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# Developments of Method of Market Completions in Mathematical Finance

• Author / Creator
Vasilev, Ilia
• In present thesis I focus on development of method of market completions and its applications to various problems in pricing and hedging of contingent claims.

Since theory of mathematical finance is well developed  on complete markets, and corresponding solutions are well understood, method of market completions emerges as a naturall idea for applying knowledge available for complete markets to the case of incomplete one.

Key approach of proposed method is to introduce family of possible “completed” versions of initially incomplete market, parametrized by the set of special auxiliary assets. This manipulation leads to multiple subproblems, that could be solved by the means of known complete market techniques. Further, having corresponding family of solutions, I demonstrate how one could come up with criteria to choose optimal solution for initial market which would not depend on auxiliary assets.

I start with discussion of reasons for market incompleteness and introduce market completions for parametrization of completed versions of the market.

Then, I demonstrate how proposed method can be applied for fundamental problems of utility maximization, including not-necessarily concave utility function and pricing of contingent claims.

Then, I move to another important group of problems in the field of mathematical finance -- hedging of contingent claims. In the modern risk management industry, however it is more common to choose partial hedging, since it allows for more flexibility and money savings. I will start with discussing application of method of market completions for fundamental problems of quantile and effective hedging and then for modern risk-measures approach. I will also provide numerical examples for solutions on incomplete market.

It is known that the key element of this problem is a risk measure chosen for assessment of risks. Two of the most widely used risk measures in the industry nowadays are Value-at-Risk (VaR) and Expected Shortfall (CVaR). However, it has been demonstrated recently that both of these measures could be incorporated into one two-parametric risk measure called Range Value-at-Risk (RVaR). I will focus on demonstration that partial hedging problem with respect to both CVaR and RVaR in incomplete market could be approached with the help of method of market completions through the Utility Maximization task embedded into RVaR optimization problem.

Conclusions and further research directions in exploring the ideas of Method of market completions are in the last chapter.

• Subjects / Keywords
Spring 2023
• Type of Item
Thesis
• Degree
Doctor of Philosophy
• DOI
https://doi.org/10.7939/r3-3d7z-p983