Why Arrow-Debreu State Prices Can Exceed One

# Can state-contingent prices be greater than $\$1$ ?(Recovery Theorem by Steve Ross, 2015)


# Can state-contingent prices be greater than $\$1$ ?(Recovery Theorem by Steve Ross, 2015)












I have been reading the paper by Steve Ross on the Recovery Theorem that was published in the Journal of Finance in 2015 (an earlier version of the paper is available on SSRN here ).

The paper basically creates a framework for recovering the real-world probability from Option prices (under the assumption of a "Representative Agent's utility function" and a Markov chain assumption for the underlying evolution).

Without diving into the detail of the theorem (which is not important for this question), I wanted to draw attention to the following specific example in the paper:

The above is supposed to represent the "State Space Transition Matrix", which Ross describes in a later section of the paper as the matrix of "Contingent State Prices".

I think that I understand the row highlighted in blue: the current underlying value is "$S_0$" (i.e. "zero" sigmas away from where we are now), and the row represents the cost of Arrow-Debreu securities (i.e. "Contingent State Prices") that pay $ \\\$1 $ in a specific future state, given that we are in state "zero" right now:

So for example, to receive $\\\$1$ in a future state where we move to a return of -2 sigmas, given that we are at $S_0$ today, would cost $\\\$0.092$ today.

I have two questions:

- What do the other rows of this matrix represent?

- How can the price of a state-contingent Arrow-Debreu claim be more than $\\\$1$, as per the highlighted cells in yellow?

## Answer by Jan Stuller (score 1)

https://quant.stackexchange.com/a/84111

Just sharing my thoughts to increase visibility and to provide a partial "phylosophical" response to my own question.

I think that intuitively and theoretically (and "philosophically"), it is possible for a state-price to be higher than \$1, if it is "extremely valuable" to receive \$1 for sure in that state in the future.

Consider that a possible future state is a state of war or a similar catastrophe: then people might be willing to pay more than \$1 now, in order to receive \$1 for sure if that catastrophic state occurs in the future (the traditional "risk-free" framework might break down, i.e. people don't trust deposits in banks to be safe, etc.).

We can draw a specific example from the past: the Templar Knights effectively served also as "bankers", offering to safely "transport" gold across the continent, whilst charging a fee for it: in this scenario, one would give the Templar Knights their gold (say 1Kg of it) in Paris, whilst being able to safely collect 0.99 Kg in Amsterdam a few days later (the Templars didn't actually physically transport the gold: in their established "centers", they had enough gold to effectively work as a "bank"). The example is akin to negative rates. Here, people were effectively willing to pay negative rates to receive money in the future for sure, rather than risking being robbed on their journey across Europe.

Mathematically, it is indeed possible to have a state-price greater than \$1 with negative interest rates: we have seen negative interest rates extensively over the past ~15 years, including yields on the Bund: here, investors were happy to pay 1000 EUR to buy the Bund whilst being guaranteed less than 1000 EUR in the future.

Mathematically, imagine only two possible future states, $\theta_1$ & $\theta_2$ one period from now, $r$ being deterministically negative and $P$ being the pay-off function, with the current notional equal to $1$; then it is possible that:

$$1=e^{-rt}\mathbb{E}^\mathbb{Q}[P(\theta)]=e^{-rt}q_1P(\theta_1)+e^{-rt}q_2P(\theta_2)$$

In the Arrow-Debreu framework, we must have that $P(\theta_1)=1$ & $P(\theta_2)=1$. We further have that $e^{-rt}>1$, and both the two risk-neutral probabilities $q_1$ & $q_2<1$. But it can be the case that one of the two quantities $e^{-rt}q_1$ or $e^{-rt}q_2$ (which are the contingent state prices) are greater that 1 (amid extremely negative rates, both could be greater than 1).

I note that this is only possible in the world of (strongly) negative rates: and those rates would have to be extremely negative for a state price to be as high as $7.5$ as in the state-contingent price matrix above taken from Ross's paper.

I would therefore still love to receive a response to the original question, pls.