2020/09/14 by Diego Zabaljauregui, Zabaljauregui, Diego
Decision Sciences · Economics, Econometrics and Finance · Energy · #Economic theories and models #Energy, Environment, and Transportation Policies #FOS: Economics and business #FOS: Mathematics #General Economics (econ.GN) #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Risk and Portfolio Optimization #Stochastic processes and financial applications #Trading and Market Microstructure (q-fin.TR)
paper · pdf · doi:10.48550/arxiv.2009.06521
openalex publication_date 2020/09/14 · openalex created_date 2021/03/29 · openalex updated_date 2026/07/28
The topics treated in this thesis are inherently two-fold. The first part\nconsiders the problem of a market maker optimally setting bid/ask quotes over a\nfinite time horizon, to maximize her expected utility. The intensities of the\norders she receives depend not only on the spreads she quotes, but also on\nunobservable factors modelled by a hidden Markov chain. This stochastic control\nproblem under partial information is solved by means of stochastic filtering,\ncontrol and PDMPs theory. The value function is characterized as the unique\ncontinuous viscosity solution of its dynamic programming equation and\nnumerically compared with its full information counterpart. The optimal full\ninformation spreads are shown to be biased when the exact market regime is\nunknown, as the market maker needs to adjust for additional regime uncertainty\nin terms of PnL sensitivity and observable order flow volatility.\n The second part deals with numerically solving nonzero-sum stochastic impulse\ncontrol games. These offer a realistic and far-reaching modelling framework,\nbut the difficulty in solving such problems has hindered their proliferation. A\npolicy-iteration-type solver is proposed to solve an underlying system of\nquasi-variational inequalities, and it is validated numerically with reassuring\nresults.\n Eventually, the focus is put on games with a symmetric structure and an\nimproved algorithm is put forward. A rigorous convergence analysis is\nundertaken with natural assumptions on the players strategies, which admit\ngraph-theoretic interpretations in the context of weakly chained diagonally\ndominant matrices. The algorithm is used to compute with high precision\nequilibrium payoffs and Nash equilibria of otherwise too challenging problems,\nand even some for which results go beyond the scope of the currently available\ntheory.\n