19th Winter school on Mathematical Finance


Peter Bank: Dealing with market frictions: some challenges for stochastic analysis and optimal control

Frictions in financial markets have proven a wonderful source for mathematical challenges at the interfaces of Finance, Optimal Control, and Stochastic Analysis. This course will present some tractable models which have been proposed in the literature on optimal trading when markets are not perfectly elastic to demand shocks. We will show how linear-quadratic optimization emerges naturally from asymptotic expansions and discuss how solutions to such problems can be interpreted concisely in practical terms. The tractability of these models also allows for a multi-agent Nash equilibrium analysis which sheds some light on price formation in dealer markets. More refined models of liquidity with transient impact lead to singular control problems for which we develop some tools from convex duality theory. Our discussion will include a number of open questions and challenges left unanswered by my works with Yan Dolinsky, Moritz Voss, Mete Soner, and David Besslich that will serve as the basic references for this course. (slides)
[1] Optimal Investment with Transient Price Impact (with Moritz Voß); To appear in SIAM Journal on Financial Mathematics, preprint arXiv 1804.07392.
[2] Continuous-time Duality for Super-replication with Transient Price Impact (with Yan Dolinsky); To appear in The Annals of Applied Probability, preprint arXiv 1808.09807.
[3] Hedging with Temporary Price Impact (with Mete Soner and Moritz Voß); Mathematics and Financial Economics, 11(2), (2017), 215-239, preprint arXiv 1510.03223.
[4] Linear quadratic stochastic control problems with singular stochastic terminal constraint (with Moritz Voß); SIAM Journal on Control Optimization, 56(2), (2018), 672-699, preprint arXiv 1611.06830.
[5] Liquidity in Competitive Dealer Markets (with Ibrahim Ekren and Johannes Muhle-Karbe); preprint arXiv 1807.08278.
[6] Modelling information flows by Meyer-σ-fields in the singular stochastic control problem of irreversible investment (with David Besslich); preprint arXiv 1810.08495.

Bruno Bouchard: Hedging in market models with linear price impact

We will study a class of financial market models in which prices are affected by trading strategies, in a seemingly linear way. This creates both market impact and illiquidity costs. We shall consider two situations: 1. The option is un-covered, meaning that the premium is paid in cash and the amount of cash and stocks to be delivered at maturity are fixed by the payoff; 2. The option is covered, meaning that the premium is paid and the payoff is delivered in cash and stocks (at their current market price) with proportions decided by the trader. In both cases, we shall show how stochastic target techniques allow one to derive pricing PDEs, and that, under additional smoothness conditions, these PDEs allow us to derive perfect hedging strategies. In case 1., the equation is semi-linear, it is fully non-linear in case 2. This shows that: a. the impact of having to jump to certain deltas at initiation and at maturity has a strong impact on the nature of the hedging strategy; b. although these markets exhibit frictions, perfect hedging is still possible (up to a face-lifting in case 2.). We will go further in the case of covered options and show some asymptotic results for asymptotically small market impact, which allows to construct approximate hedging strategies based on the traditional Black and Scholes delta hedging policy. We shall also prove a dual formulation in this case, reducing the search for a hedging strategy to the resolution of an optimal control in standard Mayer form. (slides)
[1] B. Bouchard and G. Loeper and Y. Zou; Almost-sure hedging with permanent price impact. preprint.
[2] Bruno Bouchard, Grégoire Loeper, and Yiyi Zou; Hedging of covered options with linear market impact and gamma constraint. preprint.
[3] Bruno Bouchard, Grégoire Loeper, Halil Mete Soner, Chao Zhou; Second order stochastic target problems with generalized market impact. preprint.
[4] Bruno Bouchard and Xiaolu Tan; Understanding the dual formulation for the hedging of path-dependent options with price impact. preprint.

Special invited lectures

René Aïd: Optimal electricity demand response contracting with responsiveness incentives

