Stochastic Finance @ Warwick Seminars

31st January 2024

Miha Bresar (Warwick)

Superdiffusive limits for Bessel-driven stochastic kinetics We explore the scaling of anomalous diffusion in a one-dimensional stochastic kinetic dynamics model. Our model features stochastic drift influenced by external Bessel noise and incorporates internal volatility, which has an arbitrary relationship with this external noise. We identify the superdiffusive scaling exponent for the model, and prove a weak convergence result on the corresponding scale. We show how our result extends to admit, as exogenous noise processes, not only Bessel processes but more general processes satisfying certain asymptotic conditions. We conclude by exploring the connections with stochastic interest rate models. This talk is based on joint work with Conrado Da Costa, Aleks Mijatović and Andrew Wade.

7th February 2024

Daniel Schwarz (UCL)

Radner equilibrium and systems of quadratic BSDEs with discontinuous generators We establish the existence of a Radner Equilibrium in an incomplete, continuous-time financial economy. To prove existence, we formulate a system of quadratic backward stochastic differential equations (BSDEs) which represents the equilibrium problem. Interestingly, the driver of this system of BSDEs features a discontinuity, posing a major challenge for common approaches to establish existence. Exploiting the duality between Markovian BSDEs and PDEs we use unique continuation and backward uniqueness, techniques originally used in the study of PDEs, to show that the set of discontinuity has in fact measure zero. (Joint work with Luis Escauriaza and Hao Xing.)

Corina Constantinescu (Liverpool)

Trapping Probability in Low-Income Household Capital Dynamics: Insurance and Subsidies Impact

We are examining a risk process characterized by deterministic growth and multiplicative jumps to represent the capital dynamics of low-income households. To account for the higher risk inherent in such environments, we assume that capital losses are proportional to the level of accumulated capital at the time of a jump. Our objective is to calculate the probability of a household falling below the poverty line, known as the trapping probability. "Trapping" arises when a household's capital falls below the poverty threshold, creating a situation from which escape without external help is not possible. We approach the remaining capital distribution as specific instances of the beta distribution and derive closed-form expressions for the trapping probability by analyzing the Laplace transform of the process' infinitesimal generator. Additionally, we investigate the impact of insurance on this probability, particularly when the insurance product provides proportional coverage. Our findings suggest that in certain scenarios, trapping is inevitable without external aid, typically provided in the form of subsidies.

Andrew Allan (Durham)

Rough Stochastic Analysis with Jumps Rough path theory provides a framework for the study of nonlinear systems driven by highly oscillatory (deterministic) signals. The corresponding analysis is inherently distinct from that of classical stochastic calculus, and neither theory alone is able to satisfactorily handle hybrid systems driven by both rough and stochastic noise. The introduction of the stochastic sewing lemma (Khoa Lê, 2020) has paved the way for a theory which can efficiently handle such hybrid systems. In this talk, we will discuss how this can be done in a general setting which allows for jump discontinuities in both sources of noise.

Lecture series by Ioannis Karatzas (Columbia)

More information can be found on this webpage .

1st May 2024

Dan Crisan (Imperial College London)

Global solutions for stochastically controlled models Abstract: For a class of evolution equations that possibly have only local solutions, we introduce a stochastic component that ensures that the solutions of the corresponding stochastically perturbed equations are global. The class of partial differential equations amenable for this type of treatment includes the 3D Navier-Stokes equation, the rotating shallow water equation (viscous and inviscid), 3D Euler equation (in vorticity form), 2D Burgers' equation and many other fluid dynamics models. This is based on joint work with Oana Lang (Babes-Bolyai University). https://arxiv.org/abs/2403.05923

8th May 2024

John Moriarty (Queen Mary University of London)

Optimal stopping with nonlinear expectation: geometric and algorithmic solutions We use the geometry of functions associated with martingales under nonlinear expectations to solve risk-sensitive Markovian optimal stopping problems. Generalising the linear case due to Dynkin and Yushkievich (1969), the value function is the majorant or pointwise infimum of those functions which dominate the gain function. An emphasis is placed on the geometry of the majorant and pathwise arguments, rather than exploiting convexity, positive homogeneity or related analytical properties. An algorithm is provided to construct the value function at the computational cost of a two-dimensional search. The talk is based on the preprint https://arxiv.org/abs/2306.17623 (with Tomasz Kosmala).

