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

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

Real options

Given a forecasted MetaLog distribution for the price at expiration, 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.

Financial options

In an efficient market, financial option valuation incorporates a no-arbitrage constraint.

Consider a European call option on a stock.

No-arbitrage implies:

[math]\displaystyle{ S_0 = e^{-r T} E[x] }[/math]

where

• [math]\displaystyle{ S_0 }[/math] is the current stock price.
• [math]\displaystyle{ r }[/math] is the risk-free interest rate.
• [math]\displaystyle{ T }[/math] is the time to maturity.

The example could:

• Each option bid/ask price on the market determines one quantile value.
• The no-arbitrage gives us a mean-constraint, i.e., [math]\displaystyle{ E[x] = S_0 e^{r T} }[/math].

Given:

• [math]\displaystyle{ C(S, K_i, T) }[/math] for [math]\displaystyle{ i = 1..n }[/math] strike prices.

How does each option imply the quantile?

Another approach: Breeden–Litzenberger

The Breeden–Litzenberger formula says that the second derivative of price wrt strike price equals the risk-neutral density.

[math]\displaystyle{ {{\partial^2 C(K,T)}\over{\partial K^2}} = e^{r T} f(K) }[/math]

• [math]\displaystyle{ f(x) }[/math]: Risk-neutral density function at expiration.
• [math]\displaystyle{ C(K,T) }[/math]: Call price

Given the value of an option (say from its current bid/ask), we have K, C. We actually need some way to estimate the second derivative. This could likely be done by using 3 consecutive strike prices at the same expiration.

The formula is obtained by just taking the of both sides twice of:

[math]\displaystyle{ C(S,K) = e^{-rt} \int_K^\infty (x-K) f(x) dx }[/math]

It doesn't matter whether it has a no-arbitrage or not, nor on what model is being used to value the options. It should hold regardless since [math]\displaystyle{ f(x) }[/math] is the price density. Thus, it can also be used for exotic options. You could obtain [math]\displaystyle{ f(x) }[/math], a MetaLog, from open prices, and then use an expectation:

[math]\displaystyle{ Value = e^{-r T} E_f[ h(x) ] }[/math]

for an exotic payoff function [math]\displaystyle{ h(x) }[/math]. When these can be expressed as a mixture of partial expectations, we should be able to use our closed-form formula.