A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options

Sergey Smirnov

doi:10.3390/math7121246

Abstract

This paper considers super-replication in a guaranteed deterministic problem setting with discrete time. The aim of hedging a contingent claim is to ensure the coverage of possible payoffs under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts that depend on the history of prices. The increments of the price at each moment in time must lie in the corresponding compacts. The absence of transaction costs is assumed. The game–theoretic interpretation of pricing American options implies that the corresponding Bellman–Isaacs equations hold for both pure and mixed strategies. In the present paper, we study some properties of the least favorable (for the “hedger”) mixed strategies of the “market” and of their supports in the special case of convex payoff functions.

Highlights

The present article continues the series of publications by the author [1,2,3,4,5,6,7,8]
This paper considers super-replication in a guaranteed deterministic problem setting with discrete time
We note that the perception of the risk-neutral valuation of financial instruments is of fundamental importance for the correct understanding of the phenomenon of “bubbles” in financial markets) under the assumption of no trading constraints; we find this interpretation to be quite important from an economic point of view

Summary

Introduction

The present article continues the series of publications by the author [1,2,3,4,5,6,7,8]. Let v∗t (·) denote the infimum of portfolio value at time t that guarantees coverage of current and future possible claims on the American option given the price history and appropriately chosen hedging strategy. Reference [8] obtains conditions for the upper semicontinuity of multi-valued mappings, which associate a class of optimal mixed strategies of the “market” to an asset price trajectories. We study the superhedging problem under the framework of the guaranteed deterministic approach for a case of convex payoff functions and a market model where compact sets of possible price increments that describe the uncertainty of price movements are convex. In the framework of the model with convexity assumptions, we obtain some delicate properties for optimal mixed strategies of the “market” and clarify the behavior of their supports (the general case was considered in Reference [8]). We provide a two-dimensional (We consider a rainbow option whose payoff depends on two underlying risky assets.) example that brings to light some particularities of this behavior

Convex Payoff Functions

Conclusions