Geometric Criterion for a Robust Condition of No Sure Arbitrage with Unlimited Profit
- PDF / 556,947 Bytes
- 5 Pages / 612 x 792 pts (letter) Page_size
- 41 Downloads / 181 Views
etric Criterion for a Robust Condition of No Sure Arbitrage with Unlimited Profit S. N. Smirnov* Department of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991, Russia Received August 30, 2019; in final form, December 27, 2019; accepted December 27, 2019
Abstract—A model is presented of the financial market with a discrete-time uncertain deterministic evolution of prices in which asset prices evolve under uncertainty described using a priori information on possible price increments; i.e., it is assumed that they lie in given compact sets that depend on the prehistory of prices. Trading constraints depending on the history of prices are assumed to be convex, concern only risky assets, and allow all funds to be invested in a risk-free asset. A new geometric criterion is obtained for a robust condition (i.e., the condition ensuring the structural stability of the model) under which there is no guaranteed arbitrage with unlimited profit. Keywords: deterministic market dynamics, uncertainty of price movements, trading constraints, arbitrage with unlimited profit, structural stability of the model. DOI: 10.3103/S0278641920020077
1. INTRODUCTION This work continues a series of publications [1–3] devoted to the guaranteed deterministic approach to superhedging with trading constraints. A mathematical model of the financial market with an uncertain deterministic evolution of prices and discrete time was described in [1]. The uncertainty of asset prices in this model is described by a priori information on possible price increments; i.e., it is assumed that increments of discounted prices lie in given compacts that depend on the prehistory of prices. It is shown how the guaranteed deterministic approach can be used to solve the problem of superhedging (super-replication) in the framework of the financial market model with trading constraints and an uncertain deterministic evolution of prices. The solution to this problem requires determining the minimum level of funds for an appropriate hedging strategy to guarantee the coverage of the contingent claim on a sold option, the payouts on which depend on the price history. Under rather general assumptions, the corresponding objective function satisfies the Bellman–Isaacs equations. Let us consider a financial market with n risky assets and one risk-free asset. We assume that the price of the risk-free asset is constant over time and equal to unity (after discounting; other discounted prices will also be dimensionless). We denote by Xt the vector of discount prices at current time t, where the ith component of this vector is the discounted unit price of the ith risk asset; ΔXt = Xt − Xt−1 represents the increments of discounted prices (taken in the direction of the past). Vector h describes the sizes of positions taken in risky assets; i.e., the ith component of this vector represents the (signed) number of bought or sold units of the ith asset. The main idea of the proposed approach is to formalize the uncertainty in price dynamics by assuming that incre
Data Loading...
