We are busy pricing a derivative (I think!) where the stock price is assumed to follow the usual GBM process. The strike however is not constant and follows a company's EBITDA.

Our theory is that: There is no market for a company's EBITDA and therefore we cannot setup a replicating portfolio. Something along the lines of an incomplete market (I am more than fuzzy on all the technical theory).

At the risk of being scolded I have two questions:

1. Is this theory correct and does this mean that we cannot obtain a "risk-neutral" option price?
2. If we are forced (by accounting requirements) to "value" this thing, what options (pardon the pun) do we have?

Hopefully a storm in a tea cup.

TLDR; I work for "one of the big four" audti firms and the departments are playing hot potato with this valuation. We have previously valued these instruments in the normal risk-neutral fashion but we have a new kid on the block who looked at this and asked the question.

I’ve never worked on this kind of problems specifically, but it reminds me of a class of modelling approaches I need to roll out in an incomplete uncertainty scenarios. Focus on the most relevant parameters: volatility of stock and ebitda, their correlation, etc. Analyse different scenarios: from your best shot at the parameters value to some extreme cases. This will give you sensitivity, worst-case results, possibly some instabilities, etc.
I hope I understood your problem correctly (-:

Good question.

Yes, incomplete markets very relevant. Not that most people ever worry not being able to hedge some risk! They tend to gloss over this problem. At least when the boy said that the Emperor had no clothes other people listened! So that's one approach...ignore it!

Or look at Merton's company valuation work and lots of follow ups. Real options.

I imagine that the stock price and EBITDA are highly correlated, so that could be an issue in itself.

Discuss...

Thinking about how to get a mathematically justified solution, I run into a real big problem: having to make some sort of projection of a future interest rate environment on top of everything else.

I get there by first assuming that, using market-traded derivative prices, you can derive a risk neutral future distribution for the stock price.

Good start, right?

Then you want to come up with conditional future values for the EBITDA, in the notion that somehow the risk neutral future stock price distribution can fairly simply be split up along those conditional probability lines, integrate, and call it a day.

HOWEVER, the relationship between EBITDA and stock price is almost of necessity very strongly dependent upon the interest rate environment. So stock price $100 with EBITDA $5 is very different from stock price $100 with EBITDA $10.

The risk neutral future distribution of stock price on its own essentially reflects a weighted average of different interest rate environments given a future stock price ... and joint probabilities with EBITDA suggest a need to unscramble those eggs.

So you effectively want a joint risk neutral probability distribution. Which I guess is kind of where things were left off a couple steps ago in my thinking.

But the risk neutral probability distribution of future stock price is already effectively a joint probability distribution: "What is the weighted average value today of $1 payable at time T given that the stock price at time T is X?"

This becomes, "What is the weighted average value today of $1 payable at time T given that the stock price at time T is X and the EBITDA at time T is Y?"

And I think, within the conditional distribution given that the stock price is X, you get wildly different values depending upon whether the ratio of stock price to EBITDA is 20 rather than 10.

Hope that helps!

What does the term sheet say if the EBITDA is negative?

Asset for asset option aka Margrabe option? This has been known for decades. Of course, you must have a model for EBITDA: GBM, Merton of anything you want. You must know that no model means no pricing.

Term sheet needed, pls.