# Do underlying assets have a no-arbitrage price?

Can it be shown that the Fundamental Theorem on Asset Pricing (FTAP) applies to underlying assets -- namely bonds, equities, and commodities?

FTAP says that assets have no-arbitrage prices equal to their risk-neutral expectations. A no-arbitrage price is the price which is implied by the efficient market hypothesis. However, for assets which have vague underlying/causal relationships, the belief that "the current price represents the expectation" appears to inevitably result in a tautology. Since an arbitrage-free price other than the market price for such assets can be neither be proven nor disproven, the FTAP appears to dissipate into a belief system rather quickly. Or maybe I've missed something and there is indeed a generic approach which is not also tautological?

It is not problematic for derivatives of an underlying asset to have an arbitrage-free price (and/or range of prices) when the market for the underlying is assumed to be sufficiently complete and the terminal payoff conditions based on that underlying are known. Whether or not the underlying itself is efficiently priced may not even be relevant for quantifying an efficient price of its derivative. The following passage from Chapter 1 of Baxter's and Rennie's Financial Calculus summarizes this attitude:

With markets where the stock can be bought and sold freely and arbitrarily positive and negative amounts of stock can be maintained without cost, trying to trade forward using the strong law would lead to disaster […].

[…]

But the existence of an arbitrage price, however surprising, overrides the strong law. To put it simply, if there is an arbitrage price, any other price is too dangerous to quote.

[…]

Thus maybe a strong-law price would be appropriate for a call option, and until 1973, many people would have agreed. Almost everything appeared safe to price via expectation and the strong law, and only forwards and close relations seemed to have an arbitrage price. Since 1973, however, and the infamous Black-Scholes paper, just how wrong this is has slowly come out. Nowhere in this book will we use the strong law again. […] All derivatives can be built from the underlying −− arbitrage lurks everywhere.

If arbitrage lurks everywhere, I interpret this to mean it is possible to take the FTAP view on the underlying assets themselves. It seems both fair and intuitive to think about equity and debt as being derivatives on underlying assets, and those asset in turn as being derivatives themselves. Yet, most of the field appears to be stuck on weak expectation-based models of the flavors provided by the Capital Asset Pricing Model (CAPM) and Fama-French.

Does there exist a stronger pricing model for underlying assets for which arbitrage relationships are vague or -- at very best -- probabilistic? What about things that do not neatly fit into the arbitrage framework, perhaps such as commodities?

Or, is applying FTAP in these situations ill-prescribed -- a misapplication of recent economic thinking which reduces human decision-making behaviors to mathematically convenient utility seeking functions? I can see merit in both the behavioral and no-arbitrage economic approaches to asset pricing.

Any general or specific insights on practical applications of the no-arbitrage principle on underlying assets are welcome.

• Abstract of "Non-arbitrage valuation of equities" really hits the point head on. Now, it is worth the $40? Sep 30, 2017 at 19:56 To take a somewhat trivial example of an underlying asset subject to arbitrage pricing, we can think of options on futures. But I think you have deeper examples in mind. Structural models, such as Merton's famous 1974 model, treat equities themselves as derivatives. In Merton's case the underlying was the economic value$A$of the firm's assets$A$and the strike price was the debt$D$. Here, though, it is reasonably obvious that$A$is not generally tradeable, so the argument for using arbitrage pricing theory and the associated risk-neutral processes is abstract, and only "rigorous" in a large$N$portfolio argument. The idea is basically that no single one of these untradeable securities can be arbed precisely, but that a large portfolio of them in a stable economy will take on a value expressed in effect by the market "collectively arbing" them, so that their derivatives can be treated using risk-neutral prices. It's a little thin, and makes one wonder just how large$N$is supposed to be. But it's probably the best analog to what you are looking for. Namely, a kind of, as you say, "vague arbitrage" can be collectively made risk neutral at some sufficiently large$N$. • I appreciate the insights on$N\$ needing to be large. Doesn't this favor the use of sort of a "stat arb" approach? Also, I believe that Merton's intuitions regarding valuing underlying assets have been largely under-regarded because simplifying assumptions on stochasticity were unrealistic to the point of uselessness. Do you use or do you know anyone who uses this sort of no-arbitrage approach to equity and/or debt valuation? May 1, 2017 at 20:21