Abstract
We present a new Monte Carlo methodology for the accurate estimation of the distribution of the sum of dependent log-normal random variables. The methodology delivers statistically unbiased estimators for three distributional quantities of significant interest in finance and risk management: the left tail, or cumulative distribution function; the probability density function; and the right tail, or complementary distribution function of the sum of dependent log-normal factors. For the right tail our methodology delivers a fast and accurate estimator in settings for which existing methodology delivers estimators with large variance that tend to underestimate the true quantity of interest. We provide insight into the computational challenges using theory and numerical experiments, and explain their much wider implications for Monte Carlo statistical estimators of rare-event probabilities. In particular, we find that theoretically strongly efficient estimators should be used with great caution in practice, because they may yield inaccurate results in the prelimit. Further, this inaccuracy may not be detectable from the output of the Monte Carlo simulation, because the simulation output may severely underestimate the true variance of the estimator.
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Notes
We denote the standard normal pdf with covariance \(\varSigma \) via \(\phi _\varSigma (\cdot )\) (\(\phi (\cdot )\equiv \phi _{\mathbf {I}}(\cdot )\)) and the univariate cdf and complementary cdf by \(\varPhi (\cdot )\) and \(\overline{\varPhi }(\cdot )\), respectively.
The notation \(f(x)\simeq g(x)\) as \(x\rightarrow a\) stands for \(\lim _{x\rightarrow a}f(x)/g(x)=1\). Similarly, we define \(f(x)=\mathcal {O}(g(x))\Leftrightarrow \lim _{x\rightarrow a}|f(x)/g(x)|<\mathrm {const.}<\infty \); \(f(x)=o(g(x))\Leftrightarrow \lim _{x\rightarrow a}f(x)/g(x)=0\); also, \(f(x)=\varTheta (g(x))\Leftrightarrow f(x)=\mathcal {O}(g(x))\text { and } g(x)=\mathcal {O}(f(x))\).
We remark that the GT estimator applies to the more general setting of sums and differences of log-normals. This generality of the GT estimator, however, comes at the cost of not being the most efficient estimator for sums—the case we consider here.
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Acknowledgements
We would like to thank Patrick Laub for his valuable feedback and comments on earlier drafts of this work and for sharing his computer code for pdf estimation. Zdravko Botev has been supported by the Australian Research Council Grant DE140100993. Robert Salomone has been supported by the Australian Research Council Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS), under Grant No. CE140100049.
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Appendix
Appendix
1.1 Proof of Theorem 1
Proof
To proceed with the proof we recall the following three facts. First, note that \(\ell (\gamma )=\mathbb {P}(\varvec{1}^\top \exp (\varvec{Y})\le \gamma )\), where \(\varvec{Y}=\varvec{\nu }+\mathbf {L}\varvec{Z}\). Using Jensen’s inequality, we have that for any \(\varvec{w}\in \mathcal {W}\):
Second, denote \(\bar{\varvec{w}}={{\,\mathrm{argmin}\,}}_{\varvec{w}\in \mathcal {W}} \varvec{w}^\top \varSigma \varvec{w}\) and the set \(\mathcal {C}_\gamma \equiv \{\varvec{z}: \varvec{1}^\top \exp (\mathbf {L} \varvec{z}+\varvec{\nu })\le \gamma \}\). Then, we have the asymptotic formula, proved in Gulisashvili and Tankov (2016, Formulas (13) and (63)):
where \(c_1\) is a constant, independent of \(\gamma \). Thirdly, consider the nonlinear optimization
with explicit solution
Then, we obtain the following bound on the second moment:
By substituting (19) in the last line, we obtain the upper bound
In other words, from (17) we deduce that
and therefore
which implies that the algorithm is logarithmically efficient with respect to \(\gamma \). \(\square \)
1.2 Proof of Lemma 1
Proof
Let \(N{\mathop {=}\limits ^{\mathrm {def}}}\sum _{i=1}^d\mathbb {I}{\{X_i>\gamma \}},\) so that \(\ell _1(\gamma )=\mathbb {P}(N\ge 1)\simeq \ell _\mathrm {as}\) and the residual
Note that \(\mathbb {P}(N>1)=\varTheta (r(\gamma ))\) and \(\mathbb {P}_g(N=1)=\mathbb {P}(N=1)/\ell _\mathrm {as}(\gamma )=\varTheta ( 1)\), where g is the mixture density defined in (8). We thus obtain
Therefore, since \(r(\gamma )=o(\ell _\mathrm {as}(\gamma ))\), we have:
and
Therefore, the relative error is \(\mathbb {V}\mathrm {ar}(S_n^2)/\mathbb {V}\mathrm {ar}^2(\hat{\ell }_1)=\varTheta (\ell _\mathrm {as}(\gamma )/r(\gamma ))\). By Lemma 4 there exists an \(\alpha >1\) such that
which shows that \(\frac{\ell _\mathrm {as}(\gamma )}{r(\gamma )}\) grows at least at the exponential rate \(\exp (\frac{(\alpha ^2-1)\ln ^2(\gamma )}{2\sigma ^2})\). \(\square \)
This completes the proof. The proof also fills in the omitted details for (Botev and L’Ecuyer 2017, Proposition 1).
