# Pricing Multidimensional American Options

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem-Setting and Motivating Examples

- Basket (or index) options: $\widehat{G}({x}_{1},\dots ,{x}_{n})=\omega {[}_{i=1}^{n}{w}_{i}{x}_{i}-K]$

- Arithmetic average option: $\widehat{G}({x}_{1},\dots ,{x}_{n})=\omega [\frac{1}{n}{\sum}_{i=1}^{n}{x}_{i}-K]$

- Spread options: $\widehat{G}({x}_{1},{x}_{2})=\omega [{x}_{2}-{x}_{1}-K].$

- Option on the product with random exercise price:
- $\widehat{G}({x}_{1},{x}_{2})=\omega [{x}_{1}{x}_{2}-K{x}_{2}]$1
- Power-product options: $\widehat{G}({x}_{1},\dots ,{x}_{n})=\omega [{\left({\prod}_{i=1}^{n}{x}_{i}\right)}^{p}-K]$ for some $p>1.$
- Options on the max: $\widehat{G}({x}_{1},\dots ,{x}_{n})=\omega [max({x}_{1},\dots ,{x}_{n})-K].$
- Multiple strike options: $\widehat{G}({x}_{1},\dots ,{x}_{n})=max[{\omega}_{1}({x}_{1}-{K}_{1}),\dots {\omega}_{n}({x}_{n}-{K}_{n})].$

**Proposition 1.**

**Proof.**

## 3. Valuation Formula

## 4. Probabilistic Interpretation

## 5. Discussion and Concluding Remarks

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Bessel Functions and Modified Helmoltz Equation

