Choosing put option parameters based on quantiles from the distribution of portfolio value

This paper explores how a put option changes the probability distribution of portfolio value. The paper extends the model introduced in Bell (2014) by allowing both the quantity and strike price to vary. I use the 5% quantile from the portfolio distribution to measure riskiness and compare different put options. I report a so-called ‘quantile surface’ that shows the quantile across different combinations of quantity and strike price. I find that it is possible to maximize the quantile by purchasing a put with quantity equal to one and strike deep in the money; however, the distribution with such a put option collapses to a single point because the option hedges all variation in stock price. This result is analogous to full-insurance in insurance economics, but has practical limitations. The quantile surface also shows that certain put options will decrease the quantile, which is equivalent to increasing the riskiness of the portfolio, and leads me to ask: what return will an investor receive in return for bearing that extra risk? I find that one such put option will cause the distribution to have an asymmetric shape with positive skewness, which is interesting to some speculators.