Change of Numeraire

In this chapter we discuss how a suitable change of numeraire and the corresponding change of martingale measure, can simplify the computation of pricing formula for financial derivatives. We derive a general formula for the likelihood process related to an arbitrary numeraire, and we identify the corresponding Girsanov transformation. As an example, we compute the price of an exchange option. In particular we study the class of forward measures related to zero coupon bonds and we derive a general option pricing formula. As an application of the general theory we also study the so-called numeraire portfolio.

PRICING OF UNEMPLOYMENT INSURANCE PRODUCTS WITH DOUBLY STOCHASTIC MARKOV CHAINS

This paper provides a new approach for modeling and calculating premiums for unemployment insurance products. The innovative modeling concept consists of combining the benchmark approach with its real-world pricing formula and Markov chain techniques in a doubly stochastic setting. We describe individual insurance claims based on a special type of unemployment insurance contracts, which are offered on the private insurance market. The pricing formulas are first given in a general setting and then specified under the assumption that the individual employment-unemployment process of an employee follows a time-homogeneous doubly stochastic Markov chain. In this framework, formulas for the premiums are provided depending on the ℙ-numéraire portfolio of the benchmark approach. Under a simple assumption on the ℙ-numéraire portfolio, the model is tested on its sensitivities to several parameters. With the same specification the model's employment and unemployment intensities are estimated on public data of the Federal Employment Office in Germany.

How Valuable are the Commodity Assets to Investors?

This study examines how commodity assets affect investors. Our main findings can be summarized as follows. First, the Sharpe ratio of commodity indexes is higher than that of stocks and bonds over the last ten years. Second, commodity (traditional) assets are positively (negatively) related with inflation, which implies that commodity assets provide better hedge against inflation. Third, a break-even analysis indicates that including commodity assets in diversified portfolio of stocks and bonds enhances the performance of the portfolio. Fourth, the numeraire portfolio approach of Hentschel et al.(2002) shows that, to some extent, there are gains by including commodity assets in a portfolio of stocks and bonds. For example, transaction cost of 0 to 92 basis points would keep a log-utility investor from including the Rogers International Commodities Index (RICI) in one’s portfolio. In sum, commodity assets enhance the performance of portfolio, and the performance gain is especially pronounced during the bear stock market.