Pricing Options With Curved Boundaries1

This paper provides a general valuation method for the European options whose payoff is restricted by curved boundaries contractually set on the underlying asset price process when it follows the geometric Brownian motion. Our result is based on the generalization of the Levy formula on the Brownian motion by T. W. Anderson in sequential analysis. We give the explicit probability formula that the geometric Brownian motion reaches in an interval at the maturity date without hitting either the lower or the upper curved boundaries. Although the general pricing formulae for options with boundaries are expressed as infinite series in the general case, our numerical study suggests that the convergence of the series is rapid. Our results include the formulae for options with a lower boundary by Merton (1973), for path-dependent options by Goldman, Sossin, and Gatto (1979), and for some corporate securities as special cases.     

What causes business cycles? Analysis of the Japanese industrial production data

We explore what causes business cycles by analyzing the Japanese industrial production data. The methods used are spectral analysis and factor analysis. Using the random matrix theory, we show that two largest eigenvalues are significant. Taking advantage of the information revealed by disaggregated data, we identify the first dominant factor as the aggregate demand, and the second factor as inventory adjustment. They cannot be reasonably interpreted as technological shocks. We also demonstrate that in terms of two dominant factors, shipments lead production by four months. Furthermore, out-of-sample test demonstrates that the model holds up even under the 2008–2009 recession. Because a fall of output during 2008–2009 was caused by an exogenous drop in exports, it provides another justification for identifying the first dominant factor as the aggregate demand. All the findings suggest that the major cause of business cycles is real demand shocks.