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Boundary conditions at infinity for Black-Scholes equations

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  • Yukihiro Tsuzuki

Abstract

We propose a numerical procedure for computing the prices of forward contracts, in which the underlying asset price is a Markovian local martingale. If the underlying process is a strict local martingale, multiple solutions exist for the corresponding Black-Scholes equations, and the derivative prices are characterized as the minimal solutions. When applying numerical methods that are set up on a finite grid, such as finite difference methods, additional spatial boundary conditions are required, which causes the solution to uniquely exist. In this case, the problem is reduced to specifying the boundary conditions. Our boundary condition is based on an expression of the derivative price at infinity in terms of those at finite values, and incorporates volatility behaviors beyond the boundary. The proposed procedure is demonstrated through numerical tests, which show that it outperforms the methods proposed in the literature.

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  • Yukihiro Tsuzuki, 2024. "Boundary conditions at infinity for Black-Scholes equations," Papers 2401.05549, arXiv.org, revised Sep 2024.
    • Handle: RePEc:arx:papers:2401.05549
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    References listed on IDEAS

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    1. Pal, Soumik & Protter, Philip, 2010. "Analysis of continuous strict local martingales via h-transforms," Stochastic Processes and their Applications, Elsevier, vol. 120(8), pages 1424-1443, August.
    2. Yukihiro Tsuzuki, 2023. "Pitman's Theorem, Black-Scholes Equation, and Derivative Pricing for Fundraisers," Papers 2303.13956, arXiv.org.
    3. Peter Carr & Travis Fisher & Johannes Ruf, 2012. "Why are quadratic normal volatility models analytically tractable?," Papers 1202.6187, arXiv.org, revised Mar 2013.
    4. Çetin, Umut & Larsen, Kasper, 2023. "Uniqueness in cauchy problems for diffusive real-valued strict local martingales," LSE Research Online Documents on Economics 118743, London School of Economics and Political Science, LSE Library.
    5. Leif Andersen, 2011. "Option pricing with quadratic volatility: a revisit," Finance and Stochastics, Springer, vol. 15(2), pages 191-219, June.
    6. Peter Carr & Travis Fisher & Johannes Ruf, 2014. "On the hedging of options on exploding exchange rates," Finance and Stochastics, Springer, vol. 18(1), pages 115-144, January.
    7. Qingshuo Song & Pengfei Yang, 2015. "Approximating functionals of local martingales under lack of uniqueness of the Black-Scholes PDE solution," Quantitative Finance, Taylor & Francis Journals, vol. 15(5), pages 901-908, May.
    8. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    9. Soumik Pal & Philip Protter, 2007. "Analysis of continuous strict local martingales via h-transforms," Papers 0711.1136, arXiv.org, revised Jun 2010.
    10. Erik Ekstrom & Per Lotstedt & Lina Von Sydow & Johan Tysk, 2011. "[image omitted] Numerical option pricing in the presence of bubbles," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1125-1128.
    11. Dirk Veestraeten, 2017. "On the multiplicity of option prices under CEV with positive elasticity of variance," Review of Derivatives Research, Springer, vol. 20(1), pages 1-13, April.
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