Order estimates for the exact Lugannani–Rice expansion

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.

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

$$\begin{aligned} \hat{\theta }^{(4)}_\varepsilon= & {} -\frac{K^{(5)}(\hat{\theta }_\varepsilon )}{5(K''(\hat{\theta }_\varepsilon ))^3} + \frac{K^{(3)}(\hat{\theta }_\varepsilon )K^{(4)}(\hat{\theta }_\varepsilon )}{(K''(\hat{\theta }_\varepsilon ))^4} - \frac{8(K^{(3)}(\hat{\theta }_\varepsilon ))^3}{9(K''(\hat{\theta }_\varepsilon ))^5}, \\ \hat{\theta }^{(5)}_\varepsilon= & {} -\frac{K^{(6)}(\hat{\theta }_\varepsilon )}{6(K''(\hat{\theta }_\varepsilon ))^{7/2}} + \frac{35 (K^{(4)}(\hat{\theta }_\varepsilon ))^2}{48(K''(\hat{\theta }_\varepsilon ))^{9/2}} + \frac{7K^{(3)}(\hat{\theta }_\varepsilon )K^{(5)}(\hat{\theta }_\varepsilon )}{6(K''(\hat{\theta }_\varepsilon ))^{9/2}}\\&-\, \frac{35(K^{(3)}(\hat{\theta }_\varepsilon ))^2K^{(4)}(\hat{\theta }_\varepsilon )}{8(K''(\hat{\theta }_\varepsilon ))^{11/2}} + \frac{385(K^{(3)}(\hat{\theta }_\varepsilon ))^4}{144(K''(\hat{\theta }_\varepsilon ))^{13/2}}, \\ \hat{\theta }^{(6)}_\varepsilon= & {} -\frac{K^{(7)}(\hat{\theta }_\varepsilon )}{7(K''(\hat{\theta }_\varepsilon ))^{4}} - \frac{280(K^{(3)}(\hat{\theta }_\varepsilon ))^5}{27(K''(\hat{\theta }_\varepsilon ))^{8}} + \frac{200(K^{(3)}(\hat{\theta }_\varepsilon ))^3K^{(4)}(\hat{\theta }_\varepsilon )}{9(K''(\hat{\theta }_\varepsilon ))^{7}}\\&-\, \frac{25(K^{(4)}(\hat{\theta }_\varepsilon ))^2}{3(K''(\hat{\theta }_\varepsilon ))^{6}} - \frac{20(K^{(3)}(\hat{\theta }_\varepsilon ))^2K^{(5)}(\hat{\theta }_\varepsilon )}{3(K''(\hat{\theta }_\varepsilon ))^{6}} + \frac{2K^{(4)}(\hat{\theta }_\varepsilon )K^{(5)}(\hat{\theta }_\varepsilon )}{(K''(\hat{\theta }_\varepsilon ))^{5}}\\&+\, \frac{4K^{(3)}(\hat{\theta }_\varepsilon )K^{(6)}(\hat{\theta }_\varepsilon )}{3(K''(\hat{\theta }_\varepsilon ))^{5}}, \\ \hat{\theta }^{(7)}_\varepsilon= & {} -\frac{K^{(8)}(\hat{\theta }_\varepsilon )}{8(K''(\hat{\theta }_\varepsilon ))^{9/2}} - \frac{85085(K^{(3)}(\hat{\theta }_\varepsilon ))^6}{1728(K''(\hat{\theta }_\varepsilon ))^{19/2}} - \frac{25025(K^{(3)}(\hat{\theta }_\varepsilon ))^4K^{(4)}(\hat{\theta }_\varepsilon )}{192(K''(\hat{\theta }_\varepsilon ))^{17/2}}\\&+\, \frac{5005(K^{(3)}(\hat{\theta }_\varepsilon ))^2(K^{(4)}(\hat{\theta }_\varepsilon ))^2}{64(K''(\hat{\theta }_\varepsilon ))^{15/2}} - \frac{385(K^{(4)}(\hat{\theta }_\varepsilon ))^3}{64(K''(\hat{\theta }_\varepsilon ))^{13/2}} + \frac{1001(K^{(3)}(\hat{\theta }_\varepsilon ))^3K^{(5)}(\hat{\theta }_\varepsilon )}{24(K''(\hat{\theta }_\varepsilon ))^{15/2}}\\&-\, \frac{231K^{(3)}(\hat{\theta }_\varepsilon )K^{(4)}(\hat{\theta }_\varepsilon )K^{(5)}(\hat{\theta }_\varepsilon )}{8(K''(\hat{\theta }_\varepsilon ))^{13/2}} + \frac{63(K^{(5)}(\hat{\theta }_\varepsilon ))^2}{40(K''(\hat{\theta }_\varepsilon ))^{11/2}} - \frac{77(K^{(3)}(\hat{\theta }_\varepsilon ))^2K^{(6)}(\hat{\theta }_\varepsilon )}{8(K''(\hat{\theta }_\varepsilon ))^{13/2}}\\&+\, \frac{21K^{(4)}(\hat{\theta }_\varepsilon )K^{(6)}(\hat{\theta }_\varepsilon )}{8(K''(\hat{\theta }_\varepsilon ))^{11/2}} + \frac{3K^{(3)}(\hat{\theta }_\varepsilon )K^{(7)}(\hat{\theta }_\varepsilon )}{2(K''(\hat{\theta }_\varepsilon ))^{11/2}}. \end{aligned}$$

