Financial crises prediction is an essential topic in fi nance. We propose an Early Warning Indicator (EWI) for predicting possible fi nancial crises or, more generally, market instability conditions. Our system is based on the so called price-volatility feedback rate, which is supposed to describe the ease of the market in absorbing small price perturbations.

We present an indicator of financial instability based on the computation of the decay rate for the propagation of a given market shock. The rate of variation through time of an initial perturbation of a given high frequency fi nancial time series enables us to understand if such a shock will be rapidly absorbed or, on the contrary, it will be amplifi ed by the market. The indicator combines non-linearly volatility, leverage and covariance between leverage and price and is model-free. Consistency results and other properties of the indicator under the CEV model have been investigated in [2].

A logit regression Early Warning System is employed to predict future financial crises and EWIs based on the realized variance (RV) and on the price-volatility feedback rate are considered. Our study conducted in [1] on the S&P 500 index futures reveals that, while the RV may sometimes fail in predicting crises, the EWI employing the price-volatility feedback rate is always an important predictor of fi nancial instability.

[1] Allaj, E. and Sanfelici, S., 2021 An Early Warning System for Identifying financial instability. Submitted.
[2] Mancino, M.E. and Sanfelici, S., 2020 Identifying financial instability conditions using high frequency data. J. Economic Interaction and Coordination, 15(1), 221-242.

Keywords:
Market stability · Shock propagation · Volatility · Leverage · Non-parametric estimation · Fourier transform

From the need for a more scientific approach to the problem of pricing and risk control has emerged the advantage of exploiting the availability of more advanced numerical techniques and faster computer systems.

European options are derivative contracts that give the buyer the right to buy/sell a particular asset at a fixed maturity, at a predetermined price. In the case of \barrier option”, this right gets into existence or extinguishes when the underlying asset reaches a certain barrier value.

The Black-Scholes model (BS) [1] can be considered the first of the differential models for option pricing but other models have been later introduced in the academic literature. For these more advanced models, the pricing of \barrier option” is traditionally based on Monte Carlo methods that are affected by high computational costs and inaccuracy, due to their slow convergence, or on domain methods (such as Finite Element Methods and Finite Difference methods) that have some troubles particularly in unbounded domains.

A Semi-Analytical method for Barrier Options pricing (SABO) will be presented in several frame- works [2]-[9] showing its accuracy and efficiency in the application to the numerical pricing of Euro- pean style barrier options. This new approach is based on the Boundary Element Method (BEM) well known for its high accuracy, for the implicit satisfaction of the asymptotic conditions at infinity and for the low dis- cretization costs. It is particularly advantageous in this framework because the differential problem is often defined in an unbounded domain but the data are assigned on a limited boundary (the \barrier”).

[1] Black, F. and Scholes, M. 1973, The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637{654, .
[2] Guardasoni, C. and Sanfelici, S. 2016, A boundary element approach to barrier option pricing in Black-Scholes framework. International Journal of Computer Mathematics, 93(4), 696{722.
[3] Guardasoni, C. and Sanfelici, S. 2016, Fast Numerical Pricing of Barrier Options under Stochastic Volatility and Jumps. SIAM J. Appl. Math, 76(1), 27{57.
[4] Guardasoni, C. 2018, Semi-Analytical method for the pricing of barrier options in case of time-dependent parameters (with Matlab codes). Communications in Applied and Industrial Mathematics, 9(1), 42{67,https://doi.org/10.1515/caim-2018-0004.
[5] Aimi, A. and Guardasoni, C. 2018, Collocation Boundary Element Method for the pricing of Geometric Asian Options. Engineering Analysis with Boundary Elements, 92, 90{100.
[6] Aimi, A. and Diazzi, L. and Guardasoni, C. 2018, Numerical pricing of geometric asian options with barriers. Mathematical Methods in the Applied Sciences, 41(17), 7510{7529.
[7] Aimi, A. and Diazzi, L. and Guardasoni, C. 2018, Efficient BEM-based algorithm for pricing oating strike Asian barrier options (with MATLAB code). Axioms, 7(2), https://doi.org/10.3390/axioms7020040.
[8] Semi-Analytical Method for Accurate Evaluation of Arithmetic Asian Options with Barrier, in preparation.
[9] Multi-assets Barrier Options, in preparation.

In a recent book, (Itkin, Lipton, and Mutavey, 2021) the authors aim to present new methods and approaches previously developed in a series of papers written by the authors and their associates to solve various initial-boundary value problems for PDEs with moving boundaries. Such problems often appear in mathematical finance when pricing barriers and American options, finding the hitting time probability distribution for some stochastic process, etc. As mentioned, aside from the financial interpretation, mathematically similar problems have been studied in physics for a long time. Analytical solutions to these problems require non-traditional and sometimes sophisticated methods. Our approach provides a powerful alternative to the well-known finite difference and Monte Carlo methods. We discuss various advantages of this approach, which judiciously combines semi-analytical and numerical techniques and provides a fast and accurate way of finding solutions to the corresponding equations.