Abstract: Taking uncertainty into account is crucial when making strategic decisions. To guard against the risk of adverse scenarios, traditional optimisation techniques incorporate uncertainty on the basis of prior knowledge on its distribution. In this paper, we show how, based on a limited amount of historical data, we can generate from a low-dimensional space the underlying structure of uncertainty that could then be used in such optimisation frameworks. To this end, we first exploit the correlation between the sources of uncertainty through a principal component analysis to reduce dimensionality. Next, we perform clustering to reveal the typical uncertainty patterns, and finally we generate polyhedral uncertainty sets based on a kernel density estimation (KDE) of marginal probability functions.

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.

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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.

Trading period,\( \lambda = 0 \),\( \lambda = 10^{-6} \),\( \lambda = 10^{-5} \),\( \lambda = 10^{-4} \)
1,0.2,0.4953,0.8499,0.98
2,0.2,0.2524,0.1276,0.01956
3,0.2,0.1309,0.01914,0.0003902
4,0.2,0.07248,0.002883,7.786e-6
5,0.2,0.04893,0.0004959,1.599e-7

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.

Trading period,\( \lambda = 0.0 \),\( \lambda = 0.25 \),\( \lambda = 0.5 \),\( \lambda = 0.75 \),\( \lambda = 1.0 \)
1,0.1998,0.4383,0.6449,0.8091,0.9431
2,0.2001,0.2509,0.2289,0.1542,0.05349
3,0.2002,0.1477,0.08183,0.0295,0.003255
4,0.2001,0.09342,0.03063,0.006004,0.0001604
5,0.1998,0.06968,0.01377,0.001158,2.18e-5

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.

Trading period,\( \lambda = 0.0 \),\( \lambda = 0.25 \),\( \lambda = 0.5 \),\( \lambda = 0.75 \),\( \lambda = 1.0 \)
1,0.1992,0.4074,0.5737,0.6822,0.7499
2,0.1993,0.2564,0.2655,0.2457,0.2216
3,0.1995,0.1606,0.1152,0.08082,0.05941
4,0.1997,0.1023,0.04311,0.01625,0.004082
5,0.2023,0.07331,0.002517,-0.02494,-0.03496

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.

Trading period,\( \lambda = 0.0 \),\( \lambda = 0.25 \),\( \lambda = 0.5 \),\( \lambda = 0.75 \),\( \lambda = 1.0 \)
1,-0.000596,-0.03091,-0.07115,-0.1269,-0.1932
2,-0.0008091,0.005497,0.03655,0.09144,0.1681
3,-0.0006439,0.01294,0.03337,0.05132,0.05615
4,-0.0003996,0.008844,0.01249,0.01025,0.003921
5,0.002449,0.003628,-0.01125,-0.0261,-0.03498

The electricity sector in the UK has only been deregulated relatively recently, which means that today there is still limited competition in the electricity market, although the situation has been improving in recent years. As a result of the limited competition, the electricity market is far from perfect as it offers limited liquidity and suffers from high volatility of electricity prices. Therefore, hedging against risk through electricity-related financial derivatives is of paramount importance for electricity producers and wholesalers, who wish to secure the sale or purchase of electricity from minutes to years before a given delivery period. In this thesis, we focus on hedging possibilities by considering the competition and uncertainty inherent in the electricity market. We first consider the case of the optimal execution problem under an uncertain volume target. We motivate this approach by drawing a parallel with the situation of an electricity wholesaler who must buy electricity but does not yet know the actual demand of its end-users. The trade execution framework is a good compromise between accurate modelling of market frictions and simplicity of the model. We propose two approaches to address this problem: one is based on a one-shot optimisation problem under uncertainty with a predetermined recourse rule, while the other solves a dynamic program where risk is estimated with a dynamic risk measure. We then turn our attention to the case of a market where several traders simultaneously hedge against risk. We model this situation as a non-cooperative Nash game, where the trades of one player affect the cost functions of all the other players because of market frictions and limited liquidity. We prove that if the market is viable and thus free of price manipulation, then a unique optimal strategy exists for any risk-averse player. We then show how the market equilibrium can be derived. In the last part of this thesis, we consider the problem of solving generalised variational inequality problems, which is a classical way of reformulating generalised Nash games in order to find subsets of solutions. We introduce a new numerical method that combines the dual extrapolation method with the cutting plane method. This method is shown to be efficient by means of careful numerical experiments.

In my master thesis, I have investigated different methods to tackle a problem that power system designers, such as Tractebel, face every day: finding the optimal expansion planning of a power system. We present a primal heuristic able to find not trivial expansion plans for a variety of power systems, in a reasonable amount of time, and with good convergence.