Optimizing a position typically requires two things:

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

For certain cases in finance, we have closed-form solutions for optimal positions. For instance:

• 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 am searching for a closed-form solution for option structuring, given some assumptions about the price dynamics of the underlying asset.

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

For instance, the following solutions emerge from this method:

• 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

These implications seem to be in line with general views about choosing option structures, but the main contribution of this framework is that you can exactly calibrate option structures to any type of view using a closed-form solution. That is very helpful indeed. I find these papers very useful. Contrary to my expectation, I couldn't find many references to these papers.

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