Recovery of volatility coefficient by linearization

Abstract

We study the problem of reconstruction of the asset price dependent local volatility from market prices of options with different strikes. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an explicit functional which is linear in perturbation of volatility. We obtain an integral equation for this functional and we show that under some natural conditions it can be inverted for volatility. We demonstrate the stability of the linearized problem, and we propose a numerical algorithm which is accurate for volatility functions with different properties.

Recovery of Foreign Interest Rates from Exchange Binary Options

“…A standard linearization procedure for this identification problem is used in Refs. [2,3]. Now, we consider that a price u(T,K) of a binary option with maturity T and strike price K satisfies and make the changes in variables y = ln (K/S*), τ = T-t. Then, we arrive at the following problem:…”

Calibration of the purely T-dependent Black–Scholes implied volatility

“…Now, we consider the second and the third questions. In [9,10] and in other papers in the references (see [2][3][4][5][6][7][8][9][10][11][12][13][14]18]), the dominant method of regularization is the Tikhonov one (or Tikhonov-type one as in [10]). The method was often used for the case of an L 2 -space setting (or Hilbert space setting).…”

“…In almost papers, the authors considered the problem of recovering the implied volatility which are depended on (t, X ), i.e. σ = σ (t, X ) (see [2][3][4][5][6][7][8][9][10][11][12][13][14]). In [4,8], the authors considered the implied volatility having the separable form σ = σ 0 (X )ρ(t).…”

Recover implied volatility of underlying asset from European option price

“…[3,4,7,12]). The inverse problems to recover implied volatilities are usually ill-posed, so many of the related discussions focus on the ill-posedness of the problems, the solution uniqueness and stability, and the regularization approaches in numerical experiments.…”

“…In [4], Ilia Bouchouev and Victor Isakov provide a very simple but effective method to recover implied volatility by linearization. They show the local uniqueness and stability of implied volatility and give some numerical examples.…”