VISA visitor Prof Kabanov, November 2017
7 November 2017, 10am-1pm, Short Course, E4A. Level 5, room 523, Macquarie University.
Title: Markets with Transaction Costs: Mathematical Theory
Abstract: Classical Arbitrage Theory for frictionless financial markets relates economically meaningful property of absence of arbitrage with the fundamental probabilistic concept of equivalent martingale measure. Densities processes of equivalent martingale measures plays a role of stochastic deflators. To compare the present values of assets with their future values one needs to use not the prices but the prices multiplied by stochastic deflators. The theory of markets with proportional transaction costs treats portfolios as vectors of assets without assigning to them a scalar - its monetary value. It happens that in the case of proportional transaction costs the fundamental concept is a consistent price system, a martingale evolving in the dual to the solvency cones (in physical units). In the absence of friction all such martingales can be obtained by multiplying prices by stochastic deflators. For markets with transaction costs there are several possible formalizations of absence of arbitrage and the available criteria involve consistent price systems. Surprisingly, the passage from the model with a finite number of states of the nature to the general case goes not so smoothly as in the classical theory. Several examples will be discussed in the lecture course. From mathematical point of view, the theory for market with transaction costs is a vector analog of the classical theory. It is a blend of finite dimensional geometry, geometric functional analysis and stochastic calculus. On the other hand, it feels the gap between mathematical finance and mathematical economics showing how these two disciplines are related.
9 November 2017, 12pm-2pm, Public Lecture. Macquarie City Campus, Level 24, 123 Pitt Street Sydney NSW 2000
Title: Clearing in Financial Networks
Abstract: Clearing of financial system, i.e. of a network of interconnecting banks, is a procedure of simultaneous repaying debts to reduce their total volume. The vector whose components are repayments of each bank is called clearing vector. In simple models considered by Eisenberg and Noe (2001) and, independently, by Suzuki (2002), it was shown that the clearing to the minimal value of debts accordingly to natural rules can be formulated as a fixpoint problem. The existence of their solutions, i.e. of clearing vectors, is rather straightforward and can be obtained by a direct reference to the Knaster–Tarski or Brouwer theorems. The uniqueness of clearing vectors is a more delicate problem which was solved by Eisenberg and Noe using a graph structure of the financial network. We discuss the modern state of art of the theory and, in particular, algorithmic aspects of solving clearing equations in relations with those arising in the theory of optimal stopping.
13 November, 2017 10:30am-12pm, CFR Seminar in Macquarie University, room 523
Title: Ruin probabilities with investments in a risky asset with the price given by a geometric Lévy process
Abstract: We consider a model describing the evolution of capital of a venture company selling innovations and investing its reserve into a risky asset with the price given by a geometric Lévy process. We find the exact asymptotic of the ruin probabilities. Under some natural conditions it decays as a power function. The rate of decay is a positive root of equation determined by characteristics of the price process. When the price follows a gBm the results are reduced to those of our previous works where we used the method of ODEs assuming exponentially distributed jumps. Our proofs are based on the theory of distributional equations, in particular, on a recent result by Guivarc'h and Le Page.
7 December 2017, Lecture for the UNSW-Macquarie risk workshop in UNSW 7-8 December.
Title: Hedging in Markets with Small Transaction Costs
Abstract: We discuss the concepts of approximate replication and super-replication inthe context of markets with friction. We consider a class of models where the transaction costs coefficients depend on the number n of transactions, decreasing to zero as n^(-1/2), and show that the impact of transaction costs on the super-replication price has a similar effect as a proportional increase of volatility.