The Maximal and Minimal Distributions of Wealth Processes in Black–Scholes Markets

Shuhui Liu ()
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Shuhui Liu: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

Mathematics, 2024, vol. 12, issue 10, 1-18

Abstract: The Black–Scholes formula is an important formula for pricing a contingent claim in complete financial markets. This formula can be obtained under the assumption that the investor’s strategy is carried out according to a self-financing criterion; hence, there arise a set of self-financing portfolios corresponding to different contingent claims. The natural questions are: If an investor invests according to self-financing portfolios in the financial market, what are the maximal and minimal distributions of the investor’s wealth on some specific interval at the terminal time? Furthermore, if such distributions exist, how can the corresponding optimal portfolios be constructed? The present study applies the theory of backward stochastic differential equations in order to obtain an affirmative answer to the above questions. That is, the explicit formulations for the maximal and minimal distributions of wealth when adopting self-financing strategies would be derived, and the corresponding optimal (self-financing) portfolios would be constructed. Furthermore, this would verify the benefits of diversified portfolios in financial markets: that is, do not put all your eggs in the same basket.

Keywords: self-financing portfolio; optimal investment; maximal distribution; backward stochastic differential equation; diversified portfolio (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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