Risk-Neutral Pricing, Arbitrage, and Preference-Adjusted Probabilities

# does there need to be risk-neutral agents in the market to enforce risk-neutral pricing?


# does there need to be risk-neutral agents in the market to enforce risk-neutral pricing?












I'm trying to understand a fundamental link between mathematical finance and economics. I understand that risk-neutral pricing is free of arbitrage with replicating portfolio. Does risk-neutral pricing mean that there needs to be risk-neutral agents in the economy that actually does the arbitraging/portfolio replication?

In economics, risk-neutrality translates into linear utility function, that is, there is no risk aversion. But if there are market participants who are risk-neutral and unconstrained, would we still observe risk premia? Macro-finance/representative agent models in economics almost always assume some degree of risk aversion. In these models, there could not be equity premium without risk aversion. (For instance, Consumption CAPM model has equity premium as $\gamma*cov(C_{t+1},R_{t+1})$). How do we reconcile the idea of risk-neutral pricing with representative agent models in economics that require risk aversion.

## Answer by Ian Brodrich (score 0, accepted)

https://quant.stackexchange.com/a/19553

Risk Neutral Valuation is based on arbitrage opportunities between the derivative and its underlying. So simply the arbitrage opportunity has to exist. Risk neutrality of market agents is not necessary, as their is no risk when realising the arbitrage opportunity.

## Answer by Kiwiakos (score 2)

https://quant.stackexchange.com/a/19543

No. Actually "risk neutral pricing" does not make assumptions on the risk preferences of the agents.

Securities are priced as if agents were risk neutral (that is to say as a straight expectation of discounted payoffs) but where probabilities of states of the world are not the true ones but they have been adjusted to reflect preferences.

The math:

Say that the representative agent has utility of consumption $U(c;\gamma)$ where $\gamma$ reflects risk aversion.

We are in a two-period setting. Time-zero is fixed and certain. There is a state variable that summarizes uncertainty at time-one, say $X$. Also utility is discounted with factor $\delta$.

Consumption will depend on the state of the world, as $C_1=C_1(X)$. Also we want to find the time-zero price of a security that has pay-off as a function of the state at time-one, namely $P_1=P_1(X)$.

The standard result is that in equilibrium the following is satisfied: $$ P_0\ U'(C_0) = \delta E[P_1\ U'(C_1)] = E[P_1(X)\ U'(C_1(X))] $$

For a state independent unit payoff $P_1(X)=1$ we have a bond which determines the discount factor in the economy as $$ DF = E[M(X)] \text{ for } M(X) = \delta \frac{U'(C_1(X))}{U'(C_0)} $$

For a generic payoff one can write the pricing formula as $$ P_0 = E[M(X) P_1(X)] = \int M(x) P_1(x) f(x) dx $$ The price is the expectation of the payoff times the stochastic discount factor aka marginal rate of substitution aka pricing kernel. Here $f(x)$ is the probability density of the state variable.

However, now we can define the function $$ q(x) = \frac{M(x)f(x)}{\int M(x)f(x)dx}=\frac{M(x)f(x)}{DF} $$ and confirm that this is a valid probability density function (non negative and integrates to one) of the state variable. This is obviously not the 'true' probability density, but it attaches probabilities to events and therefore can be used to compute an expectation (which we denote with $E^q$ to avoid confusion with the expectation under the true state probability density). Then we can write that when we use these preference-adjusted probabilities we can write $$ P_0=DF\ E^q[P_1(X)] $$ This is a discounted 'risk neutral' expectation, in the sense that preferences are not explicitly present. However preferences are hidden in the way 'risk neutral probabilities' are derived from the true ones.