Physical Probability Measure vs. Risk Free Probability Measure (State Contigent Claims)

Question

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 1

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 $N>K$.