20th Winter school on Mathematical Finance


Jan Obłój: Optimal transport methods in Mathematical Finance

The aim of the minicourse is to present recent advances related to optimal transport and its applications in mathematical finance, statistics, optimization and beyond. I will discuss basics of optimal transport (OT), its duality theory and properties of the induced Wasserstein distance on the space of probability measures. I will then introduce the martingale version of the problem (MOT) and discuss the rich additional structure resulting from the martingale constraint. I will touch on numerics for both problems, including the entropic relaxation of the OT. Finally, I will discuss how Wasserstein distances offer a powerful approach to distributionally robust optimisation (DRO) and present some of its applications in mathematical finance, machine learning and beyond.
Lecture notes: Notes 1, Notes 2, Notes 3, Notes 4, Notes 5, Notes 6.
Slides: Slides 1, Slides 2, Slides 3.

OT references:
[1] C. Villani (2009). Optimal Transport, Springer. Old version, New version.
[2] Filippo Santambrogio (2015). Optimal Transport for Applied Mathematicians, Springer. Old(er) version, Published version.
[3] Cedric Villani (2003). Topics in Optimal Transportation, AMS Graduate Studies in Mathematics, Volume 58.

MOT and DRO basics, a showcase of relevant topics (not all treated):
[1] Beiglböck, M., Henry-Labordère, P. & Penkner, F. (2013). Model-independent bounds for option prices - a mass transport approach. Finance Stoch. 17, 477 - 501.
[2] Beiglböck, M., Cox, A.M.G. & Huesmann, M. (2017). Optimal transport and Skorokhod embedding. Invent. math. 208, 327 - 400.
[3] Guo, G. & Obłój, J. (2019).Computational methods for martingale optimal transport problems. Ann. Appl. Probab. 29 (6), 3311 - 3347.
[4] Obłój, J & Wiesel, J. (2021). Robust estimation of superhedging prices. Ann. Statist. 49(1) 508 - 530.
[5] Blanchet, J. & Murthy, K. , (2019). Quantifying Distributional Model Risk via Optimal Transport, Mathematics of Operations Research 44:2, 565 - 600.
[6] Bartl, D. Drapeau, S. Obłój, J. and Wiesel, J. (2021). Sensitivity analysis of Wasserstein distributionally robust optimization problems. Proc. R. Soc. A. 477: 20210176.

Gilles Pagès: Functional convex ordering of stochastic processes : a constructive approach with applications to finance

We will start by some background on convex (resp. monotone convex) ordering of \(\mathbb{R}^d\)-valued random vectors, namely random vectors (or probability distributions) satisfying \[\begin{aligned} X \le_{cv} Y & \mbox{ if } \mathbb{E}\, f(X) \le \mathbb{E}\, f(Y) \\ & \mbox{ for every convex (resp. non-decreasing convex) function } f: \mathbb{R} \to \mathbb{R}\end{aligned}\] provided both expectations have a sense. In particular we will recall classical results like Strassen and Kellerer's theorems ([Str65] and [Kel72]) which make the connection between convex ordering, martingales and peacocks (for p.c.o.c. itself acronym for the French "processus croissant pour l'ordre convexe") extensively investigated by Yor and co-authors.

We will first investigate both convex orderings of the marginal for Brownian diffusions with respect to their diffusion coefficients. We will show that these results also hold in the functional sense i.e. for functionals of their whole path. Then we will extend this result to jump stochastic differential equations driven by Lévy processes but also Brownian stochastic integrals, etc., following [Pag16]. Some of these results are classical (see [Haj85]) for monotone convex ordering of diffusion) or more recently known (see Rüschendorf and co-authors [BR06, BR07, BR08], Hobson [Hob98, Hob10] among others).

We apply these results to establish in local volatility models sensitivity results of path-dependent options with respect to their volatility. Doing so, we extend a result by [EKJPS98] and [BGW96] which produce upper- and lower-bounds based on a Black-Scholes formula for a vanilla option with convex payoff when the volatility function is itself bounded and bounded away from zero.

As a second step we investigate optimal stopping theory, replacing the path-dependent functional by the Snell envelope of a "vanilla" reward process written on a martingale diffusion, with an obvious connection with American options and again an application to their sensitivity to the volatility process (see [Pag16]).

