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On a Heath-Jarrow-Morton approach for stock options

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In the Heath-Jarrow-Morton (HJM) approach in interest rate theory the whole forward rate curve rather than the short rate is considered as state variable for a stochastic model. Absence of arbitrage then leads to consistency and drift restrictions, in particular the HJM drift condition. Several attempts have been made to transfer this idea to options on a stock, cf. e.g. by Schönbucher (1999), Schweizer & Wissel (2008), Carmona & Nadtochiy (2009), Jacod & Protter (2006). Here, the underlying stock plays the role of the short rate. The implied volatility surface or a reparametrisation serves as state variable and hence as counterpart of the forward rate curve in the classical framework of HJM . Our approach to this problem resembles Carmona & Nadtochiy (2009) in that we try to preserve main features of the HJM setup. However, it is based on a different parametrisation or codebook, which allows to simplify both theory and application.

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