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- ItemOption pricing in the moderate deviations regime(Oxford [u.a.] : Wiley-Blackwell, 2017) Friz, Peter; Gerhold, Stefan; Pinter, ArpadWe consider call option prices close to expiry in diffusion models, in an asymptotic regime (“moderately out of the money”) that interpolates between the well-studied cases of at-the-money and out-of-the-money regimes. First and higher order small-time moderate deviation estimates of call prices and implied volatilities are obtained. The expansions involve only simple expressions of the model parameters, and we show how to calculate them for generic local and stochastic volatility models. Some numerical computations for the Heston model illustrate the accuracy of our results.
- ItemVarieties of Signature Tensors(Cambridge : Cambridge Univ. Press, 2019) Améndola, Carlos; Friz, Peter; Sturmfels, BerndThe signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.