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?
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.
The problem is that [math]\displaystyle{ {{\partial^2 C(K,T)}\over{\partial K^2}} }[/math] has to be estimated, and if estimated from [math]\displaystyle{ C(K_i,T) }[/math] points, is going to be really noisy, and you'll burn the two most extreme strike price points where the number of strike prices is already low. In practice, people fit a smooth curve to the [math]\displaystyle{ C(K_i,T) }[/math] points and use that. It would be nice if one fit was used for that and obtaining the MetaLog coefficients, but that doesn't look possible.
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