The specificity of our approach is to be constructive in the sense that we first establish our results in discrete time (which has its own interests), typically for a discretization scheme of the underlying process and then rely on functional limit theorems "à la Jacod-Shiryaev" [JS03] to transfer the property to the continuous time model. When dealing with numerics in Finance, it usually produces arbitrage free approximating numerical methods (as far as volatility modeling is concerned). This can be seen as a paradigm.

In view of the importance taken by McKean-Vlasov equations (for mean-field games but also for Langevin algorithm) we will apply the above paradigm to McKean-Vlasov equations (see [LP20], [LP22]) for both regular and monotone convex ordering.

A natural question on our way is to wonder whether it is possible to extend such an approach to non-Markovian dynamics. The answer is positive since this approach successfully applies to Volterra equations (see [JP22a], [JP22b]), with applications to rough stochastic volatility models. (slides)

[BGW96] Yaacov Z. Bergman, Bruce D. Grundy, and Zvi Wiener. General properties of option prices. The Journal of Finance, 51(5):1573-1610, 1996.
[BR06] Jan Bergenthum and Ludger Rüschendorf. Comparison of option prices in semimartingale models. Finance Stoch., 10(2):222-249, 2006.
[BR07] Jan Bergenthum and Ludger Rüschendorf. Comparison of semimartingales and Lévy processes. Ann. Probab., 35(1):228-254, 2007.
[BR08] Jan Bergenthum and Ludger Rüschendorf. Comparison results for path-dependent options. Statist. Decisions, 26(1):53-72, 2008.
[EKJPS98] Nicole El Karoui, Monique Jeanblanc-Picqué, and Steven E. Shreve. Robustness of the Black and Scholes formula. Math. Finance, 8(2):93-126, 1998.
[Haj85] Bruce Hajek. Mean stochastic comparison of diffusions. Z. Wahrsch. Verw. Gebiete, 68(3):315-329, 1985.
[Hob98] David Hobson. Robust hedging of the lookback option. Finance Stoch., 2(4):329-347, 1998.
[Hob10] David Hobson. Comparison results for stochastic volatility models via coupling. Finance Stoch., 14(1):129-152, 2010.
[JP22a] Benjamin Jourdain and Gilles Pagès. Convex ordering for the stochastic Volterra equation and its Euler schemes, 2022.
[JP22b] Benjamin Jourdain and Gilles Pagès. Convex ordering of solutions to one-dimensional sdes, 2022.
[JS03] Jean Jacod and Albert N. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 2003.
[Kel72] Hans G. Kellerer. Markov-Komposition und eine Anwendung auf Martingale. Math. Ann., 198:99-122, 1972.
[LP20] Yating Liu and Gilles Pagès. Functional convex order for scaled McKean-Vlasov processes, arXiv e-prints (to appear in Annals of Applied Probability), page arXiv:2005.03154, May 2020.
[LP22] Yating Liu and Gilles Pagès. Monotone convex order for McKean-Vlasov processes, Stochastic Process. Appl., 152:312?338, 2022.
[Pag16] Gilles Pagès. Convex order for path-dependent derivatives: a dynamic programming approach. In Séminaire de Probabilités XLVIII, volume 2168 of Lecture Notes in Math., pages 33-96. Springer, Cham, 2016.
[Str65] Volker Strassen. The existence of probability measures with given marginals. Ann. Math. Statist., 36:423-439, 1965.

Papers in close connection with the mini-course
[1] Gilles Pagès (2014), Convex order for path-dependent derivatives: a dynamic programming approach.
[2] Yating Liu, Gilles Pagès (2022), Functional convex order for the scaled McKean-Vlasov processes.
[3] Yating Liu, Gilles Pagès (2021), Monotone convex order for the McKean-Vlasov processes.
[4] Benjamin Jourdain, Gilles Pagès (2022), Convex ordering for stochastic Volterra equations and their Euler schemes.

Special invited lectures

José Manuel Corcuera: Path-dependent Kyle equilibrium model

We consider an auction type equilibrium model with an insider in line with the one originally introduced by Kyle in 1985 and then extended to the continuous time setting by Back in 1992. The novelty introduced in this talk is that we deal with a general price functional depending on the whole past of the aggregate demand, i.e. we work with path-dependency. By using the functional Itô calculus, we provide necessary and sufficient conditions for the existence of an equilibrium. Furthermore, we consider both the cases of a risk-neutral and a risk-averse insider.