15th May 2024

MS.02

Christa Cuchiero (University of Vienna)

Functional Itô formula and Taylor expansion for non-anticipative maps of rough paths: We rely on the approximation properties of the signature of càdlàg rough paths to derive a functional Itô formula for non-anticipative maps of rough paths. This leads to a functional extension of the Itô formula for càdlàg rough paths (by Friz and Zhang (2018)) which coincides with the change of variable formula formulated by Dupire (2009) as well as by Cont and Fournie (2010), whenever the notions of the regularity of the functionals and the integration coincide. As a byproduct, we show that sufficiently regular non-anticipative path functionals admit a functional Taylor expansion, leading to a far reaching generalization of the recently established results by Dupire and Tissot-Daguette (2022). The talk is based on ongoing joint work with Xin Guo and Francesca Primavera.

29th May 2024

NEW ROOM

MB0.08

 Aria Ahari (University of Warwick)

Boundary crossing problems and functional transformations for Ornstein-Uhlenbeck processes. We are interested in the law of the first passage time of an Ornstein-Uhlenbeck process to time-varying thresholds. We show that this problem is connected to the laws of the first passage time of the process to members of a two-parameter family of functional transformations of a time-varying boundary. For specific values of the parameters, these transformations appear in a realisation of a standard Ornstein-Uhlenbeck bridge. We provide three different proofs of this connection. The first one is based on a similar result for Brownian motion, the second uses a generalisation of the so-called Gauss-Markov processes and the third relies on the Lie group symmetry method. We investigate the properties of these transformations and study the algebraic and analytical properties of an involution operator which is used in constructing them. We also show that these transformations map the space of solutions of Sturm-Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we interpret our results through the method of images and give new examples of curves with explicit first passage time densities. This is joint work with my supervisors Dr. Larbi Alili and Dr.Massimiliano Tamborrino. Link to the pre-print: https://arxiv.org/abs/2210.01658

19th June 2024

MB0.08

Wilfried Kuissi-Kamdem (AIMS Ghana/University of Rwanda)

Pricing and hedging of (path-dependent) claims based on non-tradable assets for Epstein-Zin utility We present an Epstein-Zin utility indifference pricing and hedging for multidimensional tradable and non-tradable assets models. We provide an explicit expression for the utility indifference prices and their corresponding derivative hedges. Using forward-backward SDE method and Malliavin calculus, in a possibly non-Markovian diffusion model for the financial market, a linear indifference pricing rule is obtained. We apply our methodology to the pricing and hedging of weather derivatives, crack spread options, variance swaps, volatility swaps and options on levered exchange traded funds. (This talk is based on ongoing joint work with Olivier Menoukeu Pamen and Marcel Ndengo)

21st June 2024

Joint Session with the Guillmore Centre for Financial Technology

WBS Teaching Centre

2:00-3:30pm

Please note that registration is required.

3rd July 2024

Room MB0.02

Min Dai

Equilibrium Mean-Variance Strategy with Transaction Costs We study continuous-time mean-variance portfolio selection in the presence of proportional transaction costs, which can be formulated as a time-inconsistent singular stochastic control problem. We provide a novel definition of equilibrium solutions for the singular control problem and characterize them by a system of HJB equations. We reveal that the standard mean-variance criteria with a constant risk preference may yield absurd equilibrium strategies in the presence of transaction costs. We find that a time-varying risk preference yields a reasonable and stable equilibrium strategy. This work is jointly with Yanwei Jia and Hanqing Jin.

1st February 2023

Paolo Guasoni (Dublin City University)

General Equilibrium with Unhedgeable Fundamentals and Heterogeneous Agents We solve a general equilibrium model in which aggregate consumption has uninsurable growth shocks, rendering the market dynamically incomplete. Several long-lived agents with heterogeneous risk-aversion and time-preference make consumption and investment decisions, trading risky assets and borrowing from and lending to each other. For small growth fluctuations, we obtain closed-form expressions for stock prices, interest rates, and consumption and trading policies. Agents' stochastic discount factors depend on the history of unhedgeable shocks, agents trade assets dynamically, and the dispersion of agents' preferences impacts both the interest rate and asset prices, hence no representative agent exists.

