## Seminar Series

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## Probability and Financial Mathematics

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## School of Mathematics, University of Leeds

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## 2019/2020

Prof. Teemu PENNANEN (King's College London)

10th of October

2:00 pm

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LT 3

Roger Stevens Building

Title

Convex duality in optimal investment and contingent claim valuation in illiquid markets

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Abstract

We develop a duality theory for optimal investment and contingent claim valuation in markets where traded assets may be subject to nonlinear trading costs and portfolio constraints. Under fairly general conditions, the dual expressions decompose into three terms, corresponding to the agent's risk preferences, trading costs and portfolio constraints, respectively. The dual representations are shown to be valid when the market model satisfies an appropriate generalization of the no-arbitrage condition and the agent's utility function satisfies an appropriate generalization of asymptotic elasticity conditions. When applied to classical liquid market models or models with bid-ask spreads, we recover well-known pricing formulas in terms of martingale measures and consistent price systems. Building on the general theory of convex stochastic optimization, we also obtain optimality conditions in terms of an extended notion of a "shadow price". The results are illustrated by establishing the existence of solutions and optimality conditions for some nonlinear market models recently proposed in the literature. Our results allow for significant extensions including nondifferentiable trading costs which arise e.g. in modern limit order markets where the marginal price curve is necessarily discontinuous.

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Dr. Jing YAO (Heriot Watt University)

24th of October

2:00 pm

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LT3

Roger Stevens Building

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Title

Downside Risk Optimization with Random Targets and Portfolio Amplitude

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Abstract

In this paper, we rationalize using downside risk optimization subject to a random target in portfolio selection. In context of normality, we derive analytical solutions to the downside risk optimization with respect to random targets and investigate how the random target affects the optimal solutions. In doing so, we propose using portfolio amplitude, as a new measure in literature, to characterize the investment strategy. Particularly, we demonstrate the mechanism by which the random target inputs its impact into the system and alters the optimal portfolio selection. Our results underpin why investors prefer holding some specific assets in following random targets and provide explanations for some special investment strategies, such as constructing a stock portfolio following a bond index. Numerical examples are presented to clarify our theoretical results.

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Dr. Kathrin Glau (Queen Mary University London)

7th of November

2:00pm

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LT3

Roger Stevens Building

Title

Low-Rank Tensor Approximation for Parametric Option Pricing

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Abstract

Computationally intensive problems in finance are characterized by their intrinsic high-dimensionality which often is paired with optimizations leading to nonlinearities. While classical numerical methods typically suffer from a curse in dimensionality, machine learning approaches promise to yield fairly accurate results with a method that is scalable in the dimensions. Computational intense training phases and the required large set of training data pose some of the major challenges for the development of new and adequate numerical methods for finance. Merging classical numerical techniques with learning methods we propose a new approach to option pricing in parametric models. The work is based on [1] and ongoing research with Paolo Colusso and Francesco Statti.

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[1] Glau, K.; Kressner, D.; Statti, F.: Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing. preprint 2019

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Dr. Gonçalo dos Reis (University of Edinburgh)

14th of November

2:00 pm

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LT 3

Roger Stevens Building

Title

Itô-Wentzell for measure dependent random fields under full and conditional measure flows

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Abstract

We show several Itô-Wentzell formulae on Wiener spaces for real-valued maps depending on measures. We present both the full measure-flow and the marginal-flow cases where the measure flow derivatives are understood in the sense of Lions.

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This talk has been cancelled.

Dr. Renyuan XU (University of Oxford)

21st of November

2:00 pm

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LT3

Roger Stevens Building

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Title

A Case Study on Pareto Optimality for Collaborative Stochastic Games

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Abstract

Pareto Optimality (PO) is an important concept in game theory to measure global efficiency when players collaborate. In this talk, we start with the PO for a class of continuous-time stochastic games when the number of players is finite. The derivation of PO strategies is based on the formulation and analysis of an auxiliary N-dimensional central controller’s stochastic control problem, including its regularity property of the value function and the existence of the solution to the associated Skorokhod problem. This PO strategy is then compared with the set of (non-unique) NEs strategies under the notion of Price of Anarchy (PoA). The upper bond of PoA is derived explicitly in terms of model parameters. Finally, we characterize analytically the precise difference between the PO and the associated McKean-Vlasov control problem with an infinite number of players, in terms of the covariance structure between the optimally controlled dynamics of players and characteristics of the no-action region for the game. This is based on joint work with Xin Guo (UC Berkeley).

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Dr. Máté GERENCSER (IST Austria)

22nd of November

2:00 pm

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LT3

Roger Stevens Building

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Title

Boundary renormalisation of stochastic PDEs

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Abstract

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We discuss solution theories of singular SPDEs endowed with various boundary conditions. In several examples nontrivial boundary effects arise and another layer of renormalisation is required. We outline how these are connected to spatial singularities of simple trees in equations like the KPZ, PAM, of Phi^4.

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Dr. Luitgard VERAART (London School of Economics)

5th of December

2pm

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LT3

Roger Stevens Building

Title

When does portfolio compression reduce systemic risk?

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Abstract

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We analyse the consequences of portfolio compression on systemic risk. Portfolio compression is a post-trading netting mechanism that reduces gross positions while keeping net positions unchanged and it is part of the financial legislation in the US (Dodd-Frank Act) and in Europe (European Market Infrastructure Regulation). We derive necessary structural conditions for portfolio compression to be harmful and discuss policy implications. In particular, we show that the potential danger of portfolio compression comes from defaults of firms that conduct portfolio compression. If no such defaults occur, then portfolio compression reduces systemic risk.

