Mathematical Finance Seminar – Spring 2018

Schedule for Spring 2018

Seminars are on Thursdays
Time: 4:10pm – 5:25pm
Location: Columbia University, 903 SSW (1255 Amsterdam Ave, between 121st and 122nd Street)

Organizers: Ioannis Karatzas, Marcel Nutz, Philip Protter, Yuchong Zhang

MAFN Seminar Archive

*Tuesday 1/23/18

Hao Xing (LSE)

Title: Capital allocation under the Fundamental Review of Trading Book

Abstract: The Fundamental Review of Trading Book (FRTB) is a revised global regulatory framework on market risk proposed by the Basel Committee on Banking Supervision. It aims to replace the current market risk framework under Basel II by 2019. The FRTB shifts from Value-at-Risk to an Expected Shortfall (ES) measure. Varying liquidity horizons are also incorporated into P&L of risk positions to replace the static 10-day horizon used in the current practice. Under the new framework, banks need to allocate economic capital to each risk position in order to evaluate its performance and risk. In this talk, we will introduce two computational efficient methods for capital allocation under FRTB. Simulation results will be presented to illustrate new features of these methods under the new framework comparing to standard methods under the current framework. This is a joint work with Luting Li.

*Tuesday 1/30/18

*1025 SSW

Goncalo dos Reis (Edinburgh)

“Equilibrium pricing under relative performance concerns — old and new”

We investigate the effects of the social interactions of a finite set of agents on an equilibrium pricing mechanism. A derivative written on non-tradable underlyings is introduced to the market and priced in an equilibrium framework by agents who assess risk using convex dynamic risk measures expressed by Backward Stochastic Differential Equations (BSDE). Each agent is not only exposed to financial and non-financial risk factors, but she also faces performance concerns with respect to the other agents. Within our proposed model we prove the existence and uniqueness of an equilibrium whose analysis involves systems of fully coupled multi-dimensional quadratic BSDEs. We extend the theory of the representative agent by showing that a non-standard aggregation of risk measures is possible via weighted-dilated infimal convolution. We analyze the impact of the problem’s parameters on the pricing mechanism, in particular how the agents’ performance concern rates affect prices and risk perceptions. In extreme situations, we find that the concern rates destroy the equilibrium while the risk measures themselves remain stable. Lastly, we discuss work in progress as we analyze the mean-field approach to the problem.




Vladimir Vovk (Royal Holloway, London)

“Probability-free continuous martingales and non-stochastic portfolio theory”

In the framework of idealized financial markets with continuous price paths, I will start from stating probability-free versions of some standard definitions and results in the theory of martingales and stochastic calculus. These definitions and results will then be used to develop a probability-free version of Robert Fernholz’s stochastic portfolio theory. An important advantage of this version is that it does not depend on assumptions, apart from continuity, that are not justified economically (such as postulating a stochastic model or postulating pathwise covariations with respect to a given sequence of partitions). In particular, I will state the “master formula” (at least for smooth portfolio generating functions) and its corollaries showing the possibility of outperforming the capital-weighted index in the case of a diverse and non-degenerate market. (Parts of this talk are based on joint work with Glenn Shafer.)


Kostas Kardaras (LSE)

“Efficient estimation of present-value distributions for long- dated contracts”

ABSTRACT: Estimation of the distribution of present values for long- dated financial and insurance contracts is typically extremely slow. PDE methods will fail due to lack of information about boundary conditions; when Monte-Carlo methods are utilized, simulation for each path realization can take an prohibitive amount of time, leading to poor results. We propose an alternative simulation method, using ergodicity and time-reversal, that leads to significantly better results; in effect, reducing the simulation to a single path. For Markov chain factor models, density estimation with same rate of convergence as for the cdf is possible.


Yash Kanoria (Columbia GSB)

“The Value of State Dependent Control in Ride-sharing Systems”.


