Inverse parabolic problem with the Heaviside function arising in finance

Abstract

The inverse problem arising in finance is both mathematically and practically interesting problem. First, we attempt to solve the inverse problem arising in binary option model. Using standard linearization and contraction arguments, we prove the stability and uniqueness of the solution of the inverse problem which originates from the diffusion equation with the initial condition given by the Heaviside function. Second, we numerically identify the local volatilities from given artificial prices of the binary option. In accordance with theory we propose the effective identification around the at-the-money level.

Keywords:

• Inverse problem
• finance
• binary option

2020 Mathematics Subject Classifications:

• 35R30
• 35Q62
• 35A02
• 91-08
• 65M32

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In general case, $a(y)$ can be approximated by Holder continuous functions $a_n(y)$, $n\ge 1$. Since the fundamental solutions to the Cauchy problem (10(10) $\{\begin{array}{ll}{\displaystyle \frac{\mathrm{\partial }U}{\mathrm{\partial }\tau }=\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left(\frac{1}{2}{a}^2\frac{\mathrm{\partial }U}{\mathrm{\partial }y}\right)+(\frac{1}{2}{a}^2-r)\frac{\mathrm{\partial }U}{\mathrm{\partial }y}-rU,}& (y,\tau )\in Q;\\ U(y,\tau ){|}_{\tau =0}=H(-y),& y\in \mathbf{R},\end{array}$(10) ) for $a_n(y)$, $n\ge 1$ and their derivatives are bounded by $C{\tau }^{-\frac{3}{2}}{\mathrm{e}}^{-\frac{C{y}^2}{\tau }}$, $C{\tau }^{-\frac{p}{2}}{\mathrm{e}}^{-\frac{C{y}^2}{\tau }}$, it is not difficult to imply (30(30) $|\frac{{\mathrm{\partial }}^{2}U}{\mathrm{\partial }\tau \mathrm{\partial }y}(y,\tau )|\le C{\tau }^{-\frac{3}{2}}{\mathrm{e}}^{-\frac{C{y}^2}{\tau }};|\frac{{\mathrm{\partial }}^{p}U}{\mathrm{\partial }{y}^p}(y,\tau )|\le C{\tau }^{-\frac{p}{2}}{\mathrm{e}}^{-\frac{C{y}^2}{\tau }},p=1,2$(30) ).

Additional information

Funding

This work was supported by JSPS [grant number 18K03439].