Christoph Reisinger: Risk management of options books with arbitrage-free neural-SDE market models

Modelling joint dynamics of liquid vanilla options is crucial for arbitrage-free pricing of illiquid derivatives and managing risks of option trade books. This talk develops a nonparametric model for the European options book, respecting underlying financial constraints while being practically implementable. We derive a state space for prices which are free from static (or model-independent) arbitrage and study the inference problem where a neural SDE model is learnt from discrete time series data of stock and option prices. We validate our approach with numerical experiments using data generated from a Heston stochastic local volatility model and EURO STOXX 50 index data. A subsequent Value-at-Risk (VaR) backtesting analysis shows better coverage performance and less procyclicality than standard filtered historical simulation approaches. Finally, we derive sensitivity-based and minimum-variance-based hedging strategies. When applied to various portfolios of EURO STOXX 50 index options over usual and stressed market periods, our market models can lead to lower hedging errors than Black-Scholes delta-vega hedging, and are less sensitive to the choice of hedging instruments. (slides)
References, papers the lecture is based on
[1] S. N. Cohen, C. Reisinger, and S. Wang, Arbitrage-free neural-SDE market models, arXiv 2105.11053.
[2] S. N. Cohen, C. Reisinger, and S. Wang, Estimating risks of option books using neural-SDE market models, arXiv 2202.07148.
[3] S. N. Cohen, C. Reisinger, and S. Wang, Hedging option books using neural-SDE market models, arXiv 2205.15991.
Further background reading on static arbitrage bounds
[4] M. H. A. Davis and D. G. Hobson. The range of traded option prices. Mathematical Finance, 17(1):1-14, 2007.
[5] L. Cousot. Conditions on option prices for absence of arbitrage and exact calibration. Journal of Banking & Finance, 31(11):3377-3397, 2007.
[6] M. Schweizer and J. Wissel. Arbitrage-free market models for option prices: the multi-strike case. Finance and Stochastics, 12(4):469-505, 2008.
Further background reading on risk estimation
[7] P. Gurrola-Perez and D. Murphy. Filtered historical simulation Value-at-Risk models and their competitors. Bank of England Working Paper, (525), 2015.
Further background reading on minimum variance hedging and data-driven hedging
[8] H. Follmer and M. Schweizer. Hedging of contingent claims under incomplete information. In M. Davis and R. Elliott, editors, Applied Stochastic Analysis, pages 389-414. Stochastics Monographs, Gordon & Breach, London, 1990.
[9] H. Buehler, L. Gonon, J. Teichmann, and B. Wood. Deep hedging. Quantitative Finance, 19(8):1271-1291, 2019.
[10] J. Ruf and W. Wang. Hedging with linear regressions and neural networks. Journal of Business & Economic Statistics, pages 1-13, 2021.
Further background reading on neural SDE (martingale) models
[11] P. Gierjatowicz, M. Sabate-Vidales, D. Siska, L. Szpruch, and Z. Zuric. Robust pricing and hedging via neural SDEs, Journal of Computational Finance, to appear, arXiv 2007.04154.

Luitgard Veraart: Systemic Risk in Markets with Multiple Central Counterparties

We provide a framework for modelling risk and quantifying payment shortfalls in cleared markets with multiple central counterparties (CCPs). Building on the stylised fact that clearing membership is shared among CCPs, we show that stress in this shared membership can transmit across markets through multiple CCPs. We provide stylised examples to lay out how such stress transmission can take place, as well as empirical evidence to illustrate that the mechanisms we study could be relevant in practice. Furthermore, we show how stress mitigation mechanisms such as variation margin gains haircutting by one CCP can have spillover effects on other CCPs. The framework can be used to enhance CCP stress-testing, which currently relies on the ''Cover 2'' standard requiring CCPs to be able to withstand the default of their two largest clearing members. We show that who these two clearing members are can be significantly affected by higher-order effects arising from interconnectedness through shared clearing membership. This is joint work with Iñaki Aldasoro (BIS).
[1] Veraart, L. A. M., Aldasoro, I. (2022), Systemic Risk in Markets with Multiple Central Counterparties, BIS Working Paper No 1052.

