# EMM in incomplete markets

The simply put question is as follows: do we need to restrict ourselves to EMM exclusively when pricing European contingent claims (=option payoffs) even if markets are incomplete?

In particular, a related question just asks which EMM is better, rather than shall it be used at all? Here are some of my thoughts. For simplicity imagine that we have a market model which is not very technical, e.g. discrete-time or even tree-model. I assume that at least one EMM exists, but it may not be unique.

As far as I understood from work by Kreps, Harrison and Pliska (not sure I did completely understood it), if a contingent claim is attainable (i.e. there exists an admissible strategy with the same final payoff) then its price has to be computed as an expectation w.r.t. EMM, and in this case all EMM will give the same price over such contingent claim. Otherwise, there exists an arbitrage opportunity.

Regarding all other claims (that is, non-attainable) I've seen 2 approaches in the literature.

1. Still do $\Delta$-hedging, e.g. in Wilmott. We hedge one derivative $V_1$ with another one $V_2$, arrive to BS-like equation with an unknown function $\lambda$ - a market price of risk. This is done e.g. in stochastic volatility models, and stochastic interest rate models. One possible advantage of this approach is that $\lambda$ is the only unknown, and once fixed it allows pricing all derivatives in a consistent way.

2. Define an upper price (via upper hedge) and lower price (via lower hedge), I've seen it in Shiryaev. It appears that the upper price is exactly the maximum of expectations over all EMM, and I believe the similar would hold for the lower price.

W.r.t. mentioned approaches by question reads as follows.

1. Is pricing via market price of risk takes expectations over EMM, that is does each choice of $\lambda$ leads to some EMM? And vice-versa, does for any EMM there exists a corresponding $\lambda$? If not, why would we rely upon a $\Delta$-hedging in such case? The way it worked for complete markets may not apply in incomplete.

2. Shiryaev shows that if we price a single derivative outside of the interval formed by lower/upper prices, immediately an arbitrage opportunity arises. This argument is convincing enough not to use non-martingale measures for the pricing. However, can we price all contingent claims on the market within their intervals in a consistent way?

-