# Is there a relationship between Risk Neutral Pricing framework and Nash Equilibria?

Based on the Fundamental Theorem of Asset Pricing, the risk neutral price of a contingent claim on an asset in a liquid, arbitrage free market can be determined by switching to an equivalent $Q-$ measure: $$P(t,T)=E^{Q}_t [D(t,T)g(S(T))],$$ where $P(t,T)$ is the risk neutral price at time $t$, $S(t)$ is the asset process, $g(.)$ is the contingent claim and $D(t,T)$ the discount process.

My question (admittedly a bit vague): Is there an an equivalent interpretation of this framework from the game theory point of view which yields the "risk neutral price" as some sort of equilibrium strategy of the market participants?

• Great question! Just a quick thought to it. Risk-neutral pricing is equivalent to pricing with a stochastic discount factor (SDF). Now, if markets are complete and an SDF exists, it also is unique. If it is unique, the pricing problem can be expressed from the point of view of a representative investor &mdash; one investor. That's where the equilibrium setting would sort of break down, as there are no other players left except for that guy. – Igor Pozdeev Jul 17 '18 at 6:26
• The game is to pounce on anyone who deviates from risk-neutral prices and take their lunch money. – Lepidopterist Dec 2 '18 at 16:01
• @Lepidopterist, right. However, it seems to me that at the market opening, the risk neural price (market implied) has not been reached yet. – Ali Fathi Dec 3 '18 at 15:14
• Have a look at this paper. I believe that this paper partially answers your question: Nau, R. (2015). Risk-neutral equilibria of noncooperative games. Theory and Decision, 78(2), 171-188. – Wiles01 Mar 19 at 13:14