Technical Note—Options Portfolio Selection

2020 ◽  
Vol 68 (3) ◽  
pp. 733-740
Author(s):  
Paolo Guasoni ◽  
Eberhard Mayerhofer
Keyword(s):  
Unique Solution ◽  
Integral Equations ◽  
Portfolio Selection ◽  
Matrix Equation ◽  
Technical Note ◽  
Linear Matrix ◽  
Linear Matrix Equation ◽  
System Of Integral Equations ◽  
Discrete Representations ◽  
Hedging Demand

We develop a new method to optimize portfolios of options in a market where European calls and puts are available with many exercise prices for each of several potentially correlated underlying assets. We identify the combination of asset-specific option payoffs that maximizes the Sharpe ratio of the overall portfolio: such payoffs form the unique solution to a system of integral equations, which reduces to a linear matrix equation under discrete representations of the underlying probabilities. Even when risk-neutral volatilities are all higher than physical volatilities, it can be optimal to sell options on some assets while buying options on other assets, for which the positive hedging demand outweighs negative demand stemming from asset-specific returns.

scholarly journals A Note on the⊤-Stein Matrix Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Chun-Yueh Chiang
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Linear Matrix ◽  
Comprehensive Treatment ◽  
Backward Error ◽  
Linear Matrix Equation ◽  
Perturbation Bounds ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Linear Matrix Equations

This note is concerned with the linear matrix equationX=AX⊤B + C, where the operator(·)⊤denotes the transpose (⊤) of a matrix. The first part of this paper sets forth the necessary and sufficient conditions for the unique solvability of the solutionX. The second part of this paper aims to provide a comprehensive treatment of the relationship between the theory of the generalized eigenvalue problem and the theory of the linear matrix equation. The final part of this paper starts with a brief review of numerical methods for solving the linear matrix equation. In relation to the computed methods, knowledge of the residual is discussed. An expression related to the backward error of an approximate solution is obtained; it shows that a small backward error implies a small residual. Just like the discussion of linear matrix equations, perturbation bounds for solving the linear matrix equation are also proposed in this work.Erratum to “A Note on the⊤-Stein Matrix Equation”

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