives markets and the classic theory of valuation, we describe the changes triggered by post 2007 events. We re-discuss the valuation theory assumptions and introduce valuation under counterparty credit risk, collateral posting, initial and variation margins, and funding costs. A number of these aspects had been investigated well before 2007. We explain model dependence induced by credit effects, hybrid features, contagion, payout uncertainty, and nonlinear effects due to replacement closeout at default and possibly asymmetric borrowing and lending rates in the margin interest and in the funding strategy for the hedge of the relevant portfolio. Nonlinearity manifests itself in the valuation equations taking the form of semi-linear PDEs or Backward SDEs. We discuss existence and uniqueness of solutions for these equations. We present an invariance theorem showing that the final valuation equations do not depend on unobservable risk free rates, that become purely instrumental variables. Valuation is thus based only on real market rates and processes. We also present a high level analysis of the consequences of nonlinearities, both from the point of view of methodology and from an operational angle, including deal/entity/aggregation dependent valuation probability measures and the role of banks treasuries. Finally, we hint at how one may connect these developments to interest rate theory under multiple discount curves, thus building a consistent valuation framework encompassing most post-2007 effects.

Damiano Brigo, Joint work with Andrea Pallavicini, Daniele Perini, Marco Francischello. 

We argue that Credit Default Swap (CDS) premia for safe-haven sovereigns, like Germany and the United States, are driven to a large extent by regulatory requirements under which  derivatives dealing banks have an incentive to buy CDS to hedge counterparty credit risk of their counterparties.
We explain the mechanics of the regulatory requirements and develop a model in which derivatives dealers, who have a derivatives exposure with sovereigns, need CDS for capital relief. End users without exposure to the sovereigns sell the CDS and require a positive premium equivalent to the capital requirement. The model's predictions are confirmed using data on several sovereigns.

Joint with OMI

A well-known result of Landsberger and Meilijson says that efficient risk-sharing rules for univariate risks are characterized by a so-called comonotonicity condition. In this talk, I'll first discuss a multivariate extension of this result (joint work with R.-A. Dana and A. Galichon). Then I will discuss the restrictions (in the form of systems of nonlinear PDEs) efficient risk sharing imposes on individual consumption as a function of aggregate consumption. I'll finally give an identification result on how to recover preferences from the knowledge of the risk sharing (joint work with M. Aloqeili and I. Ekeland).

The talk is concerned with adapted solution of a multi-dimensional BSDE with a "diagonally" quadratic generator, the quadratic part of whose iith component only depends on the iith row of the second unknown variable. Local and global solutions are given. In our proofs, it is natural and crucial to apply both John-Nirenberg and reverse Holder inequalities for BMO martingales. 

We provide a result on prospect theory decision makers who are naïve about the time inconsistency induced by probability weighting. If a market offers a sufficiently rich set of investment strategies, investors postpone their trading decisions indefinitely due to a strong preference for skewness. We conclude that probability weighting in combination with naïveté leads to unrealistic predictions for a wide range of dynamic setups. Finally, I discuss recent work on the topic that invokes different assumptions on the dynamic modeling of prospect theory.

We provide a new algorithm for approximating the law of a one-dimensional diffusion M solving a stochastic differential equation with possibly irregular coefficients.
The algorithm is based on the construction of Markov chains whose laws can be embedded into the diffusion M with a sequence of stopping times. The algorithm does not require any regularity or growth assumption; in particular it applies to SDEs with coefficients that are nowhere continuous and that grow superlinearly. We show that if the diffusion coefficient is bounded and bounded away from 0, then our algorithm has a weak convergence rate of order 1/4. Finally, we illustrate the algorithm's performance with several examples.

We revisit the optimal exit problem by adding a moral hazard problem where a firm owner contracts out with an agent to run a project. We analyse the optimal contracting problem between the owner and the agent in a Brownian framework, when the latter modifies the project cash-flows with an hidden action. The analysis leads to the resolution of a constrained optimal stopping problem that we solve explicitly.

I prove that if markets are weak-form efficient, meaning current prices fully reflect all information available in past prices, then P = NP, meaning every computational problem whose solution can be verified in polynomial time can also be solved in polynomial time. I also prove the converse by showing how we can "program" the market to solveNP-complete problems. Since P probably does not equal NP, markets are probably not efficient. Specifically, markets become increasingly inefficient as the time series lengthens or becomes more frequent. An illustration by way of partitioning the excess returns to momentum strategies based on data availability confirms this prediction.

For more info please visit: http://philipmaymin.com/academic-papers#pnp