Despite the success of demand response programs in retail electricity markets in reducing average consumption, the random responsiveness of consumers to price event makes their efficiency questionnable to achieve the flexibility needed for electric systems with a large share of renewable energy. The variance of consumers' responses depreciates the value of these mechanisms and makes them poorly reliable. This paper aims at designing demand response contracts which allow to act on both the average consumption and its variance. The interaction between a risk-averse producer and a risk-averse consumer is modelled as a Principal-Agent problem, thus accounting for the moral hazard underlying demand response contracts. The producer, facing the limited flexibility of production, pays an appropriate incentive compensation in order to encourage the consumer to reduce his average consumption and to enhance his responsiveness. We provide a closed-form solution for the optimal contract in the case of constant marginal costs of energy and volatility for the producer and constant marginal value of energy for the consumer. We show that the optimal contract has a rebate form where the initial condition of the consumption serves as a baseline. Further, the consumer cannot manipulate the baseline at his own advantage. The first-best price for energy is a convex combination of the marginal cost and the marginal value of energy where the weights are given by the risk-aversion ratio, and the first-best price for volatility is the risk-aversion ratio times the marginal cost of volatility. The second-best price for energy and volatility do not share this simple structure. They are non-linear and non-constant in time. The price for energy is lower (resp. higher) than the marginal cost of energy during peak-load (resp. off-peak) periods. We illustrate the potential benefit issued from the implementation of an incentive mechanism on the responsiveness of the consumer by calibrating our model with publicly available data. We predict a significant increase of responsiveness under our optimal contract and a significant increase of the producer satisfaction. Joint work with Dylan Possamaï (Columbia University) and Nizar Touzi (Ecole Polytechnique). (slides)
[1] René Aïd, Dylan Possamaï, Nizar Touzi; Optimal electricity demand response contracting with responsiveness incentives. preprint arXiv 810.09063.

Martin Keller-Ressel: Total positivity and the shape of the yield curve

The term structure of interest rates - summarized in the form of the yield or forward curve - is one of the most fundamental economic indicators. Its shape encodes important information on the preferences for short- vs. long-term investments, the desire for liquidity and on expectations of central bank decisions and the general economic outlook. It is therefore a natural question - to be asked of any mathematical model of the term structure - which shapes of yield and forward curves the model is able to (re-)produce. For one-dimensional affine models, such as the Vašíček and the Cox-Ingersoll-Ross model, it has been known for some time that only normal, inverse and humped curves can be produced. In this talk, we provide for the first time a systematic classification of term structure shapes beyond the one-dimensional case and discuss the classification of yield and forward curve shapes in the two-dimensional Vašíček model. As expected, several additional shapes, such as a dipped curve, become attainable in the two-factor model. Our main mathematical tool is the theory of total positivity (pioneered by Schönberg, Gantmacher, Krein and Karlin in the last century), a theory linked to the variation-diminishing properties of certain matrices, function systems and integral kernels. We explain how total positivity can be applied to the shape analysis of the term structure in case of the two-factor Vasicek model and discuss possible applications to other multi-factor interest rate models as well.
[1] Keller-Ressel, Martin, and Thomas Steiner; Yield curve shapes and the asymptotic short rate distribution in affine one-factor models. Finance and Stochastics 12.2 (2008): 149-172.
[2] Keller-Ressel, Martin; Correction to: Yield curve shapes and the asymptotic short rate distribution in affine one-factor models. Finance and Stochastics 22.2 (2018): 503-510.
[3] Diez, Franziska, and Ralf Korn; Yield curve shapes of Vasicek interest rate models, measure transformations and an application for the simulation of pension products. European Actuarial Journal (2019): 1-30.
[4] Keller-Ressel, Martin; Total positivity and the classification of term structure shapes in the two-factor Vasicek model (2019). preprint arXiv:1908.04667.

Vladimir Piterbarg: The classical optimal investment problem: modern models and deep learning

We revisit the classical Merton optimal allocation problem and consider it through the lens of modern local and stochastic volatility models. We demonstrate that the adjustments to the myopic Merton ratio can be largely deduced from observed option prices. Furthermore, we investigate how deep learning techniques could help us determine a model-free optimal investment strategy. (slides)
[1] Vladimir Piterbarg (2018); Optimal Investment Problem in Stochastic and Local Volatility Models. preprint SSRN:3265987

Short lectures

Arnoud den Boer: Dynamic pricing and learning

For sellers of (online) services or commodities, efficiently learning optimal selling prices from accumulating sales data is an important challenge. This problem has received lots of research attention in the last decade, both from the operations research and computer science community. In this talk I will describe a few important and intriguing recent results. (slides)
[1] Arnoud V. den Boer, Bert Zwart; Simultaneously Learning and Optimizing Using Controlled Variance Pricing. Management Science 60(3):770-783.
[2] Arnoud V. den Boer; Dynamic Pricing with Multiple Products and Partially Specified Demand Distribution. Mathematics of Operations Research 39(3):863-888.
[3] Arnoud V. den Boer, Bert Zwart; Dynamic Pricing and Learning with Finite Inventories. Management Science 63(4):965-978.
[4] Arnoud den Boer, N. Bora Keskin; Dynamic Pricing with Demand Learning and Reference Effects. SSRN preprint 3092745.
[5] Omar Besbes, Assaf Zeevi; On the (Surprising) Sufficiency of Linear Models for Dynamic Pricing with Demand Learning. Management Science 61(4):723-739.

