Application of microlocal analysis to an inverse problem arising from financial markets

Abstract

One of the most interesting problems discerned when applying the Black-Scholes model to financial derivatives, is reconciling the deviation between expected and observed values. In our recent work, we derived a new model based on the Black-Scholes model and formulated a new mathematical approach to an inverse problem in financial markets. In this paper, we apply microlocal analysis to prove a uniqueness of the solution to our inverse problem. While microlocal analysis is used for various models in physics and engineering, this is the first attempt to apply it to a model in financial markets. First, we explain our model, which is a type of arbitrage model. Next we illustrate our new mathematical approach, and then for space-dependent real drift, we obtain stable linearization and an integral equation. Finally, by applying microlocal analysis to the integral equation, we prove our uniqueness of the solution to our new mathematical model in financial markets.

Bayesian Inference Approach to Inverse Problems in a Financial Mathematical Model

“…However, the theoretical prices of options with different strike prices which are calculated by the Black-Scholes equation differ from real market prices. In [6] and [18], taking this into account, we have extended Black (1.2) In different real markets handling the same risk and asset, an arbitrage opportunity often appears in the error µ − r. Practitioners may also request a convenience yield µ − r from commodity markets. Suppose that N arbitrage markets handle the same asset with price s i (i = 1, 2, • • • , N ) at a given time t * , we try to identify µ(S i ) − r from the measured call option prices u * (S i ), at the same time t * .…”

“…They transformed (IOP) to a Fredholm-type integral equation for f (K) and numerically determined the timeindependent local volatility function. On the other hand, Mitsuhiro and Ota [18], Korolev et al [12] and Doi and Ota [6] used the extended Black-Scholes equation (1.2) and then reconstructed the trend function by linearization method. The above studies provided point estimates of unknown parameters by exact determination or least squares optimization, without rigorously examining and considering the measurement errors in the inverse solutions.…”

Parameters Identification for Inverse Option Problems Using Markov Chain Monte Carlo Methods

“…where f is a small perturbation of the constant volatility σ 0 . Moreover, Mitsuhiro and Ota [23], Korolev et al [17] and Doi and Ota [9] used the extended Black-Scholes equation (1) and then reconstructed the trend function by linearization method. The above studies provided point estimates of unknown parameters by exact determination or least squares optimization, without rigorously examining and considering the measurement errors in the inverse solutions.…”

“…Problem If we give the data U * (x) := U(y, τ * ) on ω at τ = τ * = T − t * then identify σ 0 and µ(y) satisfying (9) However, due to the nonlinearity of this inverse problem, the uniqueness and existence of its solution are hard to prove. In this paper we attempts to reconstruct the parameters by a statistical method simultaneously estimates µ(y) and σ 0 from the measured data U * (y).…”

Calibration of Implied Volatility in Generalized Hull-White Model

“…Much research has been done on the inverse problem to reconstruct parameters from market prices [7]- [12]. At present, there are many methods to solve this kind of problems, such as regularization method, maximum likelihood method, full Bayesian method and optimization method [13]- [15]. In [16], Tang and Chen develop expansions for the bias and variance of parameter estimators for Vasicek and CIR processes, which helps to understand why the drift parameters are more difficult to estimate than the diffusion parameter, and then they study the first order approximate maximum likelihood estimator for linear drift processes.…”