# Physical Measure in Weather Derivatives — Hull

In Hull's 8ed., he states in Chapter 33, Energy and Commodity Derivatives,

The second part of the chapter considers weather and insurance derivatives. A distinctive feature of these derivatives is that they depend on variables with no systematic risk. For example, the expected value of the temperature at a certain location or the losses experienced due to hurricanes can reasonably be assumed to be the same in a risk- neutral world and the real world. This means that historical data is potentially more useful for valuing these types of derivatives than for some others.

(Boldface mine) He elaborates, in Section 33.7,

One distinctive feature of weather and insurance derivatives is that there is no systematic risk (i.e., risk that is priced by the market) in their payoffs. This means that estimates made from historical data (real-world estimates) can also be assumed to apply to the risk-neutral world. Weather and insurance derivatives can therefore be priced by

1. Using historical data to estimate the expected payoff

2. Discounting the estimated expected payoff at the risk-free rate

I'm still unclear on this. Hull is saying it's okay to use the physical probability measure in derivatives pricing for weather derivatives. His argument seems vague, however --- what does "no systematic risk" mean, and why does this imply the physical measure is valid? Also, what does this say about the market price of risk?

I am giving my interpretation and thoughts on Hull's words.

Consider a derivative of maturity $T$ with price process $(V_t)_{t \geq 0}$ written on a tradable asset with price process $(S_t)_{t \geq 0}$: for example you can take the standard example of a call option written on a stock. Letting AOA be the Absence of Arbitrage Opportunities (AOA) assumption, $\mathbb{Q}$ the risk-neutral measure and $D(t,T)$ the discount factor from $T$ to $t$, in the standard framework of derivative pricing theory we loosely have the following implication $-$ this is of course not fully rigorous:

$$\left[(S \text{ is tradable}) + (\text{AOA})\right] \quad \Rightarrow \quad V_t=\mathbb{E}^{\,\mathbb{Q}\,}[D(t,T)V_T|\mathcal{F}_t]$$

It is because the asset is tradable and that we assume there are no arbitrage opportunities that we can price the derivative as the discounted, expected payoff under the risk-neutral measure.

Now, if you are considering an underlying process $(X_t)_{t \geq 0}$ which is not tradable $-$ e.g. temperature, wind speed, etc. $-$ then the logical implication above breaks down and you should no longer be able to prove that the price of the derivative is its discounted, expected payoff under $\mathbb{Q}$: even if there are arbitrage opportunities, you cannot exploit them because the underlying asset is not tradable. In such a situation, pricing according to historical data is a reasonable choice.

The non-systematic risk part simply means that weather should be totally uncorrelated to financial markets: they are two different worlds, and the systematic risk that all financial assets (stocks, bonds, etc.) bear is not present in weather. Simply think what would happen is some abnormal, highly mortal, extremely contagious, "catastrophe-film-like" epidemic started affecting humans: financial assets around the world would become highly volatile due to the uncertain future of the human race, but you can be assured that weather wouldn't be affected by this. However, I struggle to see a straightforward connexion between the absence of systemic risk and the appropriateness of pricing under the real-world measure. Indeed, you can have derivatives that clearly have systemic risk but that you can price under the real-world measure because the underlying risk is not tradable: please refer to my question "Change of measure's impact on parameter value" for an example $-$ specifically the $1^{\text{st}}$ paragraph of section "Remarks" in @Quantuple's answer.

As a last comment, please also consider the historical dimension of the problem at hand: initially, primitive processes like temperature and wind speed might not be tradable; derivatives written on them might be issued, priced on the real-world measure. However, my view is that, once this derivatives are out there in the market, arbitrage opportunities might arise if newly issued derivatives are written at prices which are not consistent with the previously issued derivatives. Hence, although a weather derivatives market might start by pricing under the real-world measure, I think that the development of the market should end up enforcing participants to price under the risk-neutral measure to avoid arbitrage: after all, pricing models are a form of language between market participants, and you better speak the same language as others to be understood (and avoid being cheated!).