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

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## 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.

## Keywords

Barrier option price First hitting time Non-linear smooth boundary Reflection principle Symmetrization of multi-dimensional diffusion## Notes

### Acknowledgements

The authors are grateful to Professor Jiro Akahori for many valuable comments and for careful reading of the manuscript and suggesting several improvements.

## References

- Akahori, J., & Imamura, Y. (2014). On symmetrization of diffusion processes.
*Quantitative Finance*,*14*(7), 1211–1216.CrossRefGoogle Scholar - Gobet, E. (2000). Weak approximation of killed diffusion using Euler schemes.
*Stochastic Processes and Their Applications*,*87*(2), 167–197.CrossRefGoogle Scholar - Ida, Y., & Kinoshita, T. (2019). Hyperbolic symmetrization of heston type diffusion.
*Asia-Pacific Financial Markets*. https://doi.org/10.1007/s10690-019-09269-1.CrossRefGoogle Scholar - Ida, Y., Kinoshita, T., & Matsumoto, T. (2018). Symmetrization associated with hyperbolic reflection principle.
*Pacific Journal of Mathematics for Industry*,*10*, 1.CrossRefGoogle Scholar - Imamura, Y., Ishigaki, Y., & Okumura, T. (2014). A numerical scheme based on semi-static hedging strategy.
*Monte Carlo Methods and Applications*,*20*(4), 223–235.CrossRefGoogle Scholar - Revuz, D., & Yor, M. (1999).
*Continuous martingales and Brownian motion*(3rd ed.). Berlin: Springer.CrossRefGoogle Scholar

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