研究期間：10208~10307;State price densities for financial options over multiple maturities and with multiple underlying assets The state price density (SPD) is the density function under the equivalent martingale for derivative pricing and is also known as the risk-neutral density. SPD is usually calibrated from frequently traded options, such as single-asset European options at a certain maturity, and can be used for pricing illiquid options, risk management, and so on. Current studies focus on the SPD estimation based on options having the same maturities. Little has been known about the SPD estimation using options over multiple maturities or options with multiple underlying assets. This research proposal aims at proposing efficient statistical method for these two cases. For the former case, we intend to generalize the random tree based on Teng (2010) , so that a unique tree can be calibrated using options over multiple maturities. For the latter case, we consider hierarchical Archimedean copulas (HAC) for dependence modeling among multiple assets. Particularly, we are interested in the structure learning of the HAC, because current methods remain dissatisfied.