Why State Prices Do Not Identify Physical Probabilities
Summary
The document asks whether physical state probabilities can be recovered from current asset prices, state-contingent payoffs, and risk-neutral probabilities. Its central point is that state prices encode discounted, risk-adjusted probabilities; they do not by themselves reveal the physical probabilities or the state-dependent discount factors. The response cautions that recovery requires additional assumptions or information, and notes that attempts to infer probabilities from market prices under strong economic assumptions have faced empirical criticism.
It outlines one possible route when a representative investor's utility function and optimal portfolio are known. First-order conditions for utility maximization relate prices to expected marginal utility-weighted payoffs. With those inputs, the conditions can be written as a linear system in the physical probabilities and solved when the payoff information has sufficient rank. This is a conditional method, not a way to infer probabilities from prices alone; its usefulness depends on knowing the investor's preferences and portfolio choice, as well as the model assumptions.
Key ideas
- Risk-neutral probabilities reflect state prices and discounting, so they do not generally identify physical probabilities on their own.
- State-dependent discount factors cannot be recovered from the stated prices and payoffs without further information or assumptions.
- A known utility function and optimal investor portfolio can provide first-order conditions linking physical probabilities to asset prices.
- Those conditions form a linear system whose probability estimates require enough independent payoff information.
- The proposed recovery approach is conditional on a representative-investor model and is not an assumption-free procedure.
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Full text
# Physical Probability Measure vs. Risk Free Probability Measure (State Contigent Claims)
# Physical Probability Measure vs. Risk Free Probability Measure (State Contigent Claims)
currently I am working on a problem regarding state contingent claims.
I have 5 securities (one of the security is a risk-free security) and in the next period, these securities will end up in one of 4 possible states (where the cashflows differ in each state for each security).
I have the following information available:
- Current Prices of the securites
- Cashflows of the securities for the next period
I already managed to calculate the risk-free probability measures for each state and the state prices.
Now my question is, how to get the physical prob. measure for each state?
My Idea was the following:
the risk free prob. measure can be calculated with the formula:
- Prob(RF)= [m(s)/E(m)]*Prob(physical)
rerange the formula for prob(physical) gives:
- Prob(Physical)= [Prob(RF)*E(m)]/m(s)
where
- Prob(RF) = Risk free prob. measure
- Prob(physical) = Physical prob. measure
- E(m)= Sum of all state prices
- m(s)= Discount Factor
but here my problem is, that I dont know the discount factor for each state. Unfortunately, all books I´ve looked into, define the discount factor by first derivatives of utility functions. I dont have any utility functions available, so that wasnt a great help, and the reason why I´m struggling with that.
I would be very happy about an idea how to get to m(s) or another approach to get the physical prob measure.
I hope someone can give me some hints.
best regards
## Answer by Kermittfrog (score 2)
https://quant.stackexchange.com/a/59985
I think that it is not possible to recover physical probabilities from state prices / risk neutral probabilities without much more structure or information. Even if you had state transition probabilities and strong assumptions on your economy, this exercise has been empirically refuted(Ross's recovery theorem and all the follow up critique, last paper I think is Jackwerth/Menner.
If you had information on the form of the utility function and the optimal portfolio of your representative investor, you could recover the physical probabilities. The idea would be approximately as follows:
Say your investor has a budget $B$, utility function $u$; there exist $K$ states and we have $N\geq K$ assets. The agent wants to choose his asset investment levels $w=\begin{pmatrix}w_1&w_2&\ldots&w_N\end{pmatrix}$, i.e. the weight vector, such that her total expected utility is maximized, subject to $w^TP_0\leq B$, i.e. the budget restriction. $P_0$ contains today's asset prices. Finally, let $X$ denote the random payoff of the assets (vector), and with some abuse of notation let $X$ also denote the $N\times K$ matrix of payoffs, i.e. $X_{i,k}$ is the payoff of asset $i$ in state $k$. Note that $X$ must contain at least $K$ independent rows, i.e. the states are well-behaving.
Then the investor's optimization program would be something like
$$ \max_w \quad L(w)\equiv u(B_0-w^TP_0)+\delta\mathbb{E}\left(u(w^TX)\right) $$ with $\delta$ some time preference parameter. The first order condition reads
$$ \begin{align} u'(B_0-w^TP_0)P_{0,i}= \mathbb{E}\left(u'(w^TX)X_i\right) \end{align} $$
Now since we already know the optimal investment weights $w^*$ as well as the initial prices $P_0$ etc., we can back out the physical probabilities from a linear equation system. Rewrite the FOC as
$$ c=Bp $$ with $c$ the $N\times 1$-vector of elements $u'(B_0-w_*^TP_0)P_i$ and the $N \times K$ elements in $B$ defined by $B_{i,k}=u'\left(w^TX_{.,k}\right)X_{i,k}$
The vector of probabilitites is then found as $p=B^{-1}c$ if $N=K$, and by the Moore-Penrose-inverse for $N>K$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.