Thus far this book has described the use of actuarial principles for pricing weather derivatives. This chapter will now discuss the application of arbitrage pricing ideas. The main difference between actuarial and arbitrage pricing theories is that actuarial pricing is based on diversification while arbitrage pricing is based on hedging. The anticipation of following a hedging strategy can affect the prices we charge for weather contracts.

A useful context in which to explain arbitrage pricing is that of equity options. The issuer of an equity option can trade the underlying equity in order to hedge his/her risk. If the underlying equity market is liquidly traded, then many such hedging transactions can be performed between the issuance and expiry of the option, and the risk will be hedged almost perfectly. Many hedging trades are necessary because the risk from the equity option depends on the share price and hence varies as the share price fluctuates in time. The cost of the hedging combined with the distribution of payoffs on the option determines the price initially charged for the option, and this price, which we call the ‘arbitrage price’, is generally different from the price that would be charged if no such dynamic hedging were to be undertaken and the option were priced actuarially. In particular, the arbitrage price is not the expectation (although we will see below that it is possible to recover the fact that the price is the expectation, but only by redefining ‘expectation’).