7 research outputs found
Optimal discretization of hedging strategies with directional views
We consider the hedging error of a derivative due to discrete trading in the
presence of a drift in the dynamics of the underlying asset. We suppose that
the trader wishes to find rebalancing times for the hedging portfolio which
enable him to keep the discretization error small while taking advantage of
market trends. Assuming that the portfolio is readjusted at high frequency, we
introduce an asymptotic framework in order to derive optimal discretization
strategies. More precisely, we formulate the optimization problem in terms of
an asymptotic expectation-error criterion. In this setting, the optimal
rebalancing times are given by the hitting times of two barriers whose values
can be obtained by solving a linear-quadratic optimal control problem. In
specific contexts such as in the Black-Scholes model, explicit expressions for
the optimal rebalancing times can be derived
MeĢthodes asymptotiques en controĢle stochastique et applications aĢ la finance
In this thesis, we study several mathematical finance problems related to the presence of market imperfections. Our main approach for solving them is to establish a relevant asymptotic framework in which explicit approximate solutions can be obtained for the associated control problems.In the first part of this thesis, we are interested in the pricing and hedging of European options. We first consider the question of determining the optimal rebalancing dates for a replicating portfolio in the presence of a drift in the underlying dynamics. We show that in this situation, it is possible to generate positive returns while hedging the option and describe a rebalancing strategy which is asymptotically optimal for a mean-variance type criterion. Then we propose an asymptotic framework for options risk management under proportional transaction costs. Inspired by Lelandās approach, we develop an alternative way to build hedging portfolios enabling us to minimize hedging errors.The second part of this manuscript is devoted to the issue of tracking a stochastic target. The agent aims at staying close to the target while minimizing tracking efforts. In a small costs asymptotics, we establish a lower bound for the value function associated to this optimization problem. This bound is interpreted in term of ergodic control of Brownian motion. We also provide numerous examples for which the lower bound is explicit and attained by a strategy that we describe.In the last part of this thesis, we focus on the problem of consumption-investment with capital gains taxes. We first obtain an asymptotic expansion for the associated value function that we interpret in a probabilistic way. Then, in the case of a market with regime-switching and for an investor with recursive utility of Epstein-Zin type, we solve the problem explicitly by providing a closed-form consumption-investment strategy. Finally, we study the joint impact of transaction costs and capital gains taxes. We provide a system of corrector equations which enables us to unify the results in [Homogenization and asymptotics for small transaction costs, M.Soner and N. Touzi, 2013] and [Asymptotics for Merton problem with small capital gain tax and interest rate, X. Chen and M. Dai, 2013].Dans cette theĢse, nous eĢtudions plusieurs probleĢmes de matheĢmatiques financieĢres lieĢs aĢ la preĢsence dāimperfections sur les marcheĢs. Notre approche principale pour leur reĢsolution est lāutilisation dāun cadre asymptotique pertinent dans lequel nous parvenons aĢ obtenir des solutions approcheĢes explicites pour les probleĢmes de controĢle associeĢs.Dans la premieĢre partie de cette theĢse, nous nous inteĢressons aĢ lāeĢvaluation et la couverture des options europeĢennes. Nous consideĢrons tout dāabord la probleĢmatique de lāoptimisation des dates de rebalancement dāune couverture aĢ temps discret en preĢsence dāune tendance dans la dynamique du sous-jacent. Nous montrons que dans cette situation, il est possible de geĢneĢrer un rendement positif tout en couvrant lāoption et nous deĢcrivons une strateĢgie de rebalancement asymptotiquement optimale pour un criteĢre de type moyenne-variance. Ensuite, nous proposons un cadre asymptotique pour la gestion des options europeĢennes en preĢsence de couĢts de transaction proportionnels. En sāinspirant des travaux de Leland, nous deĢveloppons une meĢthode alternative de construction de portefeuilles de reĢplication permettant de minimiser les erreurs de couverture.La seconde partie de ce manuscrit est deĢdieĢe aĢ la question du suivi dāune cible stochastique. Lāobjectif de lāagent est de rester proche de cette cible tout en minimisant le couĢt de suivi. Dans une asymptotique de couĢts petits, nous deĢmontrons lāexistence dāune borne infeĢrieure pour la fonction valeur associeĢe aĢ ce probleĢme dāoptimisation. Cette borne est interpreĢteĢe en terme du controĢle ergodique du mouvement brownien. Nous fournissons eĢgalement de nombreux exemples pour lesquels la borne infeĢrieure est explicite et atteinte par une strateĢgie que nous deĢcrivons.Dans la dernieĢre partie de cette theĢse, nous consideĢrons le probleĢme de consommation et inves- tissement en preĢsence de taxes sur le rendement des capitaux. Nous obtenons tout dāabord un deĢveloppement asymptotique de la fonction valeur associeĢe que nous interpreĢtons de manieĢre probabiliste. Puis, dans le cas dāun marcheĢ avec changements de reĢgime et pour un investisseur dont lāutiliteĢ est du type Epstein-Zin, nous reĢsolvons explicitement le probleĢme en deĢcrivant une strateĢgie de consommation-investissement optimale. Enfin, nous eĢtudions lāimpact joint de couĢts de transaction et de taxes sur le rendement des capitaux. Nous eĢtablissons dans ce cadre un systeĢme dāeĢquations avec termes correcteurs permettant dāunifier les reĢsultats de [Homogenization and asymptotics for small transaction costs, M.Soner and N.Touzi, 2013] et [Asymptotics for Merton problem with small capital gain tax and interest rate, X.Chen and M.Dai, 2013]
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Hedging with Temporary Price Impact
Ā© 2016, Springer-Verlag Berlin Heidelberg. We consider the problem of hedging a European contingent claim in a Bachelier model with temporary price impact as proposed byĀ Almgren and Chriss (J Risk 3:5ā39, 2001). Following the approach ofĀ Rogers and Singh (Math Financ 20:597ā615, 2010) andĀ Naujokat and Westray (Math Financ Econ 4(4):299ā335, 2011), the hedging problem can be regarded as a cost optimal tracking problem of the frictionless hedging strategy. We solve this problem explicitly for general predictable target hedging strategies. It turns out that, rather than towards the current target position, the optimal policy trades towards a weighted average of expected future target positions. This generalizes an observation ofĀ GĆ¢rleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, 2013b) from their homogenous Markovian optimal investment problem to a general hedging problem. Our findings complement a number of previous studies in the literature on optimal strategies in illiquid markets as, e.g., GĆ¢rleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, 2013b), Naujokat and Westray (Math Financ Econ 4(4):299ā335, 2011), Rogers and Singh (Math Financ 20:597ā615, 2010), Almgren and Li (Option hedging with smooth market impact. Preprint, 2015), Moreau etĀ al. (Math Financ. doi:10.1111/mafi.12098, 2015), Kallsen and Muhle-Karbe (High-resilience limits of block-shaped order books. Preprint, 2014), Guasoni and Weber (Mathematical Financ. doi:10.1111/mafi.12099, 2015a; Nonlinear price impact and portfolio choice. Preprint, 2015b), where the frictionless hedging strategy is confined to diffusions. The consideration of general predictable reference strategies is made possible by the use of a convex analysis approach instead of the more common dynamic programming methods