Efficient tail estimation for sums of correlated lognormals

Abstract

Our focus is on efficient estimation of tail probabilities of sums of correlated lognormals. This problem is motivated by the tail analysis of portfolios of assets driven by correlated Black-Scholes models. We propose three different procedures that can be rigorously shown to be asymptotically optimal as the tail probability of interest decreases to zero. The first algorithm is based on importance sampling and is as easy to implement as crude Monte Carlo. The second algorithm is based on an elegant conditional Monte Carlo strategy which involves polar coordinates and the third one is an importance sampling algorithm that can be shown to be strongly efficient.

Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

“…In [7], a computationally expensive discretization scheme, with partition width 1/n, is used to implement the state-dependent importance sampling scheme based on the asymptotic approximation (6) in the case of regularly varying random walks. On the other hand, using (8) for the importance sampling component of an SISR procedure, whose resampling weights are proportional to w k−1 (y k−1 ), can result in a Monte Carlo estimateα B that has a bound similar to (5), which can be used to establish efficiency of the SISR procedure, as we now proceed to show. More importantly, for more complicated models, one can at best expect to have approximations of the type (7) rather than the sharp asymptotic formula (6).…”

“…, (8) in which w 0 ≡ 1 and w k−1 (y k−1 ) is a normalizing constant to make q k (· | y k−1 ) a density function for k ≥ 2. From (7), it follows that…”

“…Consider the SISR procedure with importance density (8), in which g n is given by (25), and with resampling weights (10). Consider the SISR procedure with importance density (8), in which g n is given by (25), and with resampling weights (10).…”

“…Specifically, instead of using (8) to define q k (· | y k−1 ), we define q k (· | y k−1 ) by (26) for the importance sampling step at stage k in the third paragraph of Section 2. Moreover, we now use (27) instead of (9) to define the resampling weights and perform resampling even at stage n.…”

On the Density Functions of Integrals of Gaussian Random Fields

“…The tail approximation of the finite sum of correlated log-normal random variables has been studied in [3]. The corresponding simulation is studied in [5]. There are other general studies on heavy-tailed random variables that include the sum of log-normal random variables as special cases (see [11] and [15]).…”