Minicourses

Sebastian Jaimungal: Algorithmic and high-frequency trading

This mini course focuses on how to formulate and solve various mathematical and statistical problems arising in algorithmic trading. We will look at high-frequency data, how market participants behave, investigate stochastic control problems related trade execution, high-frequency market-making, and statistical arbitrage. We will see how to solve those problems using the principles of dynamic programming, as well as how one can use machine learning techniques (including classification and reinforcement learning) to assist in building "model-free" strategies. Time permitting, we will also touch on mean-field games to account for the actions of multiple optimizing agents.
The mini-course will cover the following topics: Bayes and multi-class logistic classification, reinforcement learning, continuous time dynamic programming and Hamilton-Jacobi-Bellman Equations, limit order books, optimal execution, statistical arbitrage, incorporating order-flow, latent factors, and multiple agents through mean-field games. (see this webpage for slides and other supporting material)
References
[1] Gould, M. D., Porter, M. A., Williams, S., McDonald, M., Fenn, D. J., & Howison, S. D. (2013). Limit order books. Quantitative Finance, 13(11), 1709-1742.
[2] Cartea, Á., & Jaimungal, S. (2016). Incorporating order-flow into optimal execution. Mathematics and Financial Economics, 10(3), 339-364.
[3] Cartea, Á., & Jaimungal, S. (2015). Risk metrics and fine tuning of high-frequency trading strategies. Mathematical Finance, 25(3), 576-611.
[4] Casgrain, P., & Jaimungal, S. (2016). Trading Algorithms with Learning in Latent Alpha Models.
[5] Cartea, Á., Jaimungal, S., & Penalva, J. (2015). Algorithmic and high-frequency trading. Cambridge University Press. Also available at Amazon.co.uk.

Special invited lectures

Giulia Di Nunno: Fully dynamic risk-indifference pricing and no-good-deal bounds

In an incomplete market with no a priori assumption on the underlying price dynamics, we focus on the problem of derivative pricing from the seller's perspective. We consider risk indifference pricing as an alternative to the classical utility indifference, so that the actual evaluations are done via risk measures. In addition we propose a fully-dynamic risk-indifference criteria, in which a whole family of risk measures is considered. This is based on the concept of fully-dynamic risk measure which extends the one of dynamic risk measure by adding the actual possibility of changing the risk perspectives over time. This entails an analysis on the questions of time-consistency in the risk and then the price evaluations. The framework proposed fits well the study of both short and long term investments. In this framework we study whether the risk indifference criterion actually provides a proper convex price system. We shall see that some conditions have to be fulfilled. Then we consider the relationship of fully dynamic risk-indifference price with no-good-deal bounds. We recall that no-good-deal pricing guarantees that not only arbitrage opportunities are excluded, but also all deals that are 'too good to be true'. We shall provide necessary and sufficient conditions on the fully dynamic risk measure so that the corresponding risk-indifference prices satisfy the no-good-deal bounds. In this way no-good-deal bounds provide a way to select the risk measures to obtain a proper fully-dynamic risk-indifference price system. The presentation is based on various joint works with Jocelyne Bion-Nadal.
Main reference
[1] J. Bion-Nadal and G. Di Nunno (2017): Fully-dynamic risk-indifference pricing and no-good-deal bounds. ArXiv: 1711.05567:
Other references
[2] B. Acciaio and I. Penner. Dynamic risk measures. Advanced Mathematical Methods for Finance, Eds. G. Di Nunno and B. Øksendal. pp. 1-34, Springer, Heidelberg 2011.
[3] J. Bion-Nadal and G. Di Nunno (2017). Representation of convex operators and their static and dynamic sandwich extensions. Journal of Convex Analysis, 24 (4). A version on ArXiv: 1412.2030.
[4] J. Bion-Nadal and G. Di Nunno (2013): Dynamic no-good-deal pricing measures and extension theorems for linear operators on $L_\infty$. Finance and Stochastics, 17, pp. 587-613.
[5] S. Klöppel and M. Schweizer (2007): Dynamic utility indifference valuation via convex risk measures. Mathematical Finance, 17:599-627.
[6] D. Filipovic and G. Svinland (2012): The canonical model space for law-invariant convex risk measures is $L^1$. Mathematical Finance, 22, pp. 585-589, 2012. (Extended working paper version. No. 2, March 2008, Vienna Institute of Finance.)

