Senior lecturers
Christa Cuchiero:
Signature SDEs as affine and polynomial processes
Signature methods represent a non-parametric way for extracting characteristic features from time series data which is essential in machine learning tasks. This explains why these techniques become more and more popular in Econometrics and Mathematical Finance. Indeed, signature based approaches allow for data-driven and thus more robust model selection mechanisms, while first principles like no arbitrage can still be easily guaranteed. In view of option pricing the key quantity that one needs to compute in these models is the expected signature of some underlying process. Surprisingly this can be achieved for generic classes of jump diffusions (with possibly path dependent characteristics) via techniques from affine and polynomial processes. More precisely, we show how the signature process of these jumps diffusions can be embedded in the framework of affine and polynomial processes, which have been -- due to their tractability -- the dominating process class prior to the new era of highly over-parametrized dynamic models. In other words, this means that the infinite dimensional Feynman Kac PIDE of the signature process can be reduced to (sometimes finite dimensional) ODEs either of linear or Riccati type. We illustrate our findings by means of one dimensional diffusion processes with analytic characteristics. (slides)
Blanka Horvath:
Data-driven market simulators
and some simple applications of signature kernel methods in mathematical finance
Techniques that address sequential data have been a central theme in machine learning research in the past years. More recently, such considerations have entered the field of finance-related ML applications in several areas where we face inherently path dependent problems: from (deep) pricing and hedging (of path-dependent options) to generative modelling of synthetic market data, which we refer to as market generation. We revisit Deep Hedging from the perspective of the role of the data streams used for training and highlight how this perspective motivates the use of highly accurate generative models for synthetic data generation. From this, we draw conclusions regarding the implications for risk management and model governance of these applications, in contrast to risk-management in classical quantitative finance approaches. Indeed, financial ML applications and their risk-management heavily rely on a solid means of measuring and efficiently computing (similarity-) metrics between datasets consisting of sample paths of stochastic processes. Stochastic processes are at their core random variables with values on path space. However, while the distance between two (finite dimensional) distributions was historically well understood, the extension of this notion to the level of stochastic processes remained a challenge until recently. We discuss the effect of different choices of such metrics while revisiting some topics that are central to ML-augmented quantitative finance applications (such as the synthetic generation and the evaluation of similarity of data streams) from a regulatory (and model governance) perspective. Finally, we discuss the effect of considering refined metrics which respect and preserve the information structure (the filtration) of the market and the implications and relevance of such metrics on financial results.
Jan Obłój:
Sensitivity analysis for Wasserstein Distributionally Robust Optimization and its applications
I will showcase how methods from optimal transport and distributionally robust optimisation allow to capture and quantify sensitivity to model uncertainty for a myriad of problems. We consider a generic stochastic optimisation problem. This could be a mean-variance or a utility maximisation portfolio allocation problem, an optimised certainty equivalent or a risk measure computation, a standard regression or a deep learning problem. At the heart of the optimisation is a probability measure, or a model, which describes the system. It could come from a training set (data), simulation or a modelling effort but there is always a degree of uncertainty about it. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated measure. Our main results provide explicit formulae for the first order correction to both the value function and the optimiser. We further extend our results to optimisation under linear constraints. Our sensitivity analysis of the distributionally robust optimisation problems finds applications in statistics, machine learning, mathematical finance and uncertainty quantification. In the talk, I will discuss several financial examples anchored in a one-step financial model and compute their sensitivity to model uncertainty. These will include: option pricing (giving a non-parametric version of the so-called Vega), optimised certainty equivalent and similar risk assessments, optimal investment and a robust version of Davis' marginal utility option pricing. I will also discuss briefly other applications, such as explicit formulae for first-order approximations of square-root LASSO and square-root Ridge optimisers and measures of NN architecture robustness with respect to adversarial data. I will also showcase the link with building data-driven estimators of risk measures. Talk based on joint works with Daniel Bartl, Samuel Drapeau and Johannes Wiesel. (slides)
Gilles Pagès:
Functional convex ordering of stochastic processes : a constructive approach
After some short background on convex ordering of $\mathbb{R}^d$-valued random vectors, its connections with martingales through Kellerer's and Strassen's theorems, we show how to establish such a convex ordering for first for the marginals of martingale (or scaled) Brownian (or Lévy driven) diffusions depending on the pointwise ordering of their diffusion coefficient in one and higher dimensions. Then we extend it to convex functionals of the whole trajectories. We also consider the monotone convex order. Our approach (close to that adopted Rüschendorf and co-authors to some extent) is constructive since we first establish our results in discrete time (which has its own interests), typically for the Euler discretization scheme, and transfer the property to the continuous time process using some limit theorems « à la Jacod-Shiryaev ». In fact this is a kind of paradigm that can be e.g. applied the Snell envelope of a “vanilla” reward process in optimal stopping theory. All these results have straightforward application in Finance to analyze the sensitivity of path-dependent and/or American options to the volatility functions in local volatility models. It also provides arbitrage-free approximations. In a recent paper with Y. Liu, we extended these results to McKean-Vlasov equations. A natural question at this stage is to wonder whether it is possible to apply an approach to (non-martingale and) non-Markovian dynamics like solutions to Volterra equations. A positive answer is provided under natural convexity assumptions on the coefficients in a work in progress with B. Jourdain. This last result has applications to futures on VIX in a quadratic rough volatility model. (slides)
Mitja Stadje:
Hedging and optimal portfolio choice under endogenous permanent market impacts
