# Relationship between risk-neutral probability and subjective probability

I recently came across a Paper by a paper of Rubinstein and Jackwerth (1997): http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.5214&rep=rep1&type=pdf

where they assume that you can describe risk-neutral distribution = subjective probability $\times$ risk-aversion adjustment.

Do you know of any papers that use other approaches or go into that relationship in more detail?

Standard Finance/Utility theory dictates that all future cash-flows are priced via the marginal rate of substitution. For example, say that $X_T$ is the random variable that represents this cash-flow at a future time. Then, the value of this cash-flow today will be valued as $$X_0 = E \left[ \frac{e^{-\rho T}\ U'(W_T)}{U'(W_0)} X_T \right]$$ where $U$ is the utility function, and $W$ is the 'wealth' that serves as an argument to the utility function. $\rho$ is a rate that discount utilities (a dollar today is worth more than a dollar tomorrow.)

Now typically a utility function is upward sloping (more wealth is better) but is also concave (that is to say it increases at a decreasing rate; the millionth dollar is not as valuable as the first one). This reflects the 'risk aversion' in the following way: a loss of a dollar will cost in utility terms more than a gain of a dollar, therefore to become indifferent the odds should be better than 50-50, or equivalently I would demand a positive expected return to play a zero-sum random game.

Now back to pricing, note that a bond is just a constant cash flow $X_T=1$, which effectively gives us the bond price/ risk free rate in terms of the expected marginal rate of substitution $$B = e^{-rT} = e^{-\rho T} \frac{E[U'(W_T)]}{U'(W_0)}$$

There are two interesting cases:

1. If the utility function is linear, then $U'(W)=const$ and we are 'risk neutral'. In that case the discount rate for dollars is the same as the discount rate for utilities, $r=\rho$. The value of a random payoff is just its discounted expected value $X_0 = e^{-rT} E[X_T]$.
2. If the random payoff is practically independent of our future wealth, then the expectation of the product is the product of expectations, and its value today is again its discounted expected value $X_0 = e^{-rT} E[X_T]$. A market index which correlates with wealth has a higher premium than a small stock.

In general though, the random quantity $\mathcal{M}_T(X_T)=e^{-\rho T} \frac{U'(W_T(X_T))}{U'(W_0(X_0))}$ which has various names (pricing kernel, state price density, marginal rate of substitution, etc) summarizes risk aversion. I also put an explicit dependence of wealth on $X$ to remember that they are in general correlated.

If the distribution of this future payoff at time $T$ has a density $f_T(x)$, then the expectation that gives its price today is actually written as $$X_0 = \int \mathcal{M}_T(x)\ x\ f_T(x)\ dx = E [\mathcal{M}_T(X_T)] \int x\ \frac{\mathcal{M}_T(x)}{E [\mathcal{M}_T(X_T)]}\ f_T(x)\ dx$$

Look at the function $$g_T(x) = \frac{\mathcal{M}_T(x)}{E [\mathcal{M}_T(X_T)]}\ f_T(x)$$ This is actually always positive (because utility slopes upwards) and it also integrates to one (that's why I normalized with $E[\mathcal{M}]$). Therefore is is a valid probability density function, which is the original (real one) times a risk aversion adjustment.

The price of the claim is written as $$X_0 = e^{-rT} \int x\ g_T(x)\ dx = e^{-rT} E^g[X_T]$$ that is to say the value of the claim is the discounted expected payoff, where expectations are taken not under the real distribution $f_T$ but under the 'risk neutral' (or as I prefer to call it 'risk adjusted') one $g_T$.

(Note: The 'risk neutral' distribution is called 'risk neutral' exactly because the pricing formula above is the same as the one in the risk neutral case above.)

(Edit: Added intertemporal discount factor as OP suggested)

• Very clear answer Kiwiakos. I am not the OP but thank you anyway! I think a reasoning along these lines can be also used to answer my question here: quant.stackexchange.com/questions/8274/… . If you have time I will be very interested in your views.
– sets
Nov 10 '15 at 12:05
• Thanks a lot Kiwiakos. It helped a lot. I am still curious about the intertempral substitution. It models choices that depend on cost-benefit tradeoffs. Normally it is assumed that it lies in (0,1) and is constant. Do you know of any papers/literature which go into that in greater detaill? There is a lot of literature for the stochastic discount factor in general, e.g unc.edu/depts/econ/profs/renault/… Nov 10 '15 at 12:53
• @Alkibiades I added some explicit treatment of the intertemporal parameter. Nov 10 '15 at 15:08
• @sets It is indeed related to the transformation you are asking for. I remember that Ait-Sahalia and Lo had an old paper (1995ish) where they 'estimate' exactly this conversion function that transforms (risk neutral) distributions implied by option prices into 'real world' distributions. Nov 10 '15 at 15:09