arXiv subjects

Kevin W. Lu

Publications and source records attributed to Kevin W. Lu.

7 recordsLinked to original sources

arXiv2026

Discrete dividends after maturity adjust the stock and strike prices

The standard method to price European calls on a discrete dividend-paying stock is to subtract the present value of the dividends from the initial stock price in the Black-Scholes formula. However, when there are dividends after maturity, this is inconsistent with the model. Within the escrowed dividend model, we highlight an extension of the Black-Scholes formula in which those dividends after maturity adjust both the stock price and strike price, allowing for calls over all maturities to be priced in a model-consistent way. As a related application to American calls with a single dividend before maturity, we establish a neglected case where it is always optimal to early exercise and derive an extension of the Roll-Geske-Whaley formula when there are dividends after maturity, including fully characterising the optimal exercise policy.

q-fin.MF↗

arXiv2026

Pricing energy spread options with variance gamma-driven Ornstein-Uhlenbeck dynamics

We consider the pricing of energy spread options for spot prices following an exponential Ornstein-Uhlenbeck process driven by a sum of independent multivariate variance gamma processes, which gives rise to mean-reverting, infinite activity price dynamics. Within this class of driving processes, the Esscher transform is used to obtain an equivalent martingale measure with a focus on the weak variance alpha-gamma process. By deriving an analytical formula for the cumulant generating function of the innovation term, we obtain a pricing formula for forwards and apply the FFT method of Hurd and Zhou to price spread options. Lastly, we demonstrate how the model should be both estimated on energy prices under the real world measure and calibrated on forward or call prices, and provide numerical results for the pricing of spread options.

q-fin.MF↗

arXiv2024

Monte Carlo Simulation for Trading Under a Lévy-Driven Mean-Reverting Framework

We present a Monte Carlo approach to pairs trading on mean-reverting spreads modeled by Lévy-driven Ornstein-Uhlenbeck processes. Specifically, we focus on using a variance gamma driving process, an infinite activity pure jump process to allow for more flexible models of the price spread than is available in the classical model. However, this generalization comes at the cost of not having analytic formulas, so we apply Monte Carlo methods to determine optimal trading levels and develop a variance reduction technique using control variates. Within this framework, we numerically examine how the optimal trading strategies are affected by the parameters of the model. In addition, we extend our method to bivariate spreads modeled using a weak variance alpha-gamma driving process, and explore the effect of correlation on these trades.

q-fin.CP↗

arXiv2021

Calibration for multivariate Lévy-driven Ornstein-Uhlenbeck processes with applications to weak subordination

Consider a multivariate Lévy-driven Ornstein-Uhlenbeck process where the stationary distribution or background driving Lévy process is from a parametric family. We derive the likelihood function assuming that the innovation term is absolutely continuous. Two examples are studied in detail: the process where the stationary distribution or background driving Lévy process is given by a weak variance alpha-gamma process, which is a multivariate generalisation of the variance gamma process created using weak subordination. In the former case, we give an explicit representation of the background driving Lévy process, leading to an innovation term which is discrete and continuous mixture, allowing for the exact simulation of the process, and a separate likelihood function. In the latter case, we show the innovation term is absolutely continuous. The results of a simulation study demonstrate that maximum likelihood numerically computed using Fourier inversion can be applied to accurately estimate the parameters in both cases.

math.ST↗

arXiv2021

Necessity of weak subordination for some strongly subordinated Lévy processes

Consider the strong subordination of a multivariate Lévy process with a multivariate subordinator. If the subordinate is a stack of independent Lévy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a Lévy process, otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a Lévy process even when strong subordination does not. Here, we prove that strong and weak subordination are equal in law under the aforementioned condition. In addition, we prove that if strong subordination is a Lévy process, then it is necessarily equal in law to weak subordination in two cases: firstly, when the subordinator is deterministic and secondly, when it is pure-jump with finite activity.

math.PR↗

arXiv2019

Selfdecomposability of Weak Variance Generalised Gamma Convolutions

Weak variance generalised gamma convolution processes are multivariate Brownian motions weakly subordinated by multivariate Thorin subordinators. Within this class, we extend a result from strong to weak subordination that a driftless Brownian motion gives rise to a self-decomposable process. Under moment conditions on the underlying Thorin measure, we show that this condition is also necessary. We apply our results to some prominent processes such as the weak variance alpha-gamma process, and illustrate the necessity of our moment conditions in some cases.

math.PR↗

arXiv2018

Calibration for Weak Variance-Alpha-Gamma Processes

The weak variance-alpha-gamma process is a multivariate Lévy process constructed by weakly subordinating Brownian motion, possibly with correlated components with an alpha-gamma subordinator. It generalises the variance-alpha-gamma process of Semeraro constructed by traditional subordination. We compare three calibration methods for the weak variance-alpha-gamma process, method of moments, maximum likelihood estimation (MLE) and digital moment estimation (DME). We derive a condition for Fourier invertibility needed to apply MLE and show in our simulations that MLE produces a better fit when this condition holds, while DME produces a better fit when it is violated. We also find that the weak variance-alpha-gamma process exhibits a wider range of dependence and produces a significantly better fit than the variance-alpha-gamma process on an S&P500-FTSE100 data set, and that DME produces the best fit in this situation.

stat.ME↗