# Summary

I am a DPhil student in the Numerical Analysis Group at the University of Oxford, and I am affiliated with Worcester College. I am conducting my research under the supervision of Prof. Raphael Hauser. My research interests are related to optimisation under uncertainty and game theory applied to energy.Before my DPhil, I obtained a MSc in Mathematical Engineering from the Université catholique de Louvain (Belgium) and a MSc in Engineering from the Ecole Centrale de Paris (France). During my MSc thesis, which I undertook under the supervision of Prof. Anthony Papavasiliou, I designed a primal heuristic applied to the optimal transmission expansion planning problem.

# Publications

*Generalisation of the Diagonally Strict Convexity property to non-differentiable payoff functions,*

in preparation, 2021.

*Nash equilibrium for risk-averse investors subject to an uncertain volume target,*

in preparation, 2021.

*Optimal Trade Execution with Uncertain Volume Target,*

available on arXiv (accepted subject to minor corrections), 2018.

@article{Vaes2018, author = {Vaes, Julien and Hauser, Raphael}, title = {{Optimal Trade Execution with Uncertain Volume Target}}, journal = {arXiv 1810.11454}, year = {2018}, url = {http://arxiv.org/abs/1810.11454}, }

**Abstract:**In the seminal paper on optimal execution of portfolio transactions, Almgren and Chriss (2001) define the optimal trading strategy to liquidate a fixed volume of a single security under price uncertainty. Yet there exist situations, such as in the power market, in which the volume to be traded can only be estimated and becomes more accurate when approaching a specified delivery time. During the course of execution, a trader should then constantly adapt their trading strategy to meet their fluctuating volume target. In this paper, we develop a model that accounts for volume uncertainty and we show that a risk-averse trader has benefit in delaying their trades. More precisely, we argue that the optimal strategy is a trade-off between early and late trades in order to balance risk associated with both price and volume. By incorporating a risk term related to the volume to trade, the static optimal strategies suggested by our model avoid the explosion in the algorithmic complexity usually associated with dynamic programming solutions, all the while yielding competitive performance.

**Optimal trading under price uncertainty.**In the seminal paper on optimal execution of portfolio transactions, Almgren and Chriss (2001) define the optimal trading strategy to liquidate a fixed volume of a single security under price uncertainty. Given a fixed amount \( D_{t_b} \) of shares to be traded between the initial trading time \( t_a \) and the closing time \( t_b \), a trading strategy \( {\bf{y}} := [y_1, \dots, y_m] \) corresponds to the proportion of \( D_{t_b} \) to be traded on a discrete number of trading periods \(\tau_i := [t_{i-1}, t_i] \) for \( i \in \{1, m\} \), where \(t_i\) for \( i \in \{0, m\} \) are the decision times, with \(t_0 = t_a \) and \(t_m = t_b \). As the entire position \( D_{t_b} \) has to be traded by time \( t_b \), the following constraint must hold: \[ \sum_{i = 1}^m y_i = 1, \] and the number of shares \( n_i \) to be traded during trading period \( \tau_i \) is given by \[ n_i = y_i D_{t_b}. \] If \( S_0 \) denotes the initial security price at time \( t_a \), or equivalently at decision time \( t_0 \), Almgren and Chriss (2001) suggest that the price dynamics follows an arithmetic random walk altered by the market temporary and permanent impacts induced by the trades: \[ \begin{align} S_{i} & = S_{i-1} + \sigma_{i} \tau_{i}^{1/2} \xi_{i} + \tau_{i} g_{i}\left({\frac{n_{i}}{\tau_{i}}}\right),\\ \tilde{S}_{i} & = S_{i-1} + h_{i}\left({\frac{n_{i}}{\tau_{i}}}\right), \end{align} \] where \(S_{i}\) is the security price at decision time \(t_{i}\), \(n_{i}\) is the volume of securities bought (negative if sold) during trading period \(\tau_{i}\), \(\tilde{S}_{i}\) is the effective security price for the trades executed during trading period \(\tau_{i}\), \(\sigma_{i}\) is the volatility of the asset over trading period \(\tau_{i}\) and \(\xi_{i}\) are draws from independent continuous random variables each with zero mean and unit variance, and where \(g\) and \(h\) respectively model the permanent and temporary price impact as a function of the average trading rate over the trading interval.

