Derivatives Pricing in Non-Arbitrage Market

TL;DR

This paper develops a comprehensive framework for constructing martingale measures in non-arbitrage markets, introduces new inequalities and proofs, and derives formulas for option pricing using nonlinear processes like ARCH and GARCH.

Contribution

It presents a novel method for constructing martingale measures, provides new inequalities and a simplified proof of optional decomposition, and derives option pricing formulas based on nonlinear stochastic processes.

Findings

Constructed a family of martingale measures for risky asset evolution.

Derived conditions for the existence of equivalent martingale measures.

Obtained formulas for fair European option prices using ARCH and GARCH models.

Abstract

The general method is proposed for constructing a family of martingale measures for a wide class of evolution of risky assets. The sufficient conditions are formulated for the evolution of risky assets under which the family of equivalent martingale measures to the original measure is a non-empty set. The set of martingale measures is constructed from a set of strictly nonneg ative random variables, satisfying certain conditions. The inequalities are obtained for the non-negative random variables satisfying certain conditions. Using these inequalities, a new simple proof of optional decomposition theorem for the nonnegative super-martingale is proposed. The family of spot measures is introduced and the representation is found for them. The conditions are found under which each martingale measure is an integral over the set of spot measures. On the basis of nonlinear processes such as ARCH and GARCH, the parametric family of random processes is introduced for which the interval of non-arbitrage prices are found. The formula is obtained for the fair price of the contract with option of European type for the considered parametric processes. The parameters of the introduced random processes are estimated and the estimate is found at which the fair price of contract with option is the least.