# Pricing Double Barrier Options by Combinatorial Approaches

• Tian-Shyr Dai
• Yuh-Dauh Lyuu
Conference paper
Part of the Advances in Soft Computing book series (AINSC, volume 29)

## Summary

Double barrier options are important path-dependent derivatives in the financial market. How to price them efficiently and accurately is thus important. Until now, no simple closed-form pricing formula for double barrier options is reported. Double barrier options can be priced on a lattice that divides a certain time interval (from option initial date to maturity date) into n equal-length time steps. The pricing results obtained by the lattice algorithm converge to the true option value as n → ∞, and the results oscillate significantly especially when n is not large enough. To obtain an accurate pricing result without suffering from price oscillation, n is required to be a large number. Unfortunately, the lattice pricing algorithm runs in O(n 2) time. This paper proposes a linear-time combinatorial algorithm that can generate the same pricing results as the lattice algorithm. Thus our algorithm can handle very large n’s efficiently. This algorithm uses a novel technique based on the reflection principle and the inclusion-exclusion principle. Numerical experiments are given to verify the excellent performance of our algorithm.

## References

1. 1.
Bhagavatula RS, and Carr PP (1997) Valuing Double Barrier Options with Fourier Series. Manuscript, New York University, NY, USA.Google Scholar
2. 2.
Cox J, Ross S, Rubinstein M (1979) Option Pricing: A Simplified Approach. J. Financial Econom. 7:229–264.
3. 3.
Figlewski S, Gao B (1999) The Adaptive Mesh Model: A New Approach to Efficient Option Pricing. J. Financial Econom. 53:313–351.
4. 4.
Geman H, and Yor M (1996) Pricing and Hedging Double-Barrier Options: A Probabilistic Approach. Math. Finance 6:365–378.
5. 5.
Lyuu YD (1998) Very Fast Algorithms for Barrier Option Pricing and the Ballot Problem. J. Derivatives 5:68–79.Google Scholar
6. 6.
Ritchken P (1995) On Pricing Barrier Options. J. Derivatives 3:19–28.Google Scholar
7. 7.
Wei, JZ (1998) Valuation of Discrete Barrier Options by Interpolations. J. Derivatives, 6:51–73.