We attempt to answer two questions in this paper: (i) How did jumps in equity returns change after the 2008-2009 financial crisis-in particular, were there significant changes in jump rates or in jump sizes, or both? (ii) Can the performance of affine jumpdiffusion models be improved if jump sizes are larger, i.e., jumps with tails heavier than those of the normal distribution? To answer the second question, we find that a simple affine jump-diffusion model with both stochastic volatility and double-exponential jumps fits both the S&P 500 and the NASDAQ-100 daily returns from 1980 to 2013 well; the model outperforms existing ones (e.g., models with variance-gamma jumps or jumps in volatility) during the crisis and is at least comparable before the crisis. For the first question, on the basis of the model and the data sets, we observe that during the crisis, negative jump rate increased significantly, although there was little change in the average negative jump size.

We propose a spatial capital asset pricing model and a spatial arbitrage pricing theory (S-APT) that extend the classical asset pricing models by incorporating spatial interaction. We then apply the S-APT to study the comovements of eurozone stock indices (by extending the Fama-French factor model to regional stock indices) and the futures contracts on S&P/Case-Shiller Home Price Indices; in both cases, spatial interaction is significant and plays an important role in explaining cross-sectional correlation.

First-passage times (FPTs) of two-dimensional Brownian motion have many applications in quantitative finance. However, despite various attempts since the 1960s, there are few analytical solutions available. By solving a nonhomogeneous modified Helmholtz equation in an infinite wedge, we find analytical solutions for the Laplace transforms of FPTs; these Laplace transforms can be inverted numerically. The FPT problems lead to a class of bivariate exponential distributions which are absolute continuous but do not have the memoryless property. We also prove that the density of the absolute difference of FPTs tends to ∞ if and only if the correlation between the two Brownian motions is positive.

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