# Does risk-neutral measure have anything to deal with risk-neutrality in utility theory?

Or simply: why do we call equivalent martingale measures as risk-neutral measures?

In the utility or game theory, when we consider a person's preferences to certain outcomes, we often deal with the utility functions. For example, if we consider an investor with a utility function $U$ whose return on a portfolio $\Pi$ is $x_\Pi$, we assume that he choose a portfolio that maximizes his expected utility $$\Pi^* \quad\text{ such that }\quad\mathsf E U(x_{\Pi^*}) = \sup_{\Pi}\mathsf E U(x_\Pi).$$ In particular, we say that an investor is risk-averse (risk-seeking) whenever $U$ is concave (convex). We say that an investor is risk-neutral when $U(x) = ax + b$ is an affine function.

The risk-neutral valuation - taking expectations w.r.t. martingale measures equivalent to the real-world ones - is used in quant finance a lot for the pricing purposes. I do understand the theory behind this method, and the relation with non-arbitrage arguments. I wonder though, whether there is any relation with the risk-neutrality as in the paragraph above.

I thought of the following idea: let us think of a fair price for a contract (when we write it) as the highest one at which the agent will buy it. The agent $A$ with utility $U_A$ and expectation of prices $\mathsf E_A$ has to make a choice between the zero utility (when he does not buy contract) and $$\mathsf E_AU_A(\mathrm e^{-rT}C_T - C_0)$$ where $T$ is maturity of the contract, $r$ is a rate used to compute present value of future cashflow, $C_T$ is the payoff of the contract, $C_0$ is the price of the contract. Hence, we need to solve the equation $$\mathsf E_AU_A(\mathrm e^{-rT}C_T - C_0) = 0$$ with unknown $C_0$. Assuming that agent is risk-neutral, we obtain $$C_0 = \mathrm e^{-rT}\cdot\mathsf E_A(C_T).$$ At the same time, pricing using the $\Delta$-hedging in the Black \& Scholes framework gives us $$C_0 = \mathrm e^{-rT}\cdot\mathsf E_Q(C_T)$$ where $Q$ is a risk-neutral measure. Hence, if we assume that our agent is risk neutral, then his expectations (at least at any given time $T$) have to be given exactly by the measure $Q$.

• your notation is confusing me a bit do you mean: The investor looks for a $\Pi^*$ so that $$\mathbb{E}[ U(x_{\Pi^*})]=\max_{\Pi} \mathbb{E}[U(x_\Pi)]$$ Mar 26, 2014 at 17:57
• @Probilitator: indeed, I've modified this part - better now?
– Ilya
Mar 26, 2014 at 18:46
• yep everything fine now :D Mar 26, 2014 at 21:29

An agent with utility function $U$ values a final position $X_T$ by $E\left[U(X_T)\right]$. You can think of this as a function mapping random variables to $\mathbb{R}$, $X_T \mapsto E \left[U(X_T)\right]$.
A risk-neutral mapping should be a linear mapping of the kind above. In other words, $f$ should map some space of random variables to $\mathbb{R}$, and satisfy $$a \cdot f(X_T) + b \cdot f(Y_T) = f(a \cdot X_T + b \cdot Y_T)$$ for scalar $a,b$.
Using a martingale measure, the valuation rule is $X_T \mapsto E \left[ \frac{dQ}{dP} X_T \right]$. Note that this is a linear map. Therefore, it's like a risk-neutral agent is pricing the positions.
• Thanks, I'm assuming in your case $E = E_P$: the expectation over a market measure. In such case, the map $X_T \mapsto \mathsf E_P[X_T]$ is linear as well, so nothing distincts it from the case of a martingale measure.