- An assumption about how prices will behave in the future
- An objective function to maximize/minimize

- Markowitz portfolio optimization states that if prices follow a multivariate normal distribution, and the objective is to maximize expected log utility, then the optimal weights for the assets in our portfolio are given by w=μ/(λΣ)
- In Merton's portfolio problem, the risky asset follows a geometric Brownian motion and the investor is maximizing CRRA utility. The closed-form solution turns out to be w=(μ−r)/(λσ^2)

I find a very interesting series of papers on this subject, written by the structuring head of Citibank, Andrei N. Soklakov: Link

As a summary, he concludes with the following closed-form solution:

- If the market implied (risk-neutral) density is m, and as an investor you believe in a different density, say b
- If you are maximizing logarithmic growth rate (a.k.a Kelly optimizing),
- Then the optimal option structure, i.e. whether to buy a call option, a risk reversal, or a butterfly option, can be found by simply looking at the ratio of two densities b/m

- As an investor, if you believe that the future returns will have a higher expected value (mean) than what market-implied risk-neutral density implies, and have no view about volatility, then taking a long position in futures turns out to be optimal.
- If you have a view that volatility should be higher than implied volatility, then a long straddle is optimal.
- Similar conclusions can be derived for more complex structures like risk reversals, butterfly, call spread, ratio spread, etc

My question is, are there any other frameworks/papers that give you closed-form solutions for option structuring given an investor's own subjective views?

Thanks