Martino Grasselli : Quantization meets fourier: a new technology for pricing options

In this paper we introduce a novel pricing methodology for a broad class of models for which the characteristic function of the log-asset price can be efficiently computed. The new method avoids the numerical integration required by the Fourier-based approaches and reveals to be fast and accurate, to the point that we can calibrate the models on real data. Our approach allows to price also American-style options, as it is possible to compute the transition probabilities for the underlying. This is accomplished through an efficient multinomial lattice discretization of the asset price based on a new quantization procedure which exploits the knowledge of the Fourier transform of the process at a given time. As a motivating example, we price an American Put option in a Tempered Stable model, with constitutes the first application of quantization to a pure jump process. (joint work with Giorgia Callegaro and Lucio Fiorin)
References
[1] Giorgia Callegaro, Lucio Fiorin and Martino Grasselli. American quantized calibration in stochastic volatility.
[2] Giorgia Callegaro, Lucio Fiorin and Martino Grasselli. Quantization Meets Fourier: a New Technology for Pricing Options.
[3] Giorgia Callegaro, Lucio Fiorin and Martino Grasselli. Pricing via Quantization in Stochastic Volatility Models.
[4] Giorgia Callegaro, Lucio Fiorin and Martino Grasselli. Quantized calibration in local volatility.
[5] Gilles Pagès. Introduction to optimal vector quantization and its applications for numerics.
[6] Gilles Pagès, Abass Sagna. Recursive marginal quantization of the Euler scheme of a diffusion process.

Short lectures

Ki Wai Chau: Stochastic grid bundling method for backward stochastic differential equations

In this work, we aim to apply stochastic grid bundling method (SGBM) to numerically solve backward stochastic differential equations. SGBM is an algorithm designed to solve a backward in time dynamic programming problem with initial application in pricing Bermudan options. It takes advantage of both regress later technique and the adaptive local partition approach in order to provide better numerical result. In usual regression methods for backward in time problems, the target function values at the end of a time interval is regressed on some dependent variables measured at the beginning of the time interval, which creates a projection error. The dependent variable is regressed here on a set of basis functions at the end of the interval instead in a regress-later method, and the conditional expectation across the interval is then computed exactly for each basis function. By removing the projection error in the regression step, a regress-later method is capable of converging faster than a conventional method. With the adaptive local partition approach, a partition of the function domain is generated depending on the simulated examples and regression is performed separately at each partition. Since the support is chosen so that they contain roughly the same number of samples and is non-overlapping, SGBM is easy to scale up and can facilitate the application of parallel computing to our algorithm. (slides)
Reference
[1] Ki Wai Chau. Stochastic grid bundling method for backward stochastic differential equations (extended abstract).

Andrea Fontanari: Urn modelling of joint mortality and its impact on annuity contracts

