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Asymptotics for the Laplace transform of the time integral of the geometric Brownian motion

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  • Dan Pirjol
  • Lingjiong Zhu

Abstract

We present an asymptotic result for the Laplace transform of the time integral of the geometric Brownian motion $F(\theta,T) = \mathbb{E}[e^{-\theta X_T}]$ with $X_T = \int_0^T e^{\sigma W_s + ( a - \frac12 \sigma^2)s} ds$, which is exact in the limit $\sigma^2 T \to 0$ at fixed $\sigma^2 \theta T^2$ and $aT$. This asymptotic result is applied to pricing zero coupon bonds in the Dothan model of stochastic interest rates. The asymptotic result provides an approximation for bond prices which is in good agreement with numerical evaluations in a wide range of model parameters. As a side result we obtain the asymptotics for Asian option prices in the Black-Scholes model, taking into account interest rates and dividend yield contributions in the $\sigma^{2}T\to 0$ limit.

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  • Dan Pirjol & Lingjiong Zhu, 2023. "Asymptotics for the Laplace transform of the time integral of the geometric Brownian motion," Papers 2306.09084, arXiv.org.
    • Handle: RePEc:arx:papers:2306.09084
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    1. Ning Cai & Chenxu Li & Chao Shi, 2014. "Closed-Form Expansions of Discretely Monitored Asian Options in Diffusion Models," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 789-822, August.
    2. Alan L. Lewis & Dan Pirjol, 2022. "Proof of non-convergence of the short-maturity expansion for the SABR model," Quantitative Finance, Taylor & Francis Journals, vol. 22(9), pages 1747-1757, September.
    3. Ning Cai & Yingda Song & Steven Kou, 2015. "A General Framework for Pricing Asian Options Under Markov Processes," Operations Research, INFORMS, vol. 63(3), pages 540-554, June.
    4. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    5. Nicolas Privault & Wayne Isaac Uy, 2013. "Monte Carlo Computation of the Laplace Transform of Exponential Brownian Functionals," Methodology and Computing in Applied Probability, Springer, vol. 15(3), pages 511-524, September.
    6. Beata Stehlikova, 2014. "Approximating the zero-coupon bond price in a general one-factor model with constant coefficients," Papers 1408.5673, arXiv.org.
    7. Paolo Guasoni & Scott Robertson, 2008. "Optimal importance sampling with explicit formulas in continuous time," Finance and Stochastics, Springer, vol. 12(1), pages 1-19, January.
    8. Beáta Stehlíková & Luca Capriotti, 2014. "An Effective Approximation For Zero-Coupon Bonds And Arrow–Debreu Prices In The Black–Karasinski Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(06), pages 1-16.
    9. Dothan, L. Uri, 1978. "On the term structure of interest rates," Journal of Financial Economics, Elsevier, vol. 6(1), pages 59-69, March.
    10. Ning Cai & Steven Kou, 2012. "Pricing Asian Options Under a Hyper-Exponential Jump Diffusion Model," Operations Research, INFORMS, vol. 60(1), pages 64-77, February.
    11. Dan Pirjol & Lingjiong Zhu, 2017. "Asymptotics for the Discrete-Time Average of the Geometric Brownian Motion and Asian Options," Papers 1706.09659, arXiv.org.
    12. Fabricio Tourrucoo & Patrick S. Hagan & Gilberto F. Schleiniger, 2007. "Approximate Formulas for Zero-coupon Bonds," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(3), pages 207-226.
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    Cited by:

    1. Dan Pirjol & Lingjiong Zhu, 2024. "Short-maturity asymptotics for option prices with interest rates effects," Papers 2402.14161, arXiv.org.

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