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

Abstract

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.

Estimation of risk-neutral density surfaces

“…Then, the numerically estimated risk-neutral density was used in (15) to compute the corresponding binary call prices. Finally, we calculated the binary option prices from (16), setting σ imp , for each binary option, to the implied volatility computed from the corresponding European vanilla call option price on the S&P500 index.…”

“…Avellaneda et al [4] recovered the implied volatility surface extending the oneperiod entropy minimization, presented by Butchen and Kelly [11] and Stutzer [34], to a multi-period model. Crépey [15] considered a minimization problem with a regularization parameter based on smoothness norms for volatility functions. This approach was also followed by Lagnado and Osher [29], and also by Jackson, Suli, and Howison [26] who used splines for the regularization effect.…”

Mini-workshop: Statistical Methods for Inverse Problems

“…In the case of diffusion models, we have an inverse problem for a parabolc PDE, where the parameter is an unknown functional diffusion coefficient belonging to a suitable Sobolev/Hölder space and Tikhonov regularization methods have been succussfully employed [13,17]. A Tikhonov regularization method is used in [1] to solve an inverse free-boundary problem related to the calibration of American options.…”

“…One is the efficient computation of the gradient / Fréchet derivative of the regularized functional: this is usually done by solving, at each iteration step, an auxiliary PDE/ boundary value problem [13,10]. Also, the regularized functional is still not convex so uniqueness of the regularized solution and convergence of gradient-based methods are not obvious.…”

The Equation of Real Option Value under Trinomial Tree Model

“…Crépey [3] Calibrated the local volatility in a trinomial tree by using Tikhonov regularization. Ding and Zeng [1] researched trinomial tree model of real option value.…”