Υποδείγματα Ισορροπίας συνεχούς χρόνου σε αγορές με τριβές

Στεφανάκης, Κωνσταντίνος; Stefanakis, Konstantinos

dc.contributor.author	Στεφανάκης, Κωνσταντίνος	el
dc.contributor.author	Stefanakis, Konstantinos	en
dc.date.accessioned	2023-03-01T08:05:39Z
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/57171
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.24869
dc.description	Εθνικό Μετσόβιο Πολυτεχνείο--Μεταπτυχιακή Εργασία. Διεπιστημονικό-Διατμηματικό Πρόγραμμα Μεταπτυχιακών Σπουδών (Δ.Π.Μ.Σ.) “Μαθηματική Προτυποποίηση σε Σύγχρονες Τεχνολογίες και στα Χρηματοοικονομικά”	el
dc.rights	Default License
dc.subject	Stochastic analysis	en
dc.subject	Mean variance optimization	en
dc.subject	Price impact	en
dc.subject	Continuous time equilibria	en
dc.subject	Financial mathematics	en
dc.title	Υποδείγματα Ισορροπίας συνεχούς χρόνου σε αγορές με τριβές	el
dc.title	Continuous-time Equilibrium Returns in Markets with Frictions	en
heal.type	masterThesis
heal.classification	Χρηματοοικονομικά Μαθηματικά	el
heal.dateAvailable	2024-02-29T22:00:00Z
heal.language	en
heal.access	embargo
heal.recordProvider	ntua	el
heal.publicationDate	2022-10-31
heal.abstract	The concept of optimality with regards to investment selection is an extensively studied “problem” in Mathematical Finance. The substantial variety of approaches aiming to answer the question: “ What constitutes an optimal portfolio? ” can be traced back, at least partially, to the following: What are we trying to optimize? In other words, what would be a realistic representation of an intestor’s criterion. For this thesis, this question is the initial setting. For this, we adapt a stochastic version of the Mean-variance portfolio selection (MVPS) criterion. More precisely, we consider that the investors participating in the market invest in a non self-financing strategy aiming for maximal mean and minimal variance 1 . Moreover, after having determined the optimal strategies—according to the investors’ preferences—we study their influence on the market at the equilibrium . More precisely, the main goals of this thesis are: (I) Study the MVPS criterion when the drift of the market is derived endogenously. (II) Analyze the model and the results of [Bou+18]. (III) Introduce the price impact model of [Ant17] in continuous time without frictions. (IV) Generalize the notion of impact for a market with frictions. The second chapter of the thesis is introductory and it aims to define some mathematical concepts which are frequently used. This chapter is partially supplemented by Appendix A, which we use to further analyze and/or clarify some concepts present in the main text. To this end, throughout 1 Alternatively, as expressed in [GP16], we could say that the investors have mean-variance preferences over the change in their wealth over time. 1 the thesis, there can also be found some special blocks of text called “Closer looks”, which focus on delving somewhat deeper in a few relevant notions without disrupting the main flow of the text. In the third chapter and onwards, we work towards analyzing the main part of the thesis, that is defining and solving the portfolio optimization problem and exploring the influence of its solutions at the equilibrium, in various types of markets. Note that in Chapter 3 and 4, we essentially analyze and discuss the results of [Bou+18]. More precisely, we initially consider a financial market with 1 riskless and d risky assets, modelled by a diffusion process, where the diffusion coefficient is exogenously given, while its drift is derived endogenously by an equilibrium condition . There are N investors with heterogenous risk aversions that participate in the market and trade amongst themselves. The investors are endowed with a random wealth process consisting of two parts: their holdings on the financial market, which naturally depend on their strategy of choice, and a exogenously given random endowment which captures other sources of income, possibly correlated with the assets. The rest of the market’s participants, called noise traders, are assumed to follow strategies that are not derived through any specific optimization criterion. Having introduced the above, we assume that the trades of risky assets in the market do not incur transaction costs, i.e. a frictionless market , and define an appropriate objective function, representing the mean-variance preferences over the change of an investor’s wealth. Optimizing the aforementioned function over the space of potential strategies and for each investor independently gives us the optimal asset allocation in a frictionless market. Closing the third chapter, we concern ourselves with determining the drift coefficient of the process that drives the market. In fact, the equilibrium condition that is imposed in the market dictates that the sum of the optimal strategies of the investors matches the exogenous demand of the noise traders. Via this condition we derive endogenously the drift of the market, called equilibrium returns . In the fourth chapter, we introduce frictions in the market. We follow [Bou+18] and assume that each trade incurs a cost in the form of a transaction tax , which goes to an exogenous recipient (who does not participate in the market). To derive the respective optimal strategies in a market with frictions, we have to assume that the strategies of the investors are now absolutely continuous and incur transaction costs proportional to the square of their pointwise derivatives (that is, the 2 investors’ trading rates ). This yields a new objective function for investors’ optimal allocation, which in turn is characterized by a system of coupled but linear Forward-Backward SDEs. These equations can be solved explicitly in terms of matrix power series, leading to closed-form expressions for the liquidity premia between the equilibrium returns in a market with frictions and their frictionless counterpart. Interestingly enough, under the assumption of homogenous risk aversions and without the presence of any noise traders in the market, the frictional equilibrium returns revert to their frictionless form. In the fifth chapter we go back to the frictionless optimization problem presented in the third chapter, and we introduce a notion of price impact . Price impact can be defined as the effect that an investor has on the price of a risky asset as a result of her buying or selling it. In some sense, we could view the price of the assets as a function of an investor’s strategy. The aforementioned give us a natural way to model the concept of “impact”, through the equilibrium returns. Recall that this process drives the prices of the risky assets that exist in the market and is determined via the strategies of the investors by the equilibrium condition. Similar concepts are also studied in [Ant17] and [AK17]. Having derived a form for the price impact through the equilibrium returns, we have essentially created a new optimization problem (since the wealth of each investor depends on the assets’ return process). Its solution determines the respective optimal strategies in a frictionless market with price impact (called best-response ). When all N investors adapt the same best-response strategy, the market equilibrates at the induced fixed point, determining a Nash-equilibrium . We also note that under the assumptions of homogenous risk aversions and without the presence of any noise traders in the market, the Nash equilibrium reverts to the frictionless equilibrium returns of the third chapter. Lastly, in the sixth chapter we extend the notions already introduced during the fifth chapter, but in a market with frictions. Therein, for tractability, we consider that the investors have homogenous risk aversions. By this assumption, the solution to the new objective function in a market with frictions under the price impact of a single investor is characterized by a second order, linear, non- homogenous (random) ODE. We again note that the frictional equilibrium returns under the price impact of a single investor revert to their frictionless counterpart in both of the following cases: ( i ) 3 same risk aversions and no noise traders and ( ii ) the transaction costs, which come in the form of a transaction tax, go to zero (while the investors have homogenous risk aversions).	en
heal.advisorName	Ανθρωπέλος, Μιχαήλ	el
heal.committeeMemberName	Ανθρωπέλος, Μιχαήλ	el
heal.committeeMemberName	Παπανικολάου, Βασίλειος	el
heal.committeeMemberName	Τριανταφύλλου, Αθανάσιος	el
heal.academicPublisher	Εθνικό Μετσόβιο Πολυτεχνείο. Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών	el
heal.academicPublisherID	ntua
heal.numberOfPages	95 σ.	el
heal.fullTextAvailability	false
heal.fullTextAvailability	false