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Marsden
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Equity options where there is corporate debt

November 21st, 2021, 2:30 pm

Following up on a loose thread from earlier thought experiments.

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:

C0 = S0 - Xe-rT + P0,

where we let C0 be the leveraged company's stock price, S0 be its per share Enterprise Value, and ( Xe-rT - P0 ) 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 + Der(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 -- P0 -- 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 S0. 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 + Der(T-t) - PT 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 PT here is something I haven't worked out yet.
 
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Alan
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Re: Equity options where there is corporate debt

November 21st, 2021, 4:39 pm

Some prior research that may be of interest:
- Merton's (1974) model for equity as a call option.
- some (2003) work I did on option valuation under discrete dividends with Espen and Jorgen Haug (Back to basics). It has some overlap here. I also did a lot of (2016) follow-up on the discrete dividend problem in "Option Valuation under Stochastic Volatility II". You can think of the discrete dividend as the debt payment, and I consider the case of it being large enough to trigger a default.  The "adjust the strike price by the value of the debt payment projected" type rules can be tricky and sometimes lead to unexpected arbitrage opps!
 
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Marsden
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Re: Equity options where there is corporate debt

November 21st, 2021, 8:46 pm

Thank you Alan.

Do the "unexpected arbitrage opps" occur with European-style options?

My thinking is that dividend payments prior to expiration can be treated the same as debt payments, with the recognition that this does not reflect the "we cancelled the dividend but are still an ongoing company" possibility.
 
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Marsden
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Re: Equity options where there is corporate debt

November 22nd, 2021, 2:22 pm

So I think the solution for PT in the debt payment AFTER case comes from from the super secret, dark and evil version of Black-Scholes (which I provide here only at great personal peril):

PT = Xe-rt[1-Φ(d2)] - ST[1-Φ(d1)].

For calculating a value at time T, ST -- the Enterprise Value, for this purpose -- will be a random lognormal variable, and d1 and d2 will be functions of it.

Sadly, complications make assuming a constant r probably a computational necessity.

Beyond that, double integrals and a pretty big hairball. But somewhat tractable, I think.
 
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Alan
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Re: Equity options where there is corporate debt

November 22nd, 2021, 4:34 pm

Thank you Alan.

Do the "unexpected arbitrage opps" occur with European-style options?

My thinking is that dividend payments prior to expiration can be treated the same as debt payments, with the recognition that this does not reflect the "we cancelled the dividend but are still an ongoing company" possibility.
I'm not sure (The example in the paper is for American-style options).
 
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complyorexplain
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Re: Equity options where there is corporate debt

November 22nd, 2021, 5:19 pm

Some prior research that may be of interest:
- Merton's (1974) model for equity as a call option.

Peter Feldhuetter https://feldhutter.com/ has done some outstanding recent empirical research on the Merton model, and extensions of it.

Particularly this: https://feldhutter.com/CreditSpreadPuzzleFINAL.pdf 
 
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complyorexplain
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Re: Equity options where there is corporate debt

November 22nd, 2021, 5:24 pm

My thinking is that dividend payments prior to expiration can be treated the same as debt payments, with the recognition that this does not reflect the "we cancelled the dividend but are still an ongoing company" possibility.

I regard them as a debt on and after the announcement date, given it is very rare for a firm to renege on an announcement (although has happened, I am thinking banks last year on covid crisis).
 
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Marsden
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Re: Equity options where there is corporate debt

November 30th, 2021, 8:11 pm

I don't get the modified stock price or modified volatility models for pre-expiration dividend/debt payments: modifying the strike price seems pretty obvious. Yes, early exercise for American-style options might throw some curves, but not any more than with other methods.

What am I missing? Modifying stock price seems a bit like shortening runners legs rather than raising hurdles, and modifying volatility a bit like having a hurdles race in two feet of standing water.