# 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.
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