1.3 Proof of Lemma 2
Proof
First we show 1. To this end, recall that \(\varvec{X}=\exp (\varvec{Y})\), where \(\varvec{Y}\sim \mathsf {N}(\varvec{\nu },\varSigma )\). Further, recall the well-known property (which is strengthened in Lemma 4) that for \(i\not =j\) and \(\mathrm {Corr}(Y_i,Y_j)<1\), the pair \(Y_i,Y_j\) is asymptotically independent in the sense that
In fact, Lemma 4 shows that this decay to zero is exponential. The consequences of this are \( \mathbb {P}(\max _i Y_i>\gamma )\simeq \sum _i \mathbb {P}(Y_i>\gamma ) \) and
With these properties, we then have the lower bound:
Next, using the result \(\mathbb {P}(S>\gamma ,X_k=M<\ln \gamma )=o(\mathbb {P}(X_k>\ln \gamma ))\) from Lemma 3, we also have the analogous upper bound:
whence we conclude that \(\mathbb {P}(S>\gamma ,X_k=M)\simeq \mathbb {P}(X_k>\gamma )\).
Next, we show point 2. Using the facts that: (1) the fewer the active constraints in any solution, the closer its minimum is to zero (without constraints the minimum of (12) is zero); (2) any solution satisfies the Karush–Kuhn–Tucker (KKT) necessary conditions:
we can verify by direct substitution that \(\varvec{\mu }^*\) satisfies the KKT conditions asymptotically as \(\gamma \uparrow \infty \) and that it causes only one constraint to be active (\(g_1(\varvec{\mu }^*)=o(1)\)). Moreover, it yields the asymptotic minimum:
Finally, we show point 3, which is the linchpin of the proposed methodology. To this end, consider the \((m+1)\)-st moment with \(\varvec{\mu }\rightarrow \varvec{\mu }^*\) as \(\gamma \uparrow \infty \):
Next, notice that the measure \(\mathbb {P}_{-m\varvec{\mu }^*}\) is equivalent to first simulating
and then, given \(Y_k=y_k\), simulating all the rest of the components, denoted \(\varvec{Y}_{-k}\), from the nominal Gaussian density \(\phi _\varSigma (\varvec{y}-\varvec{\nu })\) conditional on \(Y_k=y_k\), that is, \(\varvec{Y}_{-k}\sim \phi _\varSigma (\varvec{y}-\varvec{\nu }|y_k)\). In other words, asymptotically, the effect of the change of measure induced by (12) is to modify the marginal distribution of \(X_k\) only. Thus, repeating the same argument used to prove part 1, we have
Therefore, as \(\gamma \uparrow \infty \),
Then, the part 3 of Lemma 2 follows from putting \(m=1\), and observing that
\(\square \)
Lemma 3
We have \(\mathbb {P}(S>\gamma ,X_k=M<\gamma )=o(\mathbb {P}(X_k>\gamma ))\) as \(\gamma \uparrow \infty \).
