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... 34 different strikes as in the 1997 literature), and in that sense their paper is closer in spirit to this one; see also the preprints by M. Davis and J. Obloj [8] and by M. Schweizer and J. Wissel =-=[31]-=-. Finally, we also wish to mention the recent and very interesting related work of R. Carmona and S. Nadtochiy [3], which is similar in spirit to the work of Derman, Dupire and others (in the continuo...

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... martingale diffusion which is consistent with observed call prices. (Alternative, non-diffusion models also exist, see for example, Madan and Yor [9].) The recent literature (eg Schweizer and Wissel =-=[13]-=-) has included attempts to extend the theory to allow for a stochastic local volatility surface. However, it relies on the knowledge of a double continuum of option prices, which are smooth. In contra...

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...eath, Jarrow, and Morton [9] to equity markets. An early paper is this direction is by Schönbucher [14]; see the papers of Carmona and Nadtochiy [2], Jacod and Protter [10], and Schweizer and Wissel =-=[16, 17]-=- for more recent advances. Indeed, for the purposes of hedging exotic options with portfolios of calls, it is necessary to have the joint dynamics of all the call prices available. In particular, supp...

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...NADTOCHIY for the implied volatility surface is rather natural, see also fr example [5, 6, 9], deriving no-arbitrage conditions is highly technical and could be only done in specific particular cases =-=[8, 14, 15, 16]-=-. In the present paper, we streamline some of the recent results on local volatility dynamics by employing an infinite dimensional stochastic analysis approach as advocated by the pioneering work of L...

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...n [16], [9] and [10]. This idea was then developed more thoroughly in the works of Schönbucher [32], Schweizer and Wissel [34] and Jacod and Protter [21], but the recent works of Schweizer and Wissel =-=[33]-=- and Carmona and Nadtochiy [3], [2] are more in the spirit of the market model approach that we advocate here. The first hurdle on the way to creating a stochastic dynamic model for the call price sur...

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...ction (case considered by Jacod and Protter in [7] and Schweizer and Wissel in [12]) h∈DtTANGENT MODELS 5 • Dt = {S, Ct(T, K); K > 0} for some fixed T > t (case considered by Schweizer and Wissel in =-=[13]-=-) • Dt = {S, Ct(Ti, Kj); i = 1, · · · , m, j = 1, · · · , n} (case considered by Schweizer and Wissel in [?]) • Dt = {S, Ct(T, K); T > t, K > 0} (case considered by Cont and Durrleman in [?] and Carmo...

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...d the square of the forward implied volatility as the parametrization, which is analogous to the forward interest rate in the classic HJM model. They later extended the work to a multi-strike case in =-=[42]-=-. In [42], they recognized that the no-arbitrage conditions imposed on previous parametrization could be too strict to allow for a feasible solution for the multi-strike case, so a new parametrization...

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...t(x,K) = ∂ � x ˆσ 2 t (x,K) ∂x � = ∂ ∂x (ξt(t + x,K)). (5) The idea of use such a variable in the implied volatility models was introduced for the first time, as far as we know, in [14]. In the works =-=[14,15]-=- the authors use different techniques to deal with problems strictly related to the our. They use some results about strong solutions for functional SDE proven in [17] (see also [11]) to study the cas...