The "iron butterfly" – characterizing asset price distribution from its option prices
Back in my PhD days, when I did a lot of thinking about probability distributions, a theme that I sometimes wondered about was: “When does a set of functionals fully characterize a distribution?”. In the case of expectations, “For which infinite sequences $g_i$ can you fully determine $P$ by knowing $E_P[g_i(X)]$?”.
An instance of this is the moment problem: “Do the moments $E_P[X^k]$ completely characterize $P$?”. The general answer is No, and the easiest way to see this is to think of heavy-tailed distributions, and more specifically distributions whose moments are all infinite, e.g. the Cauchy family. There are infinitely many distributions in this class, but only one moment vector for all of them: $(\infty, …, \infty)$. QED.
On the other hand, some other choices of $g$ yield positive answers, e.g. the characteristic function, $g_t(X) = e^{itX}$ suffices; or less abstractly let $g$ be the set of indicator functions for all intervals $[\infty, r]$”. Formalists may note that in these examples $g$ is not strictly a sequence since $\mathbb R$ is uncountable, but it gets close enough since you can have it range over the rationals, which is a dense countable subset.
A more applied instance of this question is: “Do option prices, under some assumptions, completely characterize the distribution of an asset’s price at a future point in time?” … But what do options have to do with expectations?, you may ask.
Let’s assume that the price p_s of an option for underlying asset u with strike s is equal to its expected payoff at expiration.
Note the shape of the payoff function. What happens if you’re long a call(s=150) and short a call(s=151) with the same expiration? This combination is known as a collar, and its payoff function looks like ___/‾‾‾‾‾ . Note that this like the derivative of the payoff function wrt to strike price (to be precise, it’s a finite difference approximation, scaled). (The fact that call options themselves are also derivatives (“derivative assets”) is only relevant here if you want to be punny or intentionally confusing)
Now let’s consider another collar slightly higher up: long call(s=151) + short call(s=152). The payoff function is of course very similar: ____/‾‾‾‾. So what happens if you subtract these two, i.e. –call(s=152) +2 call(s=151) – call(s=150)? You get this shape ____/\___ (a.k.a. the "iron butterfly"), which is like the second derivative of option price wrt strike price. It only pays off in a bounded interval around 151, and will be proportional to $P(u \in [151–1, 151+1])$ if we assume that the pdf is constant in this small interval.
The result of this is a positive answer to our question, under a few assumptions:
The price of an option is equal to its expected payoff.
The options market is rational, efficient and highly liquid, at every strike price.
The set of strike prices is granular enough and/or the pdf is smooth enough, that the $p(u)$ is roughly constant in each interval.
Note that the Put market encodes exactly the same info as the Call market, and any discrepancies between the distributions implied by these might point to potential arbitrage opportunities (let’s call them PAO#1).
[I actually did this last summer, and will add an image here once I’ve located it… long story short: the market’s “belief” about future SPY price distribution was multimodal and pretty crazy, which probably just means that way-OTM options are very illiquid and those prices can’t be taken seriously … but I have an idea about fixing that with Kalman Filters]
Note that for every strike s: call(s) = put(s) + underlying, which already implies a very straightforward potential arbitrage opportunity (PAO#2)
Q: If PAO#2 is already fully exploited, does this automatically imply that PAO#1 is no longer exploitable?
(Thanks to Marie La for inspiring me to write this up)
UPDATE: Commenter LocalVolatility claims a connection to “Breeden-Litzenberger”. Googling this led me to a 2014 New York Fed report titled “A Simple and Reliable Way to Compute Option-Based Risk-Neutral Distributions”, which credits the original result to:
Breeden, D. T. and Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices, Journal of Business 51(4): 621–651.