Short lectures

Mike Derksen: Stochastic price formation in call auctions

In modern financial markets, most stock exchanges facilitate intraday continuous trading, where buy and sell orders are immediately matched if possible. However, to start and stop trading and determine opening and closing prices, a call auction is usually conducted. In a call auction, orders are aggregated for a while without immediately giving rise to transactions, after which all possible transactions are executed against a single clearing price. While modelling continuous trading is extensively studied in the (quantitative) finance literature, the call auction has received very little attention, despite the fact that the portion of the daily volume that is transacted in the closing auction increased strongly in recent years. In this talk I will discuss a stochastic model of the call auction. The model considers random buy and sell orders, placed following order placement distributions, leading to analytical expressions for the distribution of the clearing price. Order placement distributions and distributions of bid and ask volumes are left as free parameters, permitting possibly heavy-tailed or very skewed order flow conditions. Results and open questions on the relationship between order flow and clearing price will be discussed. This talk is based on joint work with Bas Kleijn and Robin de Vilder [1, 2]. (slides)
[1] Derksen, M., Kleijn, B. and De Vilder, R. (2020), Clearing price distributions in call auctions, Quantitative Finance 20(9): 1475-1493.
[2] Derksen, M., Kleijn, B. and De Vilder, R. (2022), Heavy tailed distributions in closing auctions, Physica A: Statistical Mechanics and its Applications 593: 126959.

Jian He: A Bayesian filter based dimension reduction approach for the pricing grid

Pricing grids are widely used among financial institutions to re-evaluate portfolios when assessing the potential risks, for instance credit risk or market risks. When applying the pricing grid for the revaluation, the biggest challenge is the "curse of dimensionality". Therefore, a dimension reduction approach is usually required to project the higher dimensional risk factors to the lower dimensional factors. In this talk, we propose a Bayesian filter based dimension reduction approach and we will specially focus on the application in the credit risk calculations. (slides)

Matteo Michielon: Implied risk-neutral default probabilities via conic finance

Despite financial modelling often takes place under risk-neutral settings where trading activities are assumed to obey the law of one price, in practice quoted market prices are direction-dependent. Nonetheless, many valuation models require input parameters, implied from observed market quotes, that are consistent with the risk-neutral paradigm. In this presentation, a methodology allowing to extract risk-neutral quantities directly from bid and ask quotes, without relying on mid-quote approximations, is presented. The approach outlined relies on some monotonicity- and liquidity-related assumptions and is based on the conic finance framework, which enables to calculate bid and ask prices of financial securities by employing Choquet expectations with respect to distorted versions of the relevant pricing measures as valuation functionals. In particular, as far as the credit default swap market is concerned, it will be shown how to compute risk-neutral default probabilities from quoted bid and ask premia under well-known dynamics, and at the same time how to calculate the implied liquidity level of the market. This talk is based on [1]. (slides)
[1] Michielon, M., Khedher, A. and Spreij, P. (2021). From bid-ask credit default swap quotes to risk-neutral default probabilities using distorted expectations. International Journal of Theoretical and Applied Finance 24(3), arXiv version.

Stan Olijslagers: Discounting the Future: on Climate Change, Ambiguity Aversion and Epstein-Zin preferences

We show that deviations from standard expected time separable utility have a major impact on estimates of the willingness to pay to avoid future climate change risk. We propose a relatively standard integrated climate/economy model but add stochastic climate disasters. The model yields closed form solutions up to solving an integral, and therefore does not suffer from the curse of dimensionality of most numerical climate/economy models. We analyze the impact of substitution preferences, risk aversion (known probabilities), and specifically ambiguity aversion (unknown probabilities) on the social cost of carbon (SCC). Introducing ambiguity aversion leads to two offsetting effects on the social cost of carbon: a positive direct effect and a negative effect through discounting. Our numerical results show that for reasonable calibrations, the direct effect dominates and that ambiguity aversion gives substantially higher estimates of the SCC.
[1] Stan Olijslagers and Sweder van Wijnbergen (2022). Discounting the Future: on Climate Change, Ambiguity Aversion and Epstein-Zin preferences, working paper.

Poster presentations

  • Giovanni Amici (Polytechnic University of Turin): Multivariate Lévy Models: Calibration and Pricing
  • Benjamin Robinson (Universität Wien): Adapted Wasserstein distance between the laws of SDEs
  • Wanqing Wang (Ecole Polytechnique): Improved convergence rate of RBSDE via penalisation and its applications to American Options

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