Abstract
The Lugannani–Rice formula is a saddlepoint approximation method for estimating the tail probability distribution function, which was originally studied for the sum of independent identically distributed random variables. Because of its tractability, the formula is now widely used in practical financial engineering as an approximation formula for the distribution of a (single) random variable. In this paper, the Lugannani–Rice approximation formula is derived for a general, parametrized sequence \((X^{(\varepsilon )})_{\varepsilon >0}\) of random variables and the order estimates (as \(\varepsilon \rightarrow 0\)) of the approximation are given.
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Acknowledgments
The authors thank communications with Masaaki Fukasawa of Osaka University, who directed their attentions to the Lugannani–Rice formula. The authors also thank the anonymous referee for valuable comments and suggestions.
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Jun Sekine’s research was supported by a Grant-in-Aid for Scientific Research (C), No. 23540133, from the Ministry of Education, Culture, Sports, Science, and Technology, Japan.
Appendix: Explicit forms of higher order approximation terms
Appendix: Explicit forms of higher order approximation terms
In this section, we introduce the derivation of \(\varPsi ^\varepsilon _2(\hat{w}_\varepsilon )\) and \(\varPsi ^\varepsilon _3(\hat{w}_\varepsilon )\). First, we can inductively calculate \(\hat{\theta }^{(r)}_\varepsilon \) for \(r\ge 4\) by the same calculation as the proof of Proposition 14.
Proposition 16
Second, by continuing the differentiation in (35), we have
where g(w) and h(w) are defined as (31). Combining this with (34), Lemma 4, and Propositions 4, 8, 10, and 16, we can calculate \(\varPsi ^\varepsilon _2(\hat{w}_\varepsilon )\) and \(\varPsi ^\varepsilon _3(\hat{w}_\varepsilon )\) explicitly.
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Kato, T., Sekine, J. & Yoshikawa, K. Order estimates for the exact Lugannani–Rice expansion. Japan J. Indust. Appl. Math. 33, 25–61 (2016). https://doi.org/10.1007/s13160-015-0199-z
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DOI: https://doi.org/10.1007/s13160-015-0199-z
Keywords
- Saddlepoint approximation
- The Lugannani–Rice formula
- Order estimates
- Asymptotic expansion
- Stochastic volatility models