Georgia Institute of Technology College of Sciences

At this time, in late September, 2011 the Dow Jones Industrial Average has just suffered its worst week since October, 2008; the Standard & Poor 500 Average just completed its worst week in the past five years; and financial markets worldwide under severe stress. We think it is timely to look at aspects of the role played by "financial engineering" (also known as "mathematical finance" or "quantitative finance") in the genesis of the on-going crisis. In this talk, we examine several structured investment vehicles (SIVs) devised by financial engineers and sold worldwide to many "investors". It will be seen that these SIVs were doomed from inception. In light of these results, we are dismayed by the mathematical models propagated over the past decade by financial ``engineers'' and ``experts'' in structured finance, and it heightens our fears about the durability of the on-going worldwide financial crisis.

We develop a nonparametric test to check whether the underlying continuous time process is a diffusion, i.e., whether a process can be represented by a stochastic differential equation driven only by a Brownian motion. Our testing procedure utilizes the infinitesimal operator based martingale characterization of diffusion models, under which the null hypothesis is equivalent to a martingale difference property of the transformed processes. Then a generalized spectral derivative test is applied to check the martingale property, where the drift function is estimated via kernel regression and the diffusion function is integrated out by quadratic variation and covariation. Such a testing procedure is feasible and convenient because the infinitesimal operator of the diffusion process, unlike the transition density, has a closed-form expression of the drift and diffusion functions. The proposed test is applicable to both univariate and multivariate continuous time processes and has a N(0,1) limit distribution under the diffusion hypothesis. Simulation studies show that the proposed test has good size and all-around power against non-diffusion alternatives in finite samples. We apply the test to a number of financial time series and find some evidence against the diffusion hypothesis.

"Tail risk" refers to an 'unholy trinity' Fat Tails, Micro Correlations, and Tail Dependence, that confound traditional risk analysis and are very much under-appreciated. The talk illustrates this with some punchy data. Of great interest is the question: when does aggregation amplify tail dependence? I'll show some data and new results. Tail obesity is not well defined mathematically, we have at least three definitions, leptokurtic, regularly varying and subexponential. A measure of tail obesity for finite data sets is proposed, and some theoretical properties explored.

We consider a stochastic volatility model with Levy jumps for a log-return process Z = (Z_t )_{t\ge 0}of the form Z = U + X , where U = (U_t)_{t\ge 0}is a classical stochastic volatility model and X = (X_t)_{t\ge 0} is an independent Levy process with absolutely continuous Levy measure \nu. Small-time expansion, of arbitrary polynomial order in time t, are obtained for the tails P(Z_t\ge z), z > 0 , and for the call-option prices E( e^{z+ Z_t| - 1), z \ne 0, assuming smoothness conditions on the Levy density away from the origin and a small-time large deviation principle on U. The asymptotic behavior of the corresponding implied volatility is also given. Our approach allows for a unified treatment of general payoff functions of the form \phi(x)1_{x\ge z} for smooth function \phi and z > 0. As a consequence of our tail expansions, the polynomial expansions in t of the transition densities f_t are obtained under rather mild conditions.