I am currently working on Benth and Benth "THE VOLATILITY OF TEMPERATURE AND PRICING OF WEATHER DERIVATIVES" and i am stuck at following paragraph at page 10, which is about risk-neutral measures in incomplete markets of weather derivatives:

"In order to derive a more explicit expression for the futures price, we need to specify the risk-neutral probability $Q$. Since temperature is not a storable commodity, the futures contracts can not be hedged and the market is therefore incomplete. A risk-neutral probability is by definition a probability measure $Q\sim P$ such that all tradeable assets in the market are martingales after discounting. Thus, all equivalent probabilities $Q$ will become risk-neutral probabilities. We specify a sub-family of probability measures $Q$ using the Girsanov transform: [...]"

I do not understand why all equivalent probabilities are risk-neutral probabilities in incomplete markets?!

Thank you very much for your help!


1 Answer 1


That is my solution:

The price of weather derivatives cannot simply be calculated using a hedging strategy, since the underlying is not tradable. We are therefore in an incomplete market. If we restrict the market to weather derivatives, the treasury bill with a risk-free interest rate $r\in\mathbb{R}$ is the only tradeable asset in the market. The discounted treasury bill price is $1$ and it is therefore a trivial martingale with respect to all equivalent measures. Thus, all equivalent probability measures $\mathcal{Q}\sim \mathcal{P}$ will become risk neutral probabilities.


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