## Notes

1 | see the Ref. Broadie and Detemple (1997), for a real-world interpretation |

## References

- Ait-Sahlia, Farid, and Tze Leung Lai. 2001. Exercise boundaries and efficient approximation to American option prices and hedge parameters. Journal of Computational Finance 4: 85–104. [Google Scholar]
- Agliardi, Rossella. 2012. A comprehensive mathematical approach to exotic option pricing. Mathematical Methods in the Applied Sciences 35: 1256–68. [Google Scholar] [CrossRef]
- Boyarchenko, Svetlana I., and Sergei Z. Levendorskiĭ. 2002. Perpetual American options under Lévy processes. SIAM Journal on Control and Optimization 40: 1663–96. [Google Scholar] [CrossRef]
- Boyarchenko, Svetlana I., and Sergei Z. Levendorskiĭ. 2007. Irreversible Decisions under Uncertainty: Optimal Stopping Made Easy. Studies in Economic Theory 27. Berlin/Heidelberg: Springer. [Google Scholar]
- Brennan, Michael J., and Eduardo S. Schwartz. 1977. The valuation of the American put option. Journal of Finance 32: 449–62. [Google Scholar] [CrossRef]
- Broadie, Mark, and Jerome Detemple. 1997. The valuation of American options on multiple assets. Mathematical Finance 17: 241–86. [Google Scholar] [CrossRef]
- Carr, Peter. 1998. Randomization and the American put. Review of Financial Studies 11: 597–626. [Google Scholar] [CrossRef][Green Version]
- Chan, Raymond H., Chi-Yan Wong, and Kit-Ming Yeung. 2006. Pricing multi-asset American-style options by memory reduction Monte Carlo methods. Applied Mathematics and Computation 179: 535–44. [Google Scholar] [CrossRef]
- Dixit, Robert K., and Robert S. Pindyck. 1994. Investment Under Uncertainty. Princeton: Princeton University Press. [Google Scholar]
- Egorova, Vera N., Lucas Jodar, and Fazlollah Soleymani. 2018. A local radial basis function method for high-dimensional American option pricing problems. Mathematical Modelling and Analysis 23: 117–38. [Google Scholar]
- Erdélyi, Arthur, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi. 1953. Higher Trascendental Functions. New York: McGraw-Hill. [Google Scholar]
- Firth, Neil P. 2005. Pricing High-Dimensional American Options. Ph.D. thesis, University of Oxford, Oxford, UK. [Google Scholar]
- Geske, Robert, and Herbert E. Johnson. 1984. The American put option valued analytically. Journal of Finance 39: 1511–24. [Google Scholar] [CrossRef]
- Gukhal, Chandrasekhar Reddy. 2001. Analytical valuation of American options on jump-diffusion processes. Mathematical Finance 11: 97–115. [Google Scholar] [CrossRef]
- Hanbali, Hamza, and Daniel Linders. 2019. American-type basket option pricing: A simple two-dimensional Partial Differential Equation. Quantitative Finance 19: 1689–704. [Google Scholar] [CrossRef]
- Huang, Jing-zhi, Marti G. Subrahmanyam, and G. George Yu. 1996. Pricing and hedging American options: A recursive integration method. Review of Financial Studies 9: 277–300. [Google Scholar] [CrossRef][Green Version]
- Kim, In Joon. 1990. The analytic valuation of American options. Review of Financial Studies 3: 547–72. [Google Scholar] [CrossRef]
- Laurence, Peter, and Sandro Salsa. 2009. Regularity of the free-boundary of an American option on several assets. Communications on Pure and Applied Mathematics 62: 969–94. [Google Scholar] [CrossRef]
- Levendorskiĭ, Sergei Z. 2004. Pricing of American put under Lévy processes. International Journal of Theoretical and Applied Finance 7: 303–35. [Google Scholar] [CrossRef][Green Version]
- Longstaff, Francis A., and Eduardo S. Schwartz. 2001. Valuing American options by simulation: A simple least-square approach. Review of Financial Studies 14: 113–47. [Google Scholar] [CrossRef][Green Version]
- Mordecki, Ernesto. 2002. Optimal stopping and perpetual options under Lévy processes. Finance and Stochastics 6: 473–93. [Google Scholar] [CrossRef]
- Nielsen, Bjørn Fredrik, Ola Skavhaug, and Aslak Tveito. 2008. Penalty methods for the numerical solution pf American multi-asset problems. Journal of Computational and Applied Mathematics 222: 3–16. [Google Scholar] [CrossRef][Green Version]
- O’Sullivan, Stephen, and Conall O’Sullivan. 2011. On the acceleration of explicit finite difference methods for option pricing. Quantitative Finance 11: 1177–91. [Google Scholar] [CrossRef]
- Peskir, Goran. 2019. Continuity of the optimal stopping boundary for two-dimensional diffusions. Annals of Applied Probability 29: 505–30. [Google Scholar] [CrossRef][Green Version]
- Peskir, Goran, and Albert Shiryaev. 2006. Optimal Stopping and Free-Boundary Problems. Basel: Birkhäuser. [Google Scholar]
- Revuz, Daniel, and Marc Yor. 1999. Continuous Martingales and Browian Motion. Berlin: Springer. [Google Scholar]
- Samimi, Oldouz, and Farshid Mehrdoust. 2018. Pricing multi-asset American option under Heston stochastic volatility model. International Journal of Financial Engineering 5: 1850026. [Google Scholar] [CrossRef]
- Shiryaev, Albert N. 2008. Stochastic Modelling and Applied Probability 8. In Optimal Stopping Rules. Translated from the 1976 Russian second edition. Berlin: Springer. [Google Scholar]
- Wu, Xianbin. 2013. Accurate numerical method for pricing two-asset American put options. Journal of Function Spaces and Applications 2013: 189235. [Google Scholar] [CrossRef][Green Version]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Agliardi, E.; Agliardi, R.
Pricing Multidimensional American Options. *Int. J. Financial Stud.* **2023**, *11*, 51.
https://doi.org/10.3390/ijfs11010051

**AMA Style**

Agliardi E, Agliardi R.
Pricing Multidimensional American Options. *International Journal of Financial Studies*. 2023; 11(1):51.
https://doi.org/10.3390/ijfs11010051

**Chicago/Turabian Style**

Agliardi, Elettra, and Rossella Agliardi.
2023. "Pricing Multidimensional American Options" *International Journal of Financial Studies* 11, no. 1: 51.
https://doi.org/10.3390/ijfs11010051