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We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions. We demonstrate that complex saddle points have a natural interpretation in terms of the Picard-Lefschetz theory. Motivated in part by the semi-classical expansion of QCD with adjoint matter on R3 ⇥ S1, we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic non-degenerate minima. We find exact finite action non-BPS bounce and bion solutions to the holomorphic Newton equations. We find not only real solutions, but also complex solution with non-trivial monodromy, and finally complex multi-valued and singular solutions. Complex bions are necessary for obtaining the correct non-perturbative structure of these models. In the supersymmetric limit the complex solutions govern the ground state properties, and their contribution to the semiclassical expansion is necessary to obtain consistency with the supersymmetry algebra. The multi-valuedness of the action is either related to the hidden topological angle or to the resurgent cancellation of ambiguities. We also show that in the approximate multi-instanton description the integration over the complex quasi-zero mode thimble produces the most salient features of the exact solutions. While exact complex saddles are more difficult to construct in quantum field theory, the relation to the approximate thimble construction suggests that such solutions may be underlying some remarkable features of approximate bion saddles in quantum field theories.

This simple question does not have a simple answer. The boundary of such an interdisciplinary area is always moot and any attempt to give a formal definition is unlikely to be successful. Broadly speaking, financial econometrics is to study quantitative problems arising from finance. It uses statistical techniques and economic theory to address a variety of problems from finance. These include building financial models, estimation and inferences of financial models, volatility estimation, risk management, testing financial economics theory, capital asset pricing, derivative pricing, portfolio allocation, risk-adjusted returns, simulating financial systems, hedging strategies, among others. Technological invention and trade globalization have brought us into a new era of financial markets. Over the last three decades, enormous number of new financial products have been created to meet customers demands. For example, to reduce the impact of the fluctuations of currency exchange rates on a firms finance, which makes its profit more predictable and competitive, a multinational corporation may decide to buy the options on the future of foreign exchanges; to reduce the risk of price fluctuations of a commodity (eg lumbers, corns, soybeans), a farmer may enter into the future contracts of the commodity; to reduce the risk of weather exposures, amuse parks (too hot or too cold reduces the number of visitors) and energy companies may decide to purchase the financial derivatives based on the temperature. An important milestone is that in the year 1973, the worlds first options exchange opened in Chicago. At the very same year, Black and Scholes (1973