Calibration of the local volatility in a trinomial tree using Tikhonov regularization

Following an approach introduced by Lagnado and Osher (Lagnado R and Osher S 1997 J. Comput. Finance 1 13–25), we study the application of Tikhonov regularization to the financial inverse problem of calibrating a local volatility function from observed vanilla option prices. Moreover, we provide a unified treatment for this problem in two different settings: first, the generalized Black–Scholes model, and second, a trinomial tree discretization. We present serial and parallel implementations of the method in the discrete setting, using a probabilistic interpretation to compute, at significantly reduced cost, the gradient of the cost criterion. We illustrate the stability of this regularized calibration procedure by numerical examples. Finally we extend this methodology to the problem of calibration with American option prices.

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