In this paper we propose a new modeling approach to joint (or depen- dent) mortality, that is to say the mortality risk in couples of individuals, whose lives and survivorships are likely to be interconnected. Differently from the other works in the literature about joint mortality, we propose a constructive approach based on a special class of combinatorial stochastic processes, i.e. reinforced urn processes, or RUP in acronym (Muliere et al., 2000). Exploiting the Bayesian update mechanism embedded in RUP, the new model has the ability of continuously learning from data, even when new observations become available at a later stage. Thanks to this, the model is able to improve its predictive power over time. If prior beliefs about the dependence structure and/or the marginal survivorships are present, these can be easily embedded in the model, by acting on the urn compositions. The use of RUP allows for the exploitation of many powerful results of Bayesian nonparametrics, for example, the beta-Stacy processes, a very flexible class of random distributions to model survival functions. Empirically, the new construction can be estimated via simulation-based techniques, like Markov Chain Monte Carlo. The modeling of dependent mortality is particularly interesting when we consider annuity contracts. An annuity is a contractual guarantee, issued by insurance companies, pension plans, etc., which offers promises of providing periodic income over the lifetime of individuals, after the payment of a lump sum premium. Using the well-known data of Frees et al. (2016), we discuss how the new model can be used in practice, also showing how the strong positive de- pendence between joint lives impacts on annuity values and, by extension, on similar products.
References
[1] E. Frees, J. Carriere, E. A. Valdez (1996). Annuity valuation with dependent mortality. The Journal of Risk and Insurance 63, 229-261.
[2] E. Luciano, J. Spreeuw, E. Vigna (2008). Modelling stochastic mortality for dependent lives. Insurance: Mathematics and Economics 43, 234-244.
[3] P. Muliere, P. Secchi, S. Walker (2000). Urn schemes and reinforced ran- dom walks. Stochastic Processes and their Applications 88, 59-78.
[4] Bulla, Paolo, Pietro Muliere, and Stephen Walker. Bayesian nonparametric estimation of a bivariate survival function. Statistica Sinica (2007): 427-444.
[5] Walker, Stephen, and Pietro Muliere. Beta-Stacy processes and a generalization of the Pólya-urn scheme. The Annals of Statistics (1997): 1762-1780.

Jitze Hooijsma: Long or short: how to optimally invest in variance swaps?

The existing literature provides contradicting investment advice on the optimal investment mix for variance swaps. Some papers advocate a short-long variance swap strategy to hedge against volatility increases in the short run while benefiting from negative variance risk premia in the long run, while others prefer the opposite trading strategy. In this paper, we develop a procedure to estimate multi-factor stochastic volatility models using both equity and variance swap data. We apply our estimation methodology to several canonical variance swap models and analyze their implications for optimal portfolio choice. Our analysis dissects the complex interplay of risk factors and risk premia which is crucial for optimal variance swap investment strategies.

Rob Sperna Weiland: Feedback between credit and liquidity risk in the US corporate bond market

In this paper, we analyze the dynamic interactions between credit and liquidity risk and their impact on bond prices and risk. We propose a novel way of modeling credit-liquidity interactions by using mutually exciting processes and develop a corresponding Bayesian estimation procedure. We show, using US corporate bond transaction data, that there is evidence of feedback between credit and liquidity risk, that this feedback is asymmetric, and stronger for bonds with a low credit rating. Our model allows for a decomposition of bond yield spreads into pure credit, pure liquidity, credit-induced liquidity, and liquidity-induced credit components. We find that, on average, the credit-induced liquidity component accounts for about 8% (AAA/AA) to 17% (B and lower) of total yield spreads, but in the most distressed periods it accounts for over 40%. Our decomposition reveals that the widening of yield spreads during the financial crisis can mainly be attributed to a decrease in market liquidity, which, in turn, is for a substantial part caused by deteriorating credit conditions. Furthermore, we show that credit-liquidity interactions are responsible for a large part of Value-at-Risk bond capital requirements. Ignoring such interactions may result in a severe underestimation of required capital, especially for bonds with lower credit ratings.
References
[1] Rob C. Sperna Weiland, Roger J. A. Laeven and Frank De Jong (2017). Feedback Between Credit and Liquidity Risk in the US Corporate Bond Market.

Poster presentations

• Junbin Chen (Durham University): On nonparametric predictive inference for asset and European option trading in the binomial tree model
• Ting He (Durham University): Nonparametric predictive inference for American option pricing based on the binomial tree model
• Jakob Krause (Martin-Luther-Universität Halle-Wittenberg): The role of liquidity in electricity markets
• Pankaj Kumar (Copenhagen Business School): Hybrid statistical agent based model for financial market
• Anton Shardin (BTU Cottbus-Senftenberg): Stochastic optimal control problems under partial information for an energy storage
• Anna Sulima (Wrocław University): Predictable representation for Itô-Markov additive processes with the application to complete financial markets
• Junyi Zhang (LSE): Pricing of arithmetic Asian option under 3/2 stochastic volatility model