15th February 2023

Adrien Richou (Université de Bordeaux)

BSDEs reflected in a non convex domain: a geometric point of view In a recent paper, we have proved, with J.-F. Chassagneux and S. Nadtochiy, some existence and uniqueness results for BSDEs reflected in a non-convex domain under some restrictive assumptions on the domain and the terminal condition. All these results were obtained by tools and estimates based on the Euclidean structure of $\mathbb{R}^d$. In order to improve these results, at least in dimension $2$, it is also possible to see our domain as a flat manifold with a boundary and to take advantage of geometry tools already developed to tackle martingales in (non flat) manifolds (without boundary). In this talk, I will explain this new approach and the kind of results we are able to obtain. This is a work in progress with M. Arnaudon, J.-F. Chassagneux and S. Nadtochiy.

David Bang (University of Warwick)

Coupling of multidimensional Lévy processes and Wasserstein bounds in the small time stable domain of attraction We establish upper and lower bounds on the rate of convergence of the Wasserstein distance on the path space for a wide class of Lévy processes attracted to a multidimensional stable law in the small-time regime. In this talk, the main focus will be on the development of two novel couplings between arbitrary pure-jump Lévy processes, used to obtain upper bounds on the Wasserstein distance. We show that the rate of convergence is polynomial for the domain of normal attraction and slower than any polynomial for the domain of non-normal attraction. As an example, we will consider the class of tempered stable processes that are in the small time domain of attraction of a stable process.

Olivier Menoukeu Pamen (University of Liverpool)

Optimal consumption with labour income and borrowing constraints for recursive preferences In this talk, we present an optimal consumption and investment problem for an investor with liquidity constraints who has isoelastic recursive Epstein-Zin utility preferences and receives a stochastic stream of income. We characterise the optimal consumption strategy as well as the terminal wealth for recursive utility under dynamic liquidity constraints, which prevent the investor to borrow against his stochastic future income. Using duality and backward SDE methods in a possibly non-Markovian diffusion model for the financial market, this gives rise to an interplay of singular control and optimal stopping problems. This talk is based on a joint work with D. Becherer and W. D. Kuissi Kamdem.

Xin Zhi (University of Warwick)

Optimal Stopping with Trees In this talk, we will first review a recent method for solving high-dimensional optimal stopping problems using deep Neural Networks. Second, we propose an alternative algorithm replacing Neural Networks by CART-trees which allow for more interpretation of the estimated stopping rules. We apply our algorithm to multiple examples. We in particular compare the performance of the two algorithms with respect to the Bermudan max-call benchmark example. We also show how our algorithm can be used to plot stopping boundaries.

15th March 2023

/cancelled/

/will be rescheduled/

Daniel Schwarz (University College London)

Radner equilibrium and systems of quadratic BSDEs with discontinuous generators We establish the existence of a Radner Equilibrium in an incomplete, continuous-time financial economy. To prove existence, we formulate a system of quadratic backward stochastic differential equations (BSDEs) which represents the equilibrium problem. Interestingly, the driver of this system of BSDEs features a discontinuity, posing a major challenge for common approaches to establish existence. Exploiting the duality between Markovian BSDEs and PDEs we use unique continuation and backward uniqueness, techniques originally used in the study of PDEs, to show that the set of discontinuity has in fact measure zero. (Joint work with Luis Escauriaza and Hao Xing.)

/This talk has been cancelled and will be rescheduled in Term 3/

3rd May 2023

Eyal Neuman (Imperial College London)

Optimality and Statistical Learning of Propagator Models Price impact refers to the empirical fact that execution of a large order affects the risky asset's price in an adverse and persistent manner leading to less favourable prices. Propagator models help us to quantity the price impact. They express price moves in terms of the influence of past trades, convoluted with a price impact kernel function. We consider a class of optimal liquidation problems where the agent's transactions create transient price impact driven by a Volterra-type propagator along with temporary price impact. We formulate these problems as minimization of a revenue-risk functionals, where the agent also exploits available information on a progressively measurable price predicting signal. By using an infinite dimensional stochastic control approach, we derive analytic solutions to these equations which yields an explicit expression for the optimal trading strategy. We then consider a class of learning problems in which an agent liquidates a risky asset while creating transient price impact driven by an unknown propagator. We characterize the trader's performance as maximization of a revenue-risk functional, where the trader also exploits available information on a price predicting signal. We present a trading algorithm that alternates between exploration and exploitation phases and achieves sublinear regrets with high probability. For the exploration phase we propose a novel approach for non-parametric estimation of the price impact kernel by observing only the visible price process and derive sharp bounds on the convergence rate, which are characterised by the singularity of the propagator.

