# 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. Commented Jul 17, 2018 at 6:26
• The game is to pounce on anyone who deviates from risk-neutral prices and take their lunch money. Commented Dec 2, 2018 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. Commented Dec 3, 2018 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. Commented Mar 19, 2019 at 13:14

### Risk-Neutral Pricing as a Game-Theoretic Equilibrium

The risk-neutral pricing framework from the Fundamental Theorem of Asset Pricing can be interpreted using game theory, where market participants act strategically to reach an equilibrium price.

#### Market as a Game:

1. Players:

• Investors: Optimize portfolios based on expected returns and risk.
• Market Makers: Provide liquidity and balance order flows.
• Arbitrageurs: Eliminate arbitrage opportunities to ensure market efficiency.
2. Strategies:

• Investors: Choose asset allocations.
• Arbitrageurs: Exploit price discrepancies.
3. Equilibrium Concept:

• Nash Equilibrium: Derivative prices reach a state where no player can benefit by changing their strategy unilaterally. This is similar to the absence of arbitrage, where prices reflect all available information.
4. Risk-Neutral Measure $$( \mathbb{Q} )$$:

• It acts as a probability measure where expected returns equal the risk-free rate. This measure is consistent with equilibrium pricing and reflects market consensus, ensuring no arbitrage opportunities exist.

#### Interpretation:

• Efficiency: The risk-neutral measure represents market efficiency, where prices are aligned to prevent arbitrage.
• Equilibrium Strategy: The risk-neutral price is an equilibrium strategy, aligning with rational expectations where market participants price risk consistently.

#### Conclusion:

In this game-theoretic perspective, the risk-neutral pricing mechanism is akin to a Nash equilibrium in a strategic game among market participants, leading to efficient, arbitrage-free pricing of derivatives.