Think about how to evaluate model performance

[tyrneh]

How do we measure whether our implied pdf is accurate?

[tyrneh]

It's not possible to measure accuracy, because the true pdf is inherently unobservable. However, there are certain properties of the true pdf that we do know, so we can impose some constraints/rules on the implied pdf we generate.

To elaborate on the true pdf being inherently unobservable:

• in the present day, ex-ante to the strike date, the true distribution of the future asset price obviously cannot be known, because it's a future outcome
• using historical data so that we know both the ex-ante indicators and the future (relative to that point in time) realization of the asset price, the true distribution of the asset price still cannot be known. This is because we only observe one realization of the distribution - we would only know the true distribution if we can observe 1000 parallel universes and empirically plot all realizations to form the empirical true distribution*

To elaborate on the rules we can place on our implied pdf - a pdf has properties that must be true. We can place these rules on the implied pdf we generate

• f(x) >= 0, where the domain is {0<=x<=infinity}
• The total area under the pdf is always equal to one: ∫f(x)dx=1
• F(0) = 0: by definition of the CDF, prob that the asset price is less than or equal to 0 is 0

Footnote*:

• I guess we can actually kind of do this: using historical data, we create a pdf for each date over 1000 days, and link it to the asset price realization. Then we can choose certain parameters of generating the implied pdf that maximizes the likelihood of the realization being "correct". It's basically a method of cross validation to choose the optimal parameter.
• But this needs a lot more thinking, and sounds very complex, and probably more of an academic exercise - not sure how much value this creates