So Black-Scholes-ish closed form option prices implicitly assume no debt ... which limits their usefulness for perhaps 90% of standard equity options contracts.

(There probably is research on this that I'm unaware of, because I don't keep my nose in such things.)

If we super-impose the Modigliani-Miller Theorem onto equity prices, they behave a little bit like call option prices:

C

_{0}= S

_{0}- Xe

^{-rT}+ P

_{0,}

where we let C

_{0}be the leveraged company's stock price, S

_{0}be its per share Enterprise Value, and ( Xe

^{-rT}- P

_{0}) be a per share representation of its outstanding debt in the form of "riskless interest rate discounted value of scheduled debt payment reduced by an implicit put for the possibility of default."

This will be my general model going forward.

There are two general cases to consider in equity options: where the debt payment precedes the expiration date of an equity option; and where it is afterward.

(Everything kept dirt simple at this point: European-style options; single debt payments; no dividends; constant volatility; no kinky bullshit like stock buy-backs or new issuances of either debt or equity or corporate spin-offs or acquisitions; etc.)

I think -- and I may be wrong -- that the case where the debt payment precedes the expiration date of the equity option is not very difficult (and, in keeping with the dirt simple-ness of discussion, I assume that there is only one debt payment, although multiple payments work the same way as long as they all precede the expiration date): just use a Black-Scholes-ish closed form option price but adjust the strike price by the value of the debt payment projected -- all at riskless interest rate -- from date of payment to expiration date of option.

So if the debt payment is D and it is made on date t (the expiration date will be T), then in place of the normal strike price for the option (X) we use X + De

^{r(T-t)}.

(I take issue with using a single riskless interest rate for all future time, but I don't think that has much effect here and correcting for it isn't very interesting and would require superscripts with subscripts which are not easy to format as far as I can tell.)

The only thing particularly interesting in this is that the value of the implied put -- P

_{0}-- disappears.

Why? Because it's the value of the right of the company/shareholders to default on the debt payment ... and if we reach the expiration date for the option, post debt payment, without the company being bankrupt, implicitly the debt has not been defaulted upon.

Also -- not because it's interesting but just as a matter of clarification -- in the Black-Scholes-ish option pricing formula, we use the original per share Enterprise Value of the company as S

_{0}. After the debt payment has been made -- and remember, I have assumed that all debt is settled in one payment -- the per share Enterprise Value and the share price are the same.

For the case where the debt payment is AFTER the expiration date of the equity option, X + De

^{r(T-t)}- P

_{T}would generally be used in place of just X (there is still an implicit put included in the share price for the possibility of defaulting on the debt). Note that in this case t is greater than T, and we still have the value of an implied put included in the share price to reflect the possibility of defaulting on the debt. The value of P

_{T}here is something I haven't worked out yet.