Optimal Use of Put Options in a Stock Portfolio

I analyze 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. My model rephrases a fundamental question from insurance economics: how much coverage should a risk averse agent buy? Classic studies of rational insurance purchasing use exact algebraic analysis with a binomial probability model of portfolio value to explore this problem. In contrast, I use numerical techniques to approximate the 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 with preferences and provides examples of important concepts from the rational insurance purchasing literature: coinsurance for log utility (q*<1), full-insurance for quantile-based preferences (q*=1), and over-insurance for mean-variance utility (q*>1). I use resampling analysis to show that the optimal quantity is well defined for mean-variance and quantile-based preferences, but the optimal quantity for log utility is not stable. Although my analysis corroborates the classic result that coinsurance is optimal for log utility, I show that the specific amount of coinsurance is not well defined. In addition, the optimal quantity for mean-variance utility in my model is not allowed in a classic insurance model. By matching and extending the set of results for basic rational insurance purchasing, my research demonstrates the value of using numerical techniques to analyze the optimal use of financial derivatives in a continuous setting.