LChrisman/Possible MetaLog application examples

This is a brainstorming ideas page. A place to list ideas for possible examples (either practical or didactic) of the MetaLog distribution. In particular, I'm looking for examples that leverage the results from https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5280979 Baucells, Chrisman, Keelin & Xu (2025)].

Newsboy problem

• Q = quantity to order (Decision)
• D = Demand (a MetaLog distribution)
• Q-D = also a MetaLog for any Q.

[math]\displaystyle{ Profit = p Min( D,Q ) - c Q = p [ Min(D-Q,0) + Q ] - c Q = p Min(D-Q,0) + (p-c) Q }[/math]

[math]\displaystyle{ E[Profit] = p E[ D-Q | D-Q \lt 0 ] + (p-c) Q }[/math]

Closed form for any Q.

Finding optimal Q? The closed form for profit doesn't eliminate the need for optimization.

Solve [math]\displaystyle{ P(D \le Q) = {{p-c}\over p} }[/math] to get optimal Q:

• [math]\displaystyle{ Q^* = M_D\left({{p-c}\over p}\right) }[/math]

This is closed-form. (Wikipedia shows this solution).

This example doesn't require any of our new results, but it does show off an advantage of MetaLog.

Options pricing

• X = market price at expiration. (A MetaLog)
• K = strike price

The partial expectation: [math]\displaystyle{ E[ X-K | X\gt K] }[/math] gives the option's value at expiration. Uses our closed-form for partial expectation.

Note paper: Valentyn Khokhlov (2021), "Conditional Value at Risk and Partial Moments for the Metalog Distributions", arXiv 2102.10999.