Second, by continuing the differentiation in (35), we have

$$\begin{aligned} h^{(4)}(w)= & {} \frac{g^{(4)}(w)}{g(w)} - \frac{6(g'(w))^4}{g(w)^4} + \frac{12(g'(w))^2g''(w)}{g(w)^2} - \frac{3(g''(w))^2}{g(w)^2} - \frac{4g'(w)g^{(3)}(w)}{g(w)^2}, \\ h^{(5)}(w)= & {} \frac{g^{(5)}(w)}{g(w)} - \frac{24(g'(w))^5}{g(w)^5} - \frac{60(g'(w))^3g''(w)}{g(w)^4} + \frac{30g'(w)(g''(w))^2}{g(w)^3}\\&+\, \frac{20(g'(w))^2g^{(3)}(w)}{g(w)^2} - \frac{10g''(w)g^{(3)}(w)}{g(w)^2} - \frac{5g'(w)g^{(4)}(w)}{g(w)^2}, \\ h^{(6)}(w)= & {} \frac{g^{(6)}(w)}{g(w)} - \frac{120(g'(w))^6}{g(w)^6} + \frac{360(g'(w))^4g''(w)}{g(w)^5} - \frac{270(g'(w))^2(g''(w))^2}{g(w)^4}\\&+\, \frac{30(g''(w))^3}{g(w)^2} - \frac{120(g'(w))^3g^{(3)}(w)}{g(w)^4} + \frac{120g'(w)g''(w)g^{(3)}(w)}{g(w)^3}\\&-\, \frac{10(g^{(3)}(w))^2}{g(w)^2} + \frac{30(g'(w))^2g^{(4)}(w)}{g(w)^3} - \frac{15g''(w)g^{(4)}(w)}{g(w)^2} - \frac{6g'(w)g^{(5)}(w)}{g(w)^2}, \\ h^{(7)}(w)= & {} \frac{g^{(7)}(w)}{g(w)} + \frac{720(g'(w))^7}{g(w)^7} - \frac{2520(g'(w))^5g''(w)}{g(w)^6} + \frac{2520(g'(w))^3(g''(w))^2}{g(w)^5}\\&-\, \frac{630g'(w)(g''(w))^3}{g(w)^4} + \frac{840(g'(w))^4g^{(3)}(w)}{g(w)^5} - \frac{1260(g'(w))^2g''(w)g^{(4)}(w)}{g(w)^4}\\&+\, \frac{210(g''(w))^2g^{(3)}(w)}{g(w)^2} + \frac{140g'(w)(g^{(3)}(w))^2}{g(w)^3} - \frac{210(g'(w))^3g^{(4)}(w)}{g(w)^4}\\&+\, \frac{210g'(w)g''(w)g^{(4)}(w)}{g(w)^3} - \frac{35g^{(3)}(w)g^{(4)}(w)}{g(w)^2} + \frac{42(g'(w))^2g^{(5)}(w)}{g(w)^3}\\&-\, \frac{21g''(w)g^{(5)}(w)}{g(w)^2} - \frac{7g'(w)g^{(6)}(w)}{g(w)^2}, \end{aligned}$$

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.