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These parameters can then be used to price more complex instruments like callable bonds or callable reverse floaters. G. G. B. S. Anderssen, The linear functional strategy for improperly posed problems, in: J. Cannon, U. , Inverse Problems, Birkh¨auser, Basel, 11–30 (1986) [AFHS97] M. Avellaneda, C. Friedman, R. Holmes, D. Samperi, Calibrating Volatility Surfaces Via Relative-Entropy Minimization, Applied Mathematical Finance 4, 37–64 (1997) [BG67] G. Backus, F. Gilbert, Numerical applications of a formalism for geophysical inverse problems, Geophys.

Engl, G. Landl, Convergence rates for maximum entropy regularization, SIAM J. Numer. Anal. W. Engl, T. Langthaler, Maximum entropy regularization of nonlinear ill-posed problems, in: V. , Proceedings of the First World Congress of Nonlinear Analysts, Vol. W. Engl, T. W. Engl, H. Wacker and W. Zulehner, eds. W. Engl, A. Leitao, A Mann iterative regularization method for elliptic Cauchy problems, Numer. Func. Anal. Optim. 22, 861–884 (2001) [ELR96a] H. W. Engl, A. K. Louis and W. W. K. Louis, W. W.

K¨ ugler on these considerations, the value V of a European option can then be shown to satisfy the (by now famous) Black Scholes equation ∂V 1 ∂2V ∂V + σ2 S 2 − rV = 0. + rS ∂t 2 ∂S 2 ∂S (66) In order to completely describe the (direct) problem of calculating V for a given σ, the parabolic differential equation (66) backwards in time is augmented by an end condition and boundary conditions at zero and at infinity. For the easiest case of European call or put (together called ”vanilla”)) options, characterized by the terminal conditions V (T ) = max(S − K, 0) or V (T ) = max(K − S, 0), analytic solutions of (66) are available.

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