The trading cost \( \mathcal{C}({\bf{y}}) \) of a strategy \({\bf{y}}\) is then defined as the difference between

**(i)**the cost incurred at the end of the execution period by following the trading strategy \( {\bf{y}} \) with

**(ii)**the cost ideally obtained in an infinitely liquid market where the entire position \( D_{t_b} \) is traded at the start of the execution period. In Almgren and Chriss (2001), it is assumed that a risk-averse trader desires to minimise the trade-off of the expectation and the variance of the trading cost of their strategy,

*i.e.*\[ \begin{align} \text{minimise}_{{\bf{y}}} &&& \mathbb{E} \left[ \mathcal{C}({\bf{y}}) \right] + \lambda \, \mathbb{V} \left[ \mathcal{C}({\bf{y}}) \right],\\ \text{such that} &&& \sum_{i = 1}^m y_i = 1, \end{align} \] where \(\lambda\) is the risk-aversion parameter; the larger \(\lambda\), the more risk-averse the trader. The figure here under illustrate the optimal strategy in the mean-variance framework when the risk-aversion parameter increases; the larger \(\lambda\), the more front loading in order to avoid risk associated with the price dynamics.

As the variance of the trading cost of a trader is not of foremost importance if they are guaranteed to not pay excessive prices in adverse times, it is often assumed that a trader should evaluate their risk with the Conditional Value-at-Risk (CVaR) risk measure. This translates the idea that a trader is more interested in minimising their expected trading cost conditional to a quantile of worst case scenarios rather than minimising the variance of their trading cost over all scenarios. A trader would then solve the following optimisation problem in order to define their optimal trading strategy, \[ \begin{align} \text{minimise}_{{\bf{y}}} &&& (1-\lambda) \, \mathbb{E} \left[ \mathcal{C}({\bf{y}}) \right] + \lambda \, \text{CVaR}_{\alpha} \left[ \mathcal{C}({\bf{y}}) \right],\\ \text{such that} &&& \sum_{i = 1}^m y_i = 1, \end{align} \] where \(\text{CVaR}_{\alpha} \left[ \mathcal{C}({\bf{y}}) \right]\) can be interpreted as the expectation of the costs conditional on them belonging to the \(\alpha\)% largest trading costs. Here, a value of \(\lambda = 0\) corresponds to a risk-neutral trader, where a value of \(\lambda = 1\) corresponds to an exclusively risk-focused trader.

Given the market parameters assumed in our paper Vaes and Hauser (2018), the following graph and table present respectively the optimal strategies and their corresponding performance given different risk-aversion parameter \(\lambda\). We can notice, similarly to the Mean-Variance framework, that the more risk-averse a trader, the quicker the acquisition of the desired positions.

**Optimal strategy when considering price uncertainty together with the uncertainty associated with the volume target (Vaes and Hauser, 2018).**When the uncertainty related to the volume target is considered in the model detailed in our paper, one obtains the optimal strategies illustrated in the following figure. When comparing the values of the table here under with the previous table, one achieves significant cost reductions.

**Impact of the uncertainty associated with the volume target.**The following figure illustrates how the optimal strategies obtained when considering exclusively price uncertainty must be adapted to additionally consider the uncertainty related to the volume target. One observes that, in accordance with intuition, waiting for a better volume forecast is beneficial in terms of risk reduction.

*Optimal transmission expansion planning,*

available on UCLouvain's website, 2017.

@mastersthesis{Vaes2017, author = {Vaes, Julien and Papavasiliou, Anthony}, title = {{Optimal transmission expansion planning}}, year = {2017}, school = {Universit\'e catholique de Louvain}, url = {https://dial.uclouvain.be/memoire/ucl/en/object/thesis%3A12931}, }

# Talks and posters

**June 16, 2021:**University of Oxford, Numerical Analysis Group Internal Seminar (Oxford, UK).

**October 22, 2019:**INFORMS Annual Meeting 2019 (Seattle, USA).

**August 6, 2019:**ICCOPT 2019 - International Conference on Continuous Optimization (Berlin, Germany).

**July 31, 2019:**ICOEP - International Conference on Optimization and Equilibrium Problems (Dresden, Germany).

**February 13, 2019:**The Alan Turing Institute (London, UK).

**January 24, 2019:**University of Oxford, Mathematical Finance Internal Seminar (Oxford, UK).

**November 7, 2018:**INFORMS Annual Meeting 2018 (Phoenix, USA).

# Teaching

# Other projects

**Optimal running trajectory in Baseball.**PDF

Based on an optimization approach, we have investigated whether we could find a running trajectory for a home run in baseball that is faster than the 16.7 seconds theoretical limit stated in the following article. We assume that the runner has finite velocity and acceleration. We show that it is impossible for a runner to accomplish a home run faster than in 15.5 seconds, but we also found out that the limit of 16.7 could be beaten. The following figures illustrate the optimal paths when considering different sets of constraints. (Credit: Prof. F. Glineur)

**A tsunami simulation with the discontinuous Galerkin finite elements method.**

In 2011, Japan has been severly hit by a tsunami, which damaged tremedously the coast of the country. The idea of this project is to better understand how the tsunami propagated based on a numerical simulation. (Credit: Prof. V. Legat)