Option-Based Valuation of Corporate Debt and Risky Cash Flows

# Why aren't option pricing models more frequently used to value risky cash flows?


# Why aren't option pricing models more frequently used to value risky cash flows?












One way to think of the value of a risky firm is through expected measure theory. On the most basic level, the value of any asset is the convolution of the probability density function of its risky payoffs onto the outcome space of these risky payoffs. In this view, equity is analogous to perpetual long call option on the value of a firm, while debt is seen as a short put option on its value.

However, practitioners have not broadly adopted the use of options theory in pricing underlying corporate assets and liabilities. Real options analysis has found some acceptance among specialists', but the inputs and theory to these analyses are subjective and applications are not consistent. Why has this original use of the rational pricing of risky cash flows failed to become more broadly adopted?

In Samuelson & McKean (1965), McKean derived a closed-form solution for pricing perpetual options. The use case of such an approach was intended to value warrants, but was also intended for use in pricing more general corporate assets and liabilities.

In Merton (1973), "the possibilities for further extension of the theory [of rational option pricing] to the pricing of corporate liabilities are discussed".

In Black-Scholes (1973), the authors devote an entire section of the paper to discussing the applicability of rational pricing to corporate stocks and bonds:

> It is generally not realized that corporate liabilities other than warrants may be viewed as options. Consider, for example, a company that has common stock and bonds outstanding and whose only asset is shares of common stock of a second company... Under the conditions, it is clear that the stockholders have the equivalent of an option on their company's assets. In effect, the bond holders own the company's assets, but they have given options to the stockholders to buy the assets back. The value of the stock at the end of 10 years will be the value of the company's assets minus the face value of the bonds, or zero, whichever is greater.

In Cox-Ross (1976), the authors use discrete models to "find explicit option valuation formulas, and solutions to some previously unsolved problems involving the pricing of securities with payouts and potential bankruptcy."

In Geske (1978), the author:

> ...presents a theory for pricing options on options, or compound options. The method can be generalized to value many corporate liabilities. The compound call option formula derived herein considers a call option on stock which is itself an option on the assets of the firm. This perspective incorporates leverage effects into option pricing and consequently the variance of the rate of return on the stock is not constant as Black&Sholes assumed, but is instead a function of the level of the stock price. The Black&Sholes formula is shown to be a special case of the compound option formula. This new model for puts and calls corrects some important biases of the Black-Scholes model.

The list of academic support continues almost indefinitely, but I still very rarely see it applied.

With some much theoretical and anecdotal support, I am surprised that this topic hasn't found broader support. I can surmise a few reasons why it is not broadly used. Chief among these, markets for underlying assets are not complete, and arbitrage relationship between risky cash flows and their underlying drivers can be very firm-specific (or even totally ambiguous). There may be other reasons, but I wanted to gather some input from the community on your opinions. Maybe the theory is more broadly used than I realize, in which case, I'd also like to know where and how.

## Answer by Quantgolfer (score 3, accepted)

https://quant.stackexchange.com/a/34431

There are papers out there applying this approach. Try looking up Leland, Leland&Toft (1994 and 1996) for modelling corporate liabilities, resulting in a series of interesting results. Also, it might be worth looking into a structural credit risk modelling approach (in contrast to a reduced form approach (see Lando (2001 or 2004) for more)).

There are also papers using this approach to decide firms are over/underlevered. Otherwise there is the KMV model also using this structural approach for determining credit risk (google it, there is a ton of stuff on this). This is a very interesting area.

## Answer by zer0hedge (score 1)

https://quant.stackexchange.com/a/34556

Black-Sholes-Merton formula (and the other models) are valid under some assumptions. These assumptions are broken to some degree in financial markets and even more so for "risky cash flows" in general. You probably will not use model which assumes "A is true" when it is obvious that "A is false". Furthermore, as the title of Haug&Taleb article claims, even "option traders use (very) sophisticated heuristics, never the Black–Scholes–Merton formula".

## Answer by Dave Harris (score 0)

https://quant.stackexchange.com/a/34552

The models do not work empirically. You don't apply things that don't work. Find just ONE supporting validation study for Black-Scholes. I cannot find the citation, but there was a clever one about twenty years ago that tested the S&P 500 and determined that the Black-Scholes formula for it was uncorrelated with reality. I have run similar tests on disaggregated returns for the CRSP universe and had the same finding.