Could someone describe how risk-neutral probability measures are linked to arbitrage opportunities and also to whether or not a market is complete? I've been asked this question and am unsure how to answer it. It also asks me to 'state any results I make use of in my answer'. Not really too sure what this is getting at. All help will be appreciated!
Let's stick to a discrete market for simplicity. So, you have a finite number of states in this type of model.
The first fundamental theorem of asset pricing says that the absence of arbitrage in such markets imply the existence of (not necessarily unique) risk-neutral measure and vice-versa.
The reason it works in the second direction (the existence of a RN measure guarantees the market is artbitrage-free) is because this makes all asset prices evaluated using the risk-free asset as numéraire martingales under this measure. Intuitively, you cannot "game" a martingale process -- so, you can easily prove this by assumption an arbitrage exists and show that the process would then not be a martingale. The other direction is harder to prove.
The second fundamental theorem of asset pricing says that an artbitrage-free market with a risk-free asset is complete if and only if there exists a unique risk-neutral measure. In other words, arbitrage-free complete markets allow for one and only one risk-neutral measure -- and, conversely, if you can prove that a measure is unique, you have established that the market is complete.
How does this one play out? Intuitively, if there are enough different assets for each sources of risk to be traded, observed prices uniquely determine the change of measure. If you recall the Radon-Nikodym derivative used in the Black-Scholes-Merton model, you probably noticed that you're sort of putting a price on market risk. Specifically, you're pricing the diffusion part of the geometric Brownian motion describing the behavior of the price of the risky asset and this price happens to be the Sharp ratio. Why is it unique? One source of risk and one traded asset that completely exposes you to it.
Now, move to something like the Heston (1993) model. You have two imperfectly correlated standard brownian motions: one in the diffusion part of your risky asset and another in the diffusion part of the volatility process of that risky asset. You know how people price the equity-specific risk (the part of the first Brownian motion which is orthogonal to the other) because the risky asset is traded. What you do not know is how people price the variance-specific risk... For each price that you put here, you generate a different Radon-Nikodym derivative and, consequently, a different risk-neutral measure.
So, how do people get a unique price for Heston (1993)? Most people forget it, but Heston (1993) solved that conundrum by invoking a consumption-based model. This model has a representative investor whose first order conditions gives you a specific choice for that Radon-Nikodym derivatives. Said differently, if you make enough assumptions about preferences and individual constraints as you would do if you wrote down a complete equilibrium model, you will get a unique choice for the Radon-Nikodym derivative that is valid within that model. There is one and only way that representative investor is going to price risk given state variables. However, when you work from the martingale approach, you are trying to be agnostic about those things (as much as possible). In other words, in equilibrium models, preferences pin down a unique measure; but when you invoke theorems to price assets through a martingale approach, the reason you don't know what to do is that you're trying to be as agnostic as possible about preferences.
So, you get solutions like those of Heston (1993) that says a general type of model would give risk to the pricing kernel he chose to use and likewise for Duan (1995) in a GARCH-based option pricing model. Or, more recently, you can have recourse to more reduced-form arguments like those presented in Christoffersen, Heston and Jacobs (2013) in GARCH option pricing models. They use an exponentially quadratic picing kernel because a lot of different empirical approaches suggests this is generally what the ratio of risk-neutral to physical densities look like.