Guusje Delsing: Capital reserve management for a multi-dimensional risk model

Firms should keep capital to offer sufficient protection against the risks they are facing. In the insurance context methods have been developed to determine the minimum capital level required, but less so in the context of firms with multiple business lines including allocation. This research focusses on the calculation of finite-time ruin probabilities and capital reserves for a multi-dimensional risk model. The individual reserves of these lines of business are modelled by means of a Cramér-Lundberg model with constant incoming premiums and outgoing claims that arrive according to a Poisson process. To allow for dependence between business lines we introduce a common (latent) environmental factor. This environmental factor impacts the claim inter-occurrence times as well as the claim sizes. Considering a fixed environmental process over time, we present a novel Bayesian approach to calibrate the latent environmental state distribution based on observations concerning the claim processes. We then allow for the distribution of individual claims to change over time by using a Markov environmental process. For the latter, we present two approximations for the finite-time multivariate survival/ruin probabilities: a diffusion approximation and a single-switch approximation. Finally, we point out how to determine the (allocated) optimal initial capital of the different business lines under specific constraints on the ruin/survival probability of subsets of business lines. This research has been performed together with Erik Winands, Michel Mandjes and Peter Spreij. (slides)
[1] G. Delsing, M. Mandjes, P. Spreij, E. Winands (2019); Asymptotics and approximations of ruin probabilities for multivariate risk processes in a Markovian environment. Methodology and Computing in Applied Probability, online.
[2] G. Delsing, M. Mandjes, P. Spreij, E. Winands (2019); An optimization approach to adaptive multi-dimensional capital management. Insurance: Mathematical and Economics, 84, pp. 87-97.

Lingwei Kong: Hansen-Jagannathan distance in the presence of weak (proxy) factors

We analyse the Hansen-Jagannathan (HJ) distance statistic in a GMM framework and show its misbehaviour in the presence of weak/useless (proxy) factors. We provide a new test procedure for the specification test based on the HJ distance which is robust against the presence of weak (proxy) factors. We also show the conventional test procedure tends to over-reject correct model specification and our test procedure leads to size-correct results. Our simulation exercises support our theory. (slides)
[1] Lingwei Kong; Weak (Proxy) Factors Robust Hansen-Jagannathan Distance For Linear Asset Pricing Models, working paper.

Sofie Reyners: Machine learning for derivative pricing: Gaussian processes vs. gradient boosting

In the derivatives world, daily zillion computations need to be done. Since financial models and instruments have become more and more complex, this is not always trivial and one often has to rely on time-consuming numerical techniques. We show how machine learning algorithms can be used for speeding up classical computations, by deploying either Gaussian process regression (GPR) models or gradient boosting machine (GBM) models. The price we have to pay for this extra speed is some loss of accuracy, which is often very acceptable from a practical point of view.
In this talk, we focus on speeding up advanced pricing methods for structured products. We compare the pricing performance and behavior of GPR and GBM models. Finally, we show how Greek profiles can be computed in an efficient and smooth way. (slides)
[1] Jan De Spiegeleer, Dilip B. Madan, Sofie Reyners & Wim Schoutens (2018); Machine learning for quantitative finance: fast derivative pricing, hedging and fitting. Quantitative Finance, 18:10, 1635-1643.
[2] Davis, J., Devos, L., Reyners, S. and Schoutens, W.; Gradient Boosting for Quantitative Finance. preprint.

Poster presentations

  • Ruonan Fu (University of Tilburg): Ambiguity measures and stock market returns
  • Damian Jelito (Jagiellonian University): Optimal stopping with the risk sensitive investment criterion
  • Amit Zubier Arfan (The University of Manchester): A stochastic volatility model for optimal market-making
  • Thomas van der Zwaard (TU Delft): Valuation adjustments and the hedging thereof in practice
  • Anna Sulima (Wrocław University of Economics and Business): Completeness and arbitrage-free of regime-switching Lévy market
  • Christopher Hofmann (TU Chemnitz): Simultaneous multi-parameter choice in inverse option pricing

To the homepage of the Winter School on Mathematical Finance