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Prof. Dr. David PRÖMEL (Mannheim Universität)

12th of December

2:00 pm

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LT 3

Roger Stevens Building

Title

Martingale Optimal Transport in Robust Finance

Abstract

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Without assuming any probabilistic price dynamics, we consider a

frictionless financial market given by the Skorokhod space, on which

some financial options are liquidly traded. In this model-free setting

we show various pricing-hedging dualities and the analogue of the

fundamental theorem of asset pricing. For this purpose we study the

corresponding martingale optimal transport (MOT) problem: We obtain a

dual representation of the Kantorovich functional (super-replication

functional) defined for functions (financial derivatives) on the

Skorokhod space using quotient sets (hedging sets). Our representation

takes the form of a Choquet capacity generated by martingale measures

satisfying additional constraints to ensure compatibility with the

quotient sets. The talk is based on a joint work with Patrick

Cheridito, Matti Kiiski and H. Mete Soner.

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Dr. Martin HERDEGEN (University of Warwick)

13th of February

2:00 pm-3:00pm

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LT 05 (7.05)

Roger Stevens

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DOUBLE

SEMINAR

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Title

A Dual Characterisation of Regulatory Arbitrage for Coherent Risk Measures

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Abstract

We revisit portfolio selection in a one-period financial market under a coherent risk measure constraint, the most prominent example being Expected Shortfall (ES). Unlike in the case of classical mean-variance portfolio selection, it can happen that no efficient portfolios exist. We call this situation regulatory arbitrage. We then show that the absence of regulatory arbitrage is intimately linked to the interplay between the set of equivalent martingale measures (EMMs) for the discounted risky assets and the set of absolutely continuous measures in the dual characterisation of the risk measure. In the special case of ES, our result shows that the market does not admit regulatory arbitrage for ES at confidence level $\alpha$ if and only if there exists an EMM $Q \approx P$ such that $\Vert \frac{dQ}{dP} \Vert_\infty < \frac{1}{\alpha}$.

The talk is based on joint work with my PhD student Nazem Khan.

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Prof. Johannes MUHLE-KARBE (Imperial College London)

13th of February

3:10 pm-4:10pm

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LT 05 (7.05)

Roger Stevens

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DOUBLE

SEMINAR

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Title

Equilibrium asset pricing with frictions

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Abstract

We study how the prices of assets depend on their “liquidity”, that is, the ease with which they can be traded. We show that equilibrium prices and the corresponding optimal trading strategies can be characterised by systems of coupled forward-backward SDEs. We outline some first wellposedness results and discuss explicit formulas that arise in the limit of large liquidity.

The talk is based on joints works with Agostino Capponi, Lukas Gonon, Martin Herdegen, Dylan Possamai, and Xiaofei Shi.

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Prof. Fausto GOZZI (University LUISS Rome )

20th of February

2:00 pm

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LT 5

Roger Stevens Building

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Title

From "simple" stochastic control problems to "more realistic" ones:

an example from portfolio theory

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Abstract

In this talk we take an example from life-cycle portfolio theory

(modelled as a stochastic optimal control problem)

where we see how different economic questions, possibly arising from experimental data,

bring to formulate and study more complex problems, in particular

models which display path dependency and/or Mc Kean - Vlasov type dynamics.

Ongoing results on such problems will be presented.

We will discuss, depending on the available time, the following modelling levels.

Level 1: Lifecycle portfolio with Labor Income

Level 2: Lifecycle portfolio with Path-Dependent Labor Income (History dependent wages)

Level 3/1: Lifecycle portfolio with Path-Dependent Labor Income with uncertainty on parameters

(Taking account of estimation errors)

Level 3/2: Lifecycle portfolio with Path-Dependent Labor Income with Mc Kean-Vlasov type dynamics

(Taking account of the effect of the environment: "Keeping up with the Joneses")

Based on joint papers/work in progress with various authors

( Biagini, Biffis, Cosso, Djeiche, Kharroubi, Pham, Prosdocimi, Rosestolato, Zanco, Zanella).

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This talk has been CANCELLED.

Prof. Saul JACKA (University of Warwick)

27th of February

2:00 pm

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LT 5

Roger Stevens Building

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Title

Minimising the shuttle/commute time

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Abstract

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We consider the problem of choosing a diffusion's drift so as to minimise the expected time to commute from zero to 1 and back again.

We solve this first as a static problem and then (in a suitable sense) dynamically.

Prof. Zhenya LIU

(University Aix-Marseille and Renmin University of China)

5th of March

2:00 pm

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LT 6.142

Worsley Building

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Title

The Optimal Equity Selling Price with Endogenous Drawdown

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Abstract

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In this paper, we propose the endogenous drawdown in an investor's reward function. The endogenous drawdown is the difference between the underlying process and its maximum related process. We find the optimal selling price is a function of the historical highest price, the weights of profit and loss in investor's reward function, and the characters of the underlying stochastic process. With the data of the S&P 500 Index and SSE Composite Index, we calculate the numerical solution of the optimal selling price with consideration of endogenous drawdown. Through the out-sample test, this optimal selling price performs well to sell before the price drops sharply.

Dr. Ankush AGARWAL (University of Glasgow)

12th of March

2:00 pm

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LT 5

Roger Stevens Building

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Title

A Fourier-based Picard-iteration approach for a class of McKean-Vlasov SDEs with Lévy jumps

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Abstract

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We consider a prototype class of Lévy-driven stochastic differential equations (SDEs) with McKean-Vlasov (MK-V) interaction in the drift coefficient. It is assumed that the drift coefficient is affine in the state variable, and only measurable in the law of the solution. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated linear Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK-V SDE, and derive a discretized Picard iteration scheme that approximates the law of the solution through its characteristic function. Numerical illustrations show the effectiveness of our method, which appears to be appropriate to handle the multi-dimensional setting. This is a joint work with Stefano Pagliarani.