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), full-insurance for quantile-based preferences (q*=1), and over-insurance for mean-variance utility (q*>1). I calculate the shape of the objective function and show the optimum is well defined for mean-variance utility and quantile-based preferences in an asymptotic setting. Using resampling, I show the optimal values are stable for the mean-variance utility and the quantile-based preferences but not the log utility. For the optimal value with mean-variance 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.
