A general control variate method for option pricing under Lévy processes

On sample average approximation algorithms for determining the optimal importance sampling parameters in pricing financial derivatives on Lévy processes

“…3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Table 1: Convergence w.r.t. different initial points (n = 10 7 , ρ = 10 −8 ) λ * [1] λ * [2] λ * [3] λ * [4] λ * [5] λ * [6] λ * [7] λ * [8] λ * [9] λ * [10] λ * [11] λ * [12] Steps λ * In practice, using very large n and very small ρ is computationally inefficient, so for option pricing, we use n = 1000, ρ = 10 −4 . Using Algorithm 2, Table 2 shows the estimated variance reduction of importance sampling, where "MC price" denotes the estimate price via classical Monte Carlo simulation, "IS price" denotes the estimate price via importance sampling, and "VR ratio" is the variance reduction ratio defined as the variance of classical Monte Carlo simulation divided by the variance of importance sampling.…”

“…Although Monte Carlo simulation can solve high-dimensional problems, variance reduction techniques are often needed to improve the computational efficiency; see Glasserman [4] for a survey. For Lévy processes models, variance reduction techniques used in derivatives pricing include importance sampling (Kawai [8]), control variates (Dingeç and Hörmann [9]), stratified sampling (Kawai [10]), etc.…”

An auto-realignment method in quasi-Monte Carlo for pricing financial derivatives with jump structures

“…To demonstrate the efficiency of the auto-realignment method, variance reduction factors (VRFs) of using various PGMs in QMC with respect to the crude MC method are assessed and compared. The VRF measures the efficiency gain of QMC estimators as in many related literatures (see Dingeç and Hörmann, 2012). The VRF is defined by…”

Closed‐form approximations for spread options in Lévy markets

“…Moreover, closed‐form approximations may also allow for efficient computation of the option Greeks and implied parameters, which are of great importance for hedging purposes, as well as the pricing of new instruments. Finally, analytical approximations can also be useful in developing control variates to increase the efficiency of these Monte Carlo schemes; see the works of Kemna and Vorst, Fusai and Meucci, and Dingeç and Hörmann …”