Peter N, Bell (2014): Optimal Use of Put Options in a Stock Portfolio.
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Abstract
In this paper I consider a portfolio optimization problem where an agent holds an endowment of stock and is allowed to buy some quantity of a put option on the stock. This basic question (how much insurance to buy?) has been addressed in insurance economics through the literature on rational insurance purchasing. However, in contrast to the rational purchasing literature that uses exact algebraic analysis with a binomial probability model of portfolio value, I use numerical techniques to explore this problem. Numerical techniques allow me to approximate continuous probability distributions for key variables. Using large sample, asymptotic analysis I identify the optimal quantity of put options for three types of preferences over the distribution of portfolio value. The location of the optimal quantity varies across preferences and provides examples of important concepts from the rational purchasing literature: coinsurance for log utility (q*<1), fullinsurance for quantilebased preferences (q*=1), and overinsurance for meanvariance utility (q*>1). I calculate the shape of the objective function and show the optimum is well defined for meanvariance utility and quantilebased preferences in an asymptotic setting. Using resampling, I show the optimal values are stable for the meanvariance utility and the quantilebased preferences but not the log utility. For the optimal value with meanvariance utility I show that the put option affects the probability distribution of portfolio value in an asymmetric way, which confirms that it is important to analyze the optimal use of derivatives in a continuous setting with numerical techniques.
Item Type:  MPRA Paper 

Original Title:  Optimal Use of Put Options in a Stock Portfolio 
Language:  English 
Keywords:  Portfolio; optimization; financial derivative; put option; quantity; expected utility; numerical analysis 
Subjects:  C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C15  Statistical Simulation Methods: General C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63  Computational Techniques ; Simulation Modeling G  Financial Economics > G1  General Financial Markets > G11  Portfolio Choice ; Investment Decisions G  Financial Economics > G2  Financial Institutions and Services > G22  Insurance ; Insurance Companies ; Actuarial Studies 
Item ID:  54394 
Depositing User:  Peter N Bell 
Date Deposited:  13 Mar 2014 15:23 
Last Modified:  07 Oct 2019 20:23 
References:  Briys, E.P., & Loubergé, H. (1985). On the Theory of Rational Insurance Purchasing: A Note. The Journal of Finance, 40(2), 577581. Mossin, J. (1968). Aspects of Rational Insurance Purchasing. Journal of Political Economy, 76(4), 553568. Peters, O. (2011). The time resolution of the St. Petersburg paradox. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 369(1956). 49134931. Razin, A. (1976). Rational Insurance Purchasing. The Journal of Finance, 31(1), 133137. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/54394 
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