How Incomplete Markets Leave Risk Prices Underdetermined

# Is the market price of risk subjective?


# Is the market price of risk subjective?












Assume a model defined in a incomplete market. Assume that model contains some parameters $\theta_1 ... \theta_k$. In a risk neutral setting, one more parameter appears into the modeldynamics which is $\lambda$ (also known as the market price of risk). The market price of risk is unobservable. Does that mean it is subjective? If I am less risk averse than you, is my $\lambda$ then different from yours?

## Answer by Stéphane (score 1)

https://quant.stackexchange.com/a/54017

In an incomplete market model, some risk factors are not traded. Hence, while you have a clear idea of how the market rewards traded risk factors (assuming parameter values and the data generating process are known), you do not know exactly how untraded risk factors are priced.

One way to close the loop and fall on a unique price is to posit a representative investor and use the pricing kernel implied by their preferences. Another is to motivate a choice of a pricing kernel using reduced-form evidence. Either way, your problem with market incompleteness is that you don't have enough assets to infer a uniquely correct way of putting a price on all sources of risk, even in the absurdly optimistic setting where you know the "true" model.

Heston (1993) invokes a partial equilibrium consumption-based model to pick a pricing kernel. His model involves stochastic volatility, hence as long as the volatility disturbances aren't perfectly correlated with the equity disturbances, there is an infinite number of valid prices you can put on volatility risk using prices on options, the underlying and a risk-free proxy. But, if you are disposed to say that Heston's consumption-based argument is sound (i.e., you limit yourself to a sufficiently narrow class of equilibrium models), you have just one price and one risk-neutral measure that works. A similar thing applies to GARCH-based option pricing models as in Duan (1995), Heston and Nandi (2000), etc. with the exception that here the incompleteness stems from discrete dynamics.

Now, if you want an example of a reduced-form argument, Christoffersen, Heston and Jacobs (2013) used evidence on the shape of the pricing kernel on the S&P500. If you recall the Radon-Nikodym derivative, you essentially are looking at the ratio of the risk-neutral to the physical density. You can estimate the former using the Breeden and Litzenberger (1976) results and, for the later, you could use a parametric GARCH model and empirical errors as did, for example, Rosenberg and Engle (2002). Taking the difference of their logarithm on a year per year basis, you find clouds of points that draw a U -- i.e., your pricing kernel should be exponentially quadratic as opposed to exponentially affine as in a classical consumption-based model. CHJ2013 use it to motivate a quadratic pricing kernel for GARCH-based models. So, that's another way to pin down unique prices.

Now, from the broader perspective of what really happens in financial markets, if you think that equilibrium models are a good approximation or a good description of what is going on (in that, we just have the wrong equilibrium models, but the idea is sane), then there is a unique (though perhaps time-varying) price for every single sources of risk. The hope is that the choices you make in incomplete market models is "close enough" to that "true" price to be usable. If you rather think that those models are very crude description of otherwise chaotic and stochastic responses that emerge from the micro structure of markets, it's not obvious that anything you see in quantitative finance even makes sense. For starters, adding noise in chaotic systems can easily generate fat tailed conditional distributions and those distributions do not have finite variance, let alone finite covariances.