In June of 2001, Tokyo Electric Power Company (TEPCO) and Tokyo Gas Supply Company (TGSC) made a zero-cost risk swap contract on the average temperature of August and September of 2001 in Tokyo for their adverse situations. This is an exchange of two options on the average temperature, by which TEPCO and TGSC can respectively hedge against a cold summer and a hot summer. In this paper we develop a theoretical framework to evaluate the fairness or rational of such a zero-cost weather risk swap, derive some conditions to check the rationality and empirically evaluate the fairness of the above temperature risk swap between the two companies. Since the situation with opsions or derivatives defined on such a weather index as the average temperature is essentially incomplete in any sense, we can not simply value the options by the no-arbitrage argument and hence it is not sufficient to compare the risk-neutral expected values. In fact, no risk neutral measure exists. In other words, we have to explicitly take a risk factor into account in the evaluation.
First we define the concept of full equivalence and moment equivalance of two options on a weather index and then derive some conditions for full and moment equivalences. Thirdly using the stochastic volatility model in Kariya, Endo and Ushiyama (2003), it is shown that the options in the TEPCO-TGSC risk swap are neither fully equivalent nor moment-equivalent as they stand.
