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Subject: Lévy processes × WGL Institute: WIAS ×

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Now showing 1 - 2 of 2
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    Unified signature cumulants and generalized Magnus expansions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Friz, Peter; Hager, Paul; Tapia, Nikolas
    The signature of a path can be described as its full non-commutative exponential. Following T. Lyons we regard its expectation, the expected signature, as path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant. We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants, and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions; with motivation ranging from financial mathematics to statistical physics. From an affine process perspective, the functional relation may be interpreted as infinite-dimensional, non-commutative (``Hausdorff") variation of Riccati's equation. Many examples are given.
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    Efficient and accurate log-Levy approximations to Levy driven LIBOR models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Papapantoleon, Antonis; Schoenmakers, John G.M.; Skovmand, David
    The LIBOR market model is very popular for pricing interest rate derivatives, but is known to have several pitfalls. In addition, if the model is driven by a jump process, then the complexity of the drift term is growing exponentially fast (as a function of the tenor length). In this work, we consider a L´evy-driven LIBOR model and aim at developing accurate and efficient log-L´evy approximations for the dynamics of the rates. The approximations are based on truncation of the drift term and Picard approximation of suitable processes. Numerical experiments for FRAs, caps, swaptions and sticky ratchet caps show that the approximations perform very well. In addition, we also consider the log-L´evy approximation of annuities, which offers good approximations for high volatility regimes.