Why All Equivalent Measures Are Risk-Neutral for Weather Derivatives
Summary
The answer explains why weather derivatives can have many risk-neutral probability measures when the underlying temperature is not tradable. Without a tradable underlying, the contracts cannot be replicated or hedged using temperature itself, so the market is incomplete and no unique pricing measure is selected by replication.
If the market is restricted to weather derivatives and a risk-free treasury bill, the discounted bill price is constant. A constant price process is a martingale under every probability measure equivalent to the physical measure. Consequently, every such equivalent measure satisfies the risk-neutral condition for the assets in this restricted market. This reasoning depends on the stated market scope; adding other traded assets or constraints could narrow the set of admissible measures.
Key ideas
- Temperature is not directly tradable, so weather derivatives cannot be hedged by trading their underlying.
- Incomplete markets can admit multiple risk-neutral measures.
- With only a risk-free bill as a traded asset, its discounted value is constant.
- A constant discounted asset price is a martingale under every equivalent probability measure.
- The conclusion depends on which assets are included in the market.
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# Benth: Risk-neutral measure in incomplete markets
# Benth: Risk-neutral measure in incomplete markets
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: [...]"
Problem: I do not understand why all equivalent probabilities are risk-neutral probabilities in incomplete markets?!
Thank you very much for your help!
## Answer by Valentin (score 1)
https://quant.stackexchange.com/a/78624
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.