Proof
Let \(\beta \in (0,1)\) and \(M_{-k}=\max _{j\not =k}X_j\). Then, using the facts:
and
we obtain \(\overline{\varPhi }(\ln (\gamma -\gamma ^\beta ))\simeq \overline{\varPhi }(\ln \gamma )\) for any \(\beta \in (0,1)\). More generally,
Then, we have \(\mathbb {P}(S>\gamma ,X_k=M<\gamma )=\)
Since for large enough \(\gamma \) there exists a \(\beta '\in (\beta ,1)\) such that \((d-1)\gamma ^\beta <\gamma ^{\beta '}\), we have
The proof will then be complete if we can find a \(\beta \in (0,1)\), such that (\(u=\ln \gamma \))
Since \(\mathbb {P}(\max _{j\not =k}Y_j>\beta u,Y_k>\beta u)=\mathcal {O}\left( \sum _{j\not =k}\mathbb {P}(Y_j>\beta u,Y_k>\beta u)\right) \), the last is equivalent to showing that the bivariate normal probability \( \mathbb {P}(Y_j>\beta u,Y_k>\beta u)=o(\mathbb {P}(Y_k>u)) \) for some \(\beta \in (0,1)\). This last part then follows from Lemma 4, which completes the proof. \(\square \)
Lemma 4
(Gaussian Tail Probability) Let \(Y_1\sim \mathsf {N}(\nu _1,\sigma _1^2)\) and \(Y_2\sim \mathsf {N}(\nu _2,\sigma _2^2)\) be jointly bivariate normal with correlation coefficient \(\rho \in (-1,1)\). Then, there exists an \(\alpha >1\) such that
where \(a\wedge b\) stands for \(\min \{a,b\}\).
Proof
Without loss of generality, we may assume that \(\sigma _1>\sigma _2\), so that
Define the convex quadratic program:
where \(\varSigma _{11}=\sigma _1^2,\varSigma _{12}=\varSigma _{21}=\rho \sigma _1\sigma _2,\varSigma _{22}=\sigma _2^2\). Denote the solution as \(\varvec{y}^*\). Then, we have the following asymptotic result Hashorva and Hüsler (2003):
where \(d_1\in \{1,2\}\) is the number of active constraints in (20). Next, consider the quadratic programing problem which is the same as (20), except that we drop the first constraint (that is, we drop \(y_1\ge \gamma -\nu _1\)). The minimum of this second quadratic programing problem is \(\frac{(\gamma -\nu _2)^2}{2\sigma _2^2}\), and is achieved at the point \( \tilde{\varvec{y}}= \left( (\gamma -\nu _1)\rho \sigma _2/\sigma _1,\gamma -\nu _2 \right) ^\top \). Note that since \(\tilde{y}_1<\gamma -\nu _1\), we have dropped an active constraint. Since dropping an active constraint in a convex quadratic minimization achieves an even lower minimum, we have the strict inequality between the minima of the two quadratic minimization problems:
for any large enough \(\gamma >\nu _2\). Hence, after rearrangement of the last inequality, we have
and therefore there clearly exists an \(\alpha \) in the range
For such an \(\alpha \) (in the above range), we have
Therefore, \( \exp (-\frac{(\varvec{y}^*)^\top \varSigma ^{-1}\varvec{y}^*}{2})=o\left( \exp (-\frac{(\alpha \gamma -\nu _2)^2}{2\sigma _2^2})\right) ,\; \gamma \uparrow \infty \), and the exponential rate of decay of \(\mathbb {P}(Y_1>\gamma ,Y_2>\gamma )\) is greater than that of \(\mathbb {P}(Y_2>\alpha \gamma )\). This completes the proof. \(\square \)
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Botev, Z.I., Salomone, R. & Mackinlay, D. Fast and accurate computation of the distribution of sums of dependent log-normals. Ann Oper Res 280, 19–46 (2019). https://doi.org/10.1007/s10479-019-03161-x
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DOI: https://doi.org/10.1007/s10479-019-03161-x