Risk-Neutral Probabilities and Discounted Asset Pricing

# Risk Neutral Pricing, a quick question


# Risk Neutral Pricing, a quick question












I am a newbie.

The risk neutral pricing has the following formulation:

$$P=\frac{\hat{E(d)}}{R}$$,

But the discounted expected value has the formulation of:

$$P=\frac{E(d)}{R}$$.

The text book says $\hat{E}(.)$ is the expectation over risk-neutral probability.

What exactly is that risk-neutral probability?

## Answer by Stéphane (score 3, accepted)

https://quant.stackexchange.com/a/54020

When you look at actual data from the stock market, the probability distribution that you have in mind and that would describe the likelihood of different scenarios occuring going forward is what we call the "physical" probability distribution.

Intuitively the risk-neutral probability distribution is a "distorted" version of the physical probability distribution that accounts for risk aversion. Essentially, the issue is that if you buy a stock, you expose yourself to more risk than if you put money in a bank account or buy a government bond from a country like the US. It is possible that the price drops dramatically in the future for a stock, but you are unlikely to loose the money you put in the bank and it is also very unlikely that the US government will default payment on your bond. The reason some people nonetheless buy stocks is that they expect the roller-coaster ride to pay off more on average -- that's the idea behind risk-aversion: you need to compensate people for bearing additional risks by giving them higher returns on average.

The risk-neutral probability is just a very convenient way of saying the same thing. Instead of looking at the actual world and thinking that people want compensation for risk, you could say that people tend to exaggerate the likelihood what's bad for them and to downplay the likelihood of what's good for them. In that world, they do not demand compensation for risk -- they are risk-neutral investors in a world where they get to distort probabilities. The reason risk-neutral probabilities are so useful is because you can summarize all the information about preferences with regard to risk into the change of measure you apply to move from physical to risk-neutral probabilities. After that, you can think about everything in terms of simple statistics -- you no longer have to think about preferences.

If you want a formal version of the idea, I could write down a consumption model and show you how the Euler equation (one of the first order condition for expected utility maximization) of a representative investor in an equilibrium model can be re-arranged to get a risk-neutral distribution out of it, but it wouldn't add much to the discussion if you're not familiar with economic theory.

EDIT

Let there be a representative investor in a market economy. We assume that he lives two periods (don't worry, in an infinite horizon model, we'd get exactly the same solution here) and he receives a random endowment $e_t$ in each period. He can save some of his endowment for the next period by purchasing a risky asset at the price of $S_t$ for an uncertain payoff of $x_{t+1}$ in the next period. His expected utility maximization problem is thus \begin{align} \max_\xi \left\{ u(c_t) + \beta E_t( u(c_{t+1}) \right\} \\ \text{s.t.} \begin{cases} c_t = e_t - \xi S_t \\ c_{t+1} = e_{t+1} + \xi x_t \end{cases}. \end{align}

We assume that the condition probability distribution of the future endowment and payoff processes is such that the conditional expectation exists and, further, we assume that the instantaneous utility function $u: \mathbb{R_+} \rightarrow \mathbb{R}$ is twice continuously differentiable in the non-negative real line with $u'(c) > 0, u''(c) < 0$ for all $c \in \mathbb{R}_+$ and that $\beta \in (0,1)$. Under these assumptions, this problem is well posed. If we further assume the Inada condition $\lim_{c \downarrow 0} u'(c) = \infty$, we are garanteed that we have an interior solution whih must satify \begin{align} 0 &= - u'(c_t) S_t + E_t \left( \beta u(c_{t+1} x_{t+1} \right) \\ \leftrightarrow S_t &= E \left[ \beta \frac{u'(c_{t+1})}{u'(c_t)} x_{t+1} \bigg| F_t \right] \end{align} where I highlighted the fact that we are conditioning the expectation on the natural filtration of the economy, the sigma algebra given by $F_t$. This equation is what we economists call the Euler equation. There is an equation like this in all representative agent models for every asset in the economy. Now, let $m_{t+1} := \beta u'(c_{t+1})/u'(c_t)$. We're going to call this the "stochastic discount factor" because it "discounts" $x_{t+1}$ differently in different states of the world. Specifically, here you "don't like" stuff that varies in the same direction as your consumption at the same time as your consumption, but you do like the opposite (that would be a sort of insurance).

Now, the risk-neutral measure. We introduce a new asset in the economy, a bond $B_t$. That bond is without risk, hence it earns you a risk-free rate $B_{t+1}/B_t = (1 + R_{ft})$. This security is subject to the equation above, hence \begin{align} B_t &= E_t \left( m_{t+1} B_{t+1} \right) \\ 1 &= E_t \left( m_{t+1} (1 + R_{ft}) \right) \\ \frac{1}{1 + R_{ft}} &= E_t(m_{t+1}) \\ \frac{1}{1 + R_{ft}} &= \int_{\omega \in \Omega} m_{t+1}(\omega) dP(\omega) \end{align} where $\omega$ is an event, $\Omega$ is the sample space and $dP(\omega) := p(\omega | F_t) d\omega$. Essentially, I just applied the definition of an expectation. Here, $P$ is what we call a probability measure. Now, we'll do a bit of algebra: \begin{align} S_t &= \int_{\omega \in \Omega} m_{t+1}(\omega) x_{t+1}(\omega) dP(\omega) \\ &= \int_{\omega \in \Omega} \frac{1+R_{ft}}{1+R_{ft}} m_{t+1}(\omega) x_{t+1} (\omega) dP(\omega) \\ &= \frac{1}{1+R_{ft}} \int_{\omega \in \Omega} x_{t+1}(\omega) m_{t+1}(\omega)(1+R_{ft}) dP(\omega) \\ &= \frac{1}{1+R_{ft}} \int_{\omega \in \Omega} x_{t+1}(\omega) dQ(\omega) \\ &:= \frac{1}{1+R_{ft}} E^Q \left[ x_{t+1} | F_t \right] \end{align} where I basically defined $dQ := (1 + R_{ft}) m_{t+1} dP$. You can write this as a Radon-Nikodym derivative of $Q$ with respect to $P$. Given $u'(c) > 0$ and $\beta \in (0,1)$, you can conclude that $m_{t+1} > 0$. Since $P$ is a probability measure, it is non-negative, hence $Q$ will also be non-negative. I have also shown that under $P$, $m_{t+1}(1+R_{ft})$ must integrate to one since the expected value of $m_{t+1}$ is $1/(1+R_{ft})$. Hence, $Q$ is something always nonnegative that must sum to 1 -- i.e., it is a probability measure and this why you can view this equation as involving a "distorted" expectation.