We consider hedging and expected utility maximization problems of a large investor who is allowed to make transactions on a tradable asset in a financial market with endogenous permanent market impacts. The asset price is assumed to follow a nonlinear price curve quoted in the market as the utility indifference curve of a representative liquidity supplier. Under this market impact model, we introduce a completeness condition under which any derivative can be perfectly replicated by a dynamic trading strategy. For the optimal portfolio choice problem, we show that optimality can be fully characterized via a system of coupled forward-backward stochastic differential equations (FBSDEs) which is equivalent to a highly non-linear backward stochastic partial differential equation (BSPDE). We show existence and uniqueness solutions for FBSDEs in the case where the driver function of the representative market maker grows at least quadratic or the utility function of the large investor falls faster than quadratic or is exponential. Explicit examples are provided when the market is complete or the driver function is positively homogeneous. This talk is based on joint works with Masaaki Fukasawa and Thai Nguyen. (slides) |
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Ioannis Anagnostou:
Financial market community detection and an application to portfolio risk modelling
One of the most challenging aspects in the analysis and modelling of financial markets is the presence of an emergent, intermediate level of structure standing in between the microscopic dynamics of individual financial entities and the macroscopic dynamics of the market as a whole. This mesoscopic level of organisation is often sought for via factor models that ultimately decompose the market according to geographic regions and economic industries. However, at a more general level, the presence of mesoscopic structure might be revealed in an entirely data-driven approach, looking for a modular and possibly hierarchical organisation of the empirical correlation matrix between financial time series. The crucial ingredient in such an approach is the definition of an appropriate null model for the correlation matrix. Recent research showed that community detection techniques developed for networks become intrinsically biased when applied to correlation matrices. For this reason, a method based on Random Matrix Theory has been developed, which identifies the optimal hierarchical decomposition of the system into internally correlated and mutually anti-correlated communities. Building upon this technique, here we resolve the mesoscopic structure of the CDS market and identify groups of issuers that cannot be traced back to standard industry/region taxonomies, thereby being inaccessible to standard factor models. We use this decomposition to introduce a novel default risk model that is shown to outperform more traditional alternatives.
Thijs Kamma:
Dual formulation of the optimal consumption problem with ratio habit formation
This paper provides a dual formulation of the optimal consumption problem with internal ratio habit formation. In this problem, the agent derives utility from the ratio of consumption to the internal habit component. Due to this multiplicative specification of the habit model, standard Lagrangian techniques fail to supply a candidate for the dual problem. Using a slight modification of the conventional Legendre transform, we manage to identify a candidate formulation and prove that it specifies a well-posed dual problem. This formulation discloses the analytical links between optimal consumption, the habit level and the portfolio process. On the basis of these links, we propose two analytical approximations to optimal consumption. Small duality gaps demonstrate the potential accuracy of these approximations. (slides)
Sven Karbach:
An affine stochastic volatility model in Hilbert spaces with state-dependent jumps
We present a flexible and tractable stochastic volatility model in Hilbert spaces with constant and state-dependent jumps. The model consists of a Hilbert-valued linear SDE joined by an affine Markov process, which we use to model the operator-valued instantaneous variance process of the former. The linear SDE admits for a possibly unbounded drift operator and infinite-dimensional Wiener noise perturbed by the stochastic volatility dynamics. The variance process itself is recruited from a class of pure-jump Markov processes with values in the positive self-adjoint Hilbert-Schmidt operators and such that its Laplace transform exhibits an exponential-affine form in the initial value of the process. We show that this desired affine property inherits to the characteristic function of the joint model and we discuss applications of our infinite-dimensional stochastic volatility model to commodity forward markets, where the dynamics of forward price curves can be specified by a SPDE in the Heath-Jarrow-Morton-Musiela modeling framework, which, formulated as a linear SDE on some Hilbert space containing the forward curves, fits into our model setting.
Ioana Neamțu:
Risk-taking and uncertainty: do contingent convertible (CoCo) bonds increase the risk appetite of banks?
We assess the impact of contingent convertible (CoCo) bonds and the wealth transfers they imply conditional on conversion on the risk-taking behaviour of the issuing bank. We also test for regulatory arbitrage: do banks try to maintain risk-taking incentives by issuing CoCo bonds, when regulators reduce them through higher capitalization ratios? While we test for, and reject sample selection bias, we show that CoCo bonds issuance has a strong positive effect on risk-taking behaviour, and so do conversion parameters that reduce dilution of existing shareholders upon conversion. Higher economic volatility amplifies the impact of CoCo bonds on risk-taking. [Joint work with Mahmoud Fatouh and Sweder van Wijnbergen.]
Anna Sulima:
Completeness, arbitrage and optimal portfolio strategy in an Itô-Markov additive market
We study a market with the prices of financial assets described by Itô-Markov additive processes, which combine Lévy processes and regime switching models. Such a process evolves as an Itô-Lévy process between changes of states of a Markov chain, that is, its parameters depend on the current state of the Markov chain. Thus, the model takes into account two sources of risk: the jump diffusion risk and the regime switching risk. Due to the presence of these sources of risk, our market model is incomplete. We show how to complete the Itô-Markov additive market model by adding Markovian jump securities, Markovian power-jump securities and impulse regime switching securities. Using these securities, all contingent claims can be replicated by a self-financing portfolio. Moreover, we give conditions for the market to be asymptotic-arbitrage-free, namely, we find a martingale measure under which all the discounted price processes are martingales. We also consider the problem of identifying the optimal strategy that maximizes the expected value of the utility function of the wealth process at the end of some fixed period. The analysis is conducted for the logarithmic and power utility functions. (slides) References: 1. Palmowski, Z.; Stettner, Ł.; Sulima, A. (2019). Optimal portfolio selection in an Itô-Markov additive market. Risks 7(1): 34. 2. Palmowski, Z.; Stettner, Ł.; Sulima, A. (2018). A note on chaotic and predictable representations for Itô-Markov additive processes. Stochastic Analysis and Applications 36: 622-638. |