On the Spanning Property of Risk Bonds Priced by Equilibrium

We propose a method of pricing financial securities written on nontradable underlyings such as temperature or precipitation levels. To this end, we analyze a financial market where agents are exposed to financial and nonfinancial risk factors. The agents hedge their financial risk in the stock market and trade a risk bond issued by an insurance company. From the issuer's point of view the bond's primary purpose is to shift insurance risks related to noncatastrophic weather events to financial markets. As such, its terminal payoff and yield curve depend on an underlying climate or temperature process whose dynamics are independent of the randomness driving stock prices. We prove that if the bond's payoff function is monotone in the external risk process, it can be priced by an equilibrium approach. The equilibrium market price of climate risk and the equilibrium price process are characterized as solutions of nonlinear backward stochastic differential equations (BSDEs). Transferring the BSDEs into partial differential equations (PDEs), we represent the bond prices as smooth functions of the underlying risk factors. Our analytical results make the model amenable to a numerical analysis.