Lattice approximation schemes for skew diffusions: financial and actuarial applications

This study proposes a recombining lattice-based methodology useful to discretize skew diffusions. The introduction of skew stochastic processes to evaluate financial and insurance products allows to cap-ture more accurately the dynamics of the underlying activity, i.e., stock prices in finance or reference funds in actuarial markets, thus overcoming the biases introduced by the Black-Scholes model, and permits to obtain more accurate evaluations of the analyzed products. The lattice-based discretization is flexible and applicable to a wide range of skew diffusions as, for instance, skew geometric Brownian motions, skew Cox-Ingersoll-Ross processes, skew constant-elasticity-of-variance processes, to name just a few, and is suitable for evaluating financial derivatives, bonds, interest-sensitive claims and, particularly, when American-style contracts are considered. In this perspective, it is useful not only in financial but also in actuarial applications where such claims are embedded in several structured insurance policies, e.g., equity-linked policies with surrender options. The proposed binomial lattice model is generated directly starting from the original skew process, which is made piecewise tractable by simply applying the Harrison-Shepp (1981) methodology to manage the local time component. The method is able to force a heteroske-dastic process to be homoscedastic to obtain a recombinant and com-putationally simple lattice in which the number of nodes grows up linearly with the number of time steps. The lattice is constructed starting from the computation of the node values on the edges, while the values for the inner nodes are simply determined by generating horizontal layers of nodes starting from the ones located on the two edges. Transition probabilities are computed by imposing that the two successor points for each node bracket the expected process val-ue in the next time interval, resorting to multiple upward or down-ward jumps when necessary to obtain a legitimate probability. Other lattice methods presented in the literature both of the binomial and trinomial-type, as in Zhuo et al. (2017) and Menoukeu-Pamen et al. (2023), need a preliminary transformation of the Nelson-Ramaswamy (1990) type to make the skew process homoskedastic before applying the Harrison-Shepp (1981) methodology. Hence, with respect to the existing lattice-based models, the main advantages of the proposed approach rely on its flexibility and direct applicability to the original skew process without resorting to any preliminary process transfor-mation, other than the one used to manage the local time component. An alternative procedure to approximate skew diffusions has been presented by Ding et al. (2021). They propose a continuous-time Markov chain approximation and obtain an explicit closed-form ap-proximation of the transition density of a general skew diffusion pro-cess, which facilitates the valuation of various financial contracts. Numerical comparisons with the existing models show the superiority of the proposed lattice approach in terms of computational cost. Fur-thermore, additional experiments demonstrate that it is accurate and efficient, and recovers various benchmark results in the literature when applied to evaluate European and American call and put op-tions, equity-linked policies with or without surrender options, bonds, and interest-sensitive claims. Future developments of the proposed approach will be based on the investigation of its applicability to more complex skew frameworks, like the doubly skewed Cox-Ingersoll-Ross process, and to skew stochastic volatility model in which the latent stochastic variance follows a skew diffusion process, as in the skew SABR model and skew Heston-SABR model, to name just a few. The latter is more challenging because it requires the con-struction of a bivariate lattice environment. Indeed, the proposed lat-tice model would be useful to discretize the skew stochastic volatility process but, then, it needs to manage the volatility component ap-pearing in the underlying activity process and the correlation affect-ing the two processes. Hence, new numerical techniques will be de-veloped.