A Numerical Scheme for Expectations with First Hitting Time to Smooth Boundary

Abstract

In the present paper, we propose a numerical scheme to calculate expectations with first hitting time to a given smooth boundary, in view of the application to the pricing of options with non-linear barriers. To attack the problem, we rely on the symmetrization technique in Akahori and Imamura (Quant Finance 14(7):1211–1216, 2014) and Imamura et al. (Monte Carlo Methods Appl 20(4):223–235, 2014), with some modifications. To see the effectiveness, we perform some numerical experiments.

Appendix: Hyperbolic Reflection Principle

In the complex coordinate $Z = X + i Y$, its law is invariant under the action of $SL (2, {\mathbb {R}})$; for

$$\begin{aligned}&\begin{pmatrix} a &\quad b \\ c &\quad d \end{pmatrix} =:A \in SL (2, {\mathbb {R}}), \\&(Z_t) \overset{\text {law}}{=} \left( \frac{a Z_t + b}{c Z_t + d} \right) =: \Phi _A (Z_t) \end{aligned}$$

provided $Z_0 = (a Z_0 + b)/(c Z_0 + d)$. More generally, it is invariant under isometry (a map preserving distance of any two given points) of the hyperbolic space ${\mathbb {H}}^2 := \{ z \in {\mathbb {C}}: \mathrm {Im} z > 0 \}$, which is equipped with the distance given by

$$\begin{aligned} d (z_1, z_2) = \mathrm {arcosh} \left( 1 + \frac{|z_1-z_2|^2}{2 \mathrm {Im} (z_1) \mathrm {Im} (z_2) }\right) . \end{aligned}$$

Thus the Hyperbolic Brownian motion to the hyperbolic space ${\mathbb {H}}^2$ is the standard Brownian motion to the Euclidean space. We can then define “reflection” in ${\mathbb {H}}^2$ and then associated reflection principle holds. Associated symmetrization has been introduced by Y. Ida and her collaborators in Ida and Kinoshita (2019), see also Ida et al. (2018).