大阪大学　数理・データ科学教育研究センター
Center for Mathematical Modeling and Data Science,Osaka University

Implied volatility from local volatility: A path integral approach

Tai-Ho Wang (Baruch College, The City University of New York)

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2. セミナー・集中講義
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4. Implied volatility from local volatility: A path integral approach

大阪大学 金融・保険セミナーシリーズ 第48回(CSFI-CRESTジョイントセミナー)

Implied volatility from local volatility: A path integral approach

Tai-Ho Wang (Baruch College, The City University of New York)

We derive an exact Brownian bridge representation for the transition density in a local volatility model, which then leads to an exact expression for the transition density in terms of a path integral. In the time homogeneous case, we recover the heat kernel expansion by Taylor-expanding around the most-likely-path. Repeating the same procedure in the time inhomogeneous case leads to a new and natural approximation to the transition density which differs from the conventional heat kernel expansion. We show that by suitably approximating the path integral representation, we recover the results obtained in our previous work. Applying the methodology to higher dimension models we obtain a Bessel bridge representation for the heat kernel in the hyperbolic space. In particular, the closed form expression in the case of 3 dimensional hyperbolic space is recovered.