We study the design of state-dependent control for a closed queueing network model of ride-sharing systems. We focus on the dispatch policy, where the platform can choose which vehicle to assign when a passenger request comes in, and assume that this is the exclusive control lever available. The vehicle once again becomes available at the destination after dropping the customer. We consider the proportion of dropped demand in steady state as the performance measure.

We propose a family of simple and explicit policies called Scaled MaxWeight (SMW) policies and prove that under the complete resource pooling (CRP) condition (analogous to the condition in Hall’s marriage theorem), each SMW policy leads to exponential decay of demand-dropping probability as the number of vehicles scales to infinity. We further show that there is an SMW policy that achieves the optimal exponent among all non-idling policies, and analytically specify this policy in terms of the passenger request arrival rates for all source-destination pairs. The optimal SMW policy protects structurally under-supplied locations.

Joint work with Sid Banerjee (Cornell) and Pengyu Qian (Columbia Business School).

Yash Kanoria is an Associate Professor in Decision, Risk and Operations at Columbia Business School, working primarily on matching markets and the design and operations of marketplaces. Previously, he obtained a BTech from IIT Bombay in 2007, a PhD in Electrical Engineering from Stanford in 2012, and spent a year at Microsoft Research New England during 2012-13 as a Schramm postdoctoral fellow. He received an NSF CAREER Award in 2017, a Simons-Berkeley Research Fellowship in 2015 and an INFORMS JFIG paper competition second prize in 2014.

Harvey Stein (Bloomberg)
“A Unified Framework for Default Modeling”
Abstract: Credit risk models largely bifurcate into two classes — the structural models and the reduced form models.  Attempts have been made to reconcile the two approaches by adjusting filtrations to restrict information (Cetin, Jarrow, Protter, and Yldrm [CJPY04], Jarrow and Protter [JP04], and Giesecke [Gie06]) but they are technically complicated and tend to approach filtration modification in an ad-hock fashion. 
Here we propose a reconciliation inspired by actuarial science’s approach to survival analysis.  Extending the work of Chen [Che07], we model the survival and hazard rate curves themselves as a stochastic processes.  This puts default models in a form resembling the HJM framework for interest rates (Heath, Jarrow, and Morton [HJM92]), and yields a unified framework for default modeling.



Ruimeng Hu (UC Santa Barbara)

“Portfolio Optimization Under Fractional Stochastic Environments”

Abstract: Rough stochastic volatility models have attracted a lot of attention recently, in particular for the linear option pricing
problem. In this talk, I will start with power utilities, and propose to use a martingale distortion representation of the optimal value
function for the nonlinear asset allocation problem in a (non-Markovian) fractional stochastic environment (for all Hurst index
H \in (0, 1)). A first-order approximation of the optimal value is rigorously established, where the return and volatility of the
underlying asset are functions of a stationary slowly varying fractional Ornstein-Uhlenbeck process. We prove that this
approximation can be also generated by a fixed zeroth order trading strategy providing an explicit strategy which is asymptotically
optimal in all admissible controls. Similar results are also obtained under fast mean-reverting fractional stochastic environment.
Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this fixed strategy in a
specific family of admissible strategies.

3/15/18 No seminar (Spring Recess)

Kasper Larsen (Rutgers)

Title: Smart TWAP Trading in Continuous-Time Equilibria

Abstract: This paper presents a continuous-time equilibrium model of liquidity provision in a market with multiple strategic investors with intraday trading targets.  We show analytically that there are infinitely many Nash equilibria. We solve for the welfare-maximizing equilibrium and the competitive equilibrium, and we illustrate that these equilibria are different.  The model is easily computed numerically and we provide a number of numerical illustrations.

