Efficient Inverse Z-Transform and Pricing Barrier and Lookback Options With Discrete Monitoring

Efficient Evaluation of Double-Barrier Options and Joint CPDF of a Levy Process and Its Two Extrema

“…The iteration procedure in the present paper can be applied in a similar vein to double barrier options with discrete monitoring and calculation of joint probability distributions in stable Lévy models. See [27,26] for the single barrier case. The procedure can be applied to improve the performance of pricing of barrier and American options in regime-switching Lévy models, approximations of models with stochastic volatility and interest rates with regime-switching models, where the technique of the EPV-operators was used [18,19,16,29,7,21].…”

“…We leave to the future the systematic study of accuracy of methods based on time discretization and calculations in the state space. Numerical examples in [27] show that, in the case of options with one barrier, a moderately good accuracy can be achieved using time discretization provided the calculations are in the dual space.…”

“…We denote the expectation operator under Q by E. In a number of publications, various methods were applied to calculation of expectations V (f ; T ; x 1 , x 2 ) of functions of spot value x 1 of X and its running maximum or minimum x 2 and related optimal stopping problems, standard examples being barrier and American options, and lookback options with barrier and/or American features. See, e.g., [35,13,14,15,44,5,6,40,39,7,9,10,42,41,8,20,36,34,37,48,33,47,21,25,27] and the bibliographies therein. Options with discrete and continuous monitoring were considered.…”

Efficient inverse Z-transform: sufficient conditions

“…In [11], we suggested to alleviate this problem deforming the contour of integration {q = re iϕ | − π < ϕ < π} in (1.2) into a contour of the form L L;σ ℓ ,b ℓ ,ω ℓ = χ L;σ ℓ ,b ℓ ,ω ℓ (R), where the conformal map χ L;σ ℓ ,b ℓ ,ω ℓ is of the form (1.3) χ L;σ ℓ ,b ℓ ,ω ℓ (y) = σ ℓ + ib ℓ sinh(iω ℓ + y);…”

“…and truncate the sum. Numerical examples in [11] demonstrate the advantages of the sinhacceleration vs trapezoid rule in applications to the evaluation of the joint cumulative probability distribution function (cpdf) of a Lévy process and its extremum. In the present paper, we explain how to apply the new realization of the inverse Z-transform to more general situations than considered in [11].…”