10th May 2023

Max Nendel (Bielefeld University)

A Parametric Approach to the Estimation of Convex Risk Functionals based on Wasserstein Distance We explore a static setting for the assessment of risk in the context of mathematical finance and actuarial science that takes into account model uncertainty in the distribution of a possibly infinite-dimensional risk factor. We allow for perturbations around a baseline model, measured via Wasserstein distance, and we investigate to which extent this form of probabilistic imprecision can be parametrized. The aim is to come up with a convex risk functional that incorporates a safety margin with respect to nonparametric uncertainty and still can be approximated through parametrized models.The particular form of the parametrization allows to develop a numerical method, based on neural networks, which gives both the value of the risk functional and the optimal perturbation of the reference measure. Moreover, we study the problem under additional constraints on the perturbations, namely, a mean and a martingale constraint. We show that, in both cases, under suitable conditions on the loss function, it is still possible to estimate the risk functional by passing to a parametric family of perturbed models, which again allows for a numerical approximation via neural networks. The talk is based on joint work with Alessandro Sgarabottolo.

24th May 2023

Room MS.04

Umut Cetin (LSE)

Speeding up the Euler scheme for killed diffusions Let X be a linear diffusion taking values in (l,r) and consider the standard Euler scheme to compute an approximation to E[g(X(T);T<u]] for="" a="" given="" function="" g="" and="" deterministic="" t,="" where="" u="" is="" the="" first="" time="" that="" x="" exits="" (l.r).="" it="" well-known="" presence="" of="" killing="" introduces="" loss="" accuracy..="" we="" introduce="" drift-implicit="" euler="" method="" to="" bring="" convergence="" rate="" back="" optimal="" can="" be="" obtained="" in="" absence="" killing,="" using="" theory="" recurrent="" transformations="" developed="" recently.="" although="" current="" setup="" assumes="" one-dimensional="" setting,="" multidimensional="" extension="" within="" reach="" as="" soon="" systematic="" treatment="" available="" higher="" dimensions.<="" p=""> </u]]>

21st June 2023

Room MS.03

Tolulope Fadina (Essex)

Measures of Risk under Uncertainty A risk analyst assesses potential financial losses based on multiple sources of information. Often, the assessment does not only depend on the specification of the loss random variable, but also various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for risk evaluation principles which quantifies jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalized risk measure and present a series of relevant theoretical results. The worst-case, coherent, and robust generalized risk measures are characterized via different sets of intuitive axioms. We establish the equivalence between a few natural forms of law invariance in our framework, and the technical subtlety therein reveals a sharp contrast between our framework and the traditional one. Moreover, coherence and strong law invariance are derived from a combination of other conditions, which provides additional support for coherent risk measures such as Expected Shortfall over Value-at-Risk, a relevant issue for risk management practice.

14th July 2023

(On Friday)

Room MS.03

Samuel Cohen (Oxford)

Stability and approximation of projection filters Nonlinear filtering is a central mathematical tool in understanding how we process information. Sadly, the equations involved are often very high dimensional, which may lead to difficulties in applications. One possible resolution (due to D. Brigo and collaborators) is to replace the filter by a low-dimensional approximation, with hopefully small error. In this talk we will see how, in the case where the underlying process is a finite-state Markov Chain, results on the stability of filters can be strengthened to show that this introduces a well-controlled error, leveraging tools from information geometry. (Based on joint work with Eliana Fausti)

Ruiqi Liu (University of Warwick)

The Optimal Control of Inventory and Production for a Hybrid Energy ProducerLink opens in a new window

25th Feb 2022

(online only)

Huy Chau (University of Manchester)

Super-replication with transaction costs under model uncertainty for continuous processesLink opens in a new window

18th Mar 2022

(online only)

Mikko Pakkanen (Imperial College London)

Rough volatility - Re-examining empirical evidence using the generalised method of momentsLink opens in a new window

Harto Saarinen (University of Turku)

Optimal control problems of one-dimensional diffusions with random intervention timesLink opens in a new window

Urvashi Behary Paray (University of Warwick)

Convexity Corrections via a Markov-functional approachLink opens in a new window

Wed 15th June 2022

3pm in MS.04

Cosimo Andrea Munari (University of Zurich)

Fundamental theorem of asset pricing with acceptable risk in markets with frictions Link opens in a new window

A dynamic dual representation of the buyer's price of American options in a nonlinear

incomplete market