# Risk-neutral pricing in incomplete markets

I know that in order to use the risk-neutral valuation principle, that is, pricing options as their payoff function under a risk neutral measure, one has to have a complete market.

But in the context of incomplete markets: What does the risk-neutral price represent if the option is not replicable? Am I right in assuming it would still be arbitrage-free prices, but without the possibility to eliminate the risk of the option by a hedging strategy?

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Q: What does the risk-neutral price represent if the option is not replicable?

In an incomplete market, there is no unique martingale measure but instead a set $Q$ of equivalent martingale measures. Consequently, there is an interval of arbitrage-free prices:

$\Big( inf_{\mathbf{Q} \in Q} E_{\mathbf{Q}}[DX], sup_{\mathbf{Q} \in Q} E_{\mathbf{Q}}[DX] \Big)$, where $E_{\mathbf{Q}}[DX]$ is the expected discounted payoff.

This interval may be too broad to serve us in our pricing or market making.

Q: Am I right in assuming it would still be arbitrage-free prices?

Here I will follow "Arbitrage Theory in Continous Time" by T. Björk. Björk focuses on an incomplete market where there are more random sources than there are traded assets. It may be otherwise the case that market frictions and liquidity issues forbid the replication of payoffs.

Quoting Björk (page 209):

In particular, if we take the price of one particular "benchmark" derivative as a priori given, then the prices of all other derivatives will be uniquely determined by the price of the benchmark

If we have two claims $Y$ and $Z$ whose dynamcs follow : $$\Pi(t;Y) = F(t, X(t))$$ $$\Pi(t,Z) = G(t, X(t)$$

We can still put together a portfolio based on $F$ and $G$.

$$dF = \alpha_F F dt + \sigma_F F dW$$ $$dG = \alpha_G G dt + \sigma_G G dW$$

where

$$\alpha_F = \frac{F_t + \mu F_x + \frac{1}{2} \sigma^2F_{xx}}{F}$$ $$\sigma_F = \frac{\sigma F_x}{F}$$

and correspondingly for $\alpha_G$ and $\sigma_G$

The dynamics of the self-financing portfolio then follow:

$$dV = V \big( u_F \alpha_F + u_G \alpha_G \big) dt + V \big( u_F \sigma_F + u_G \sigma_G \big) dW$$

We can make the portfolio locally riskless imposing: $$u_F + u_G = 1$$ $$u_F \sigma_F + u_G \sigma_G = 0$$

$$dV = V \frac{\alpha_G \sigma_F - \alpha_F \sigma_G}{\sigma_F - \sigma_G} dt$$

We know that:

$$\frac{\alpha_G \sigma_F - \alpha_F \sigma_G}{\sigma_F - \sigma_G} = r$$

Hence:

$$\frac{\alpha_F -r}{\sigma_F} = \frac{\alpha_G -r}{\sigma_G}$$

This means that all the derivatives products share the same market price of risk. Therefore, it may not be possible to replicate the payoff but it may be possible to extract information (the market price of risk) from the derivatives market.

What if no "benchmark" derivative is available? There are different strategies whose main goal is to bound the interval of allowed arbitrage free prices so that it would be possible to define a bid-ask spread. For example:

• Indifference prices: the idea is to identify a payoff that, under a preference function, would be indifferent for the investor
• Minimize the hedging risk with a portfolio under a specific utility function

See for example this and this.

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