Joint work with Jin Hyuk Choi (UNIST) and Duane J. Seppi (CMU)


Alexander Schied (Waterloo)

Title: Currency target zone models, price impact, and singular stochastic control

Abstract: We study optimal buying and selling strategies in target zone models. Such models describe situations in which a currency exchange rate is kept above or below a certain barrier due to central bank intervention.
We first consider an optimal portfolio liquidation problem for an investor for whom prices are optimal at the barrier and who creates temporary price impact. This problem will be formulated as the minimization of a cost-risk functional over strategies that only trade when the price process is located at the barrier. We solve the corresponding singular stochastic control problem by means of a scaling limit of critical branching particle systems, which is known as a catalytic superprocess.
Then we consider a stochastic game between a trader and the central bank in a situation in which both players also generate permanent price impact. We first solve the central bank’s control problem by means of the Skorokhod map and then derive the trader’s optimal strategy by solving a sequence of approximating control problems, thus establishing a Stackelberg equilibrium.
This is joint work with Eyal Neuman.


Sebastian Jaimungal (Toronto)

Title: Algorithmic Trading and Mean-Field Games with Latent Factors
Abstract: Financial markets are often driven by latent factors which traders cannot observe. Here, we address an algorithmic trading problem with collections of heterogeneous agents who aim to perform statistical arbitrage, where all agents filter the latent states of the world, and their trading actions have permanent and temporary price impact. This leads to a large stochastic game with heterogeneous agents. We solve the stochastic game by investigating its mean-field game (MFG) limit, with sub-populations of heterogenous agents, and, using a convex analysis approach, we show that the solution is characterized by a vector-valued forward-backward stochastic differential equation (FBSDE). We demonstrate that the FBSDE admits a unique solution, obtain it in closed-form, and characterize the optimal behaviour of the agents in the MFG equilibrium. Moreover, we prove the MFG equilibrium provides an ϵ-Nash equilibrium for the finite player game. We conclude by illustrating the behaviour of agents using the optimal MFG strategy through simulated examples. (Joint work with Philippe Casgrain) 


Jaime San Martin (Universidad de Chile)

Title: M-matrices, trees and ultra metric matrices

M-matrices play an important role in many applications like: Large sparse systems, discretization of differential elliptic operators, linear complementarity problems in linear and quadratic programming. One of the most interesting applications of M-matrices is the Leontief’s input-output analysis in economic systems. We will study the connection of M-matrices and inverse M-matrices with Markov chains, in particular we are interested in random walks on trees. We introduce the notion of ultrametric matrix and we show they are, in a way, fundamental blocks for the theory of M-matrices.






Ed Kaplan (Yale)

Joint Probability Colloquium
4:15-5:30 in Uris 303
(Followed by a reception)

Speaker: Ed Kaplan (Yale)

Title:  Approximating The FCFS Stochastic Matching Model With Ohm’s Law

Abstract:  The FCFS stochastic matching model, where each server in an infinite sequence is matched to the first eligible customer from a second
infinite sequence, developed from queueing problems addressed by Kaplan (1984) in the context of public housing assignments. The
goal of this model is to determine the matching rates between eligible customer- and server-types, that is, the fraction of all matches
that occur between type-i customers and type-j servers. This model was solved in a beautiful paper by Adan and Weiss (2012), but the
resulting equation for the matching rates is quite complicated, involving the sum of permutation-specific terms over all permutations of
the server-types. Here we develop an approximation for the matching rates based on Ohm’s Law that in some cases reduces to exact results,
and via analytical, numerical, and simulation examples is shown to be highly accurate. As our approximation only requires solving a system
of linear equations, it provides an accurate and tractable alternative to the exact solution. (This is joint work with Mohammad Fazel-Zarandi)




Xiaolu Tan (Paris Dauphine)

Title: Some applications of the randomization approach in robust finance.

Abstract: We discuss some applications of the randomization technique in robust finance. In a first case, we consider a super-replication problem of the American option; in a second case, we study the super-replication and utility maximization problem under proportional transaction cost. The randomization technique allows to reduce the initial problems to that of the European option in a frictionless market. This allows one to apply some classical results and arguments to study the new problem.