SUITABLE MEASURES OF JUMPS IN STOCHASTIC MODELS FOR STOCK MARKET INDICES

Abstract

The dynamics of the stock market indices log-returns (∆(lnS~t)) have been characterised by non-normal features such as upward and downward jumps of different measures, asymmetric and leptokurtic features. The Bi-Power Variation (BPV) process has been used to develop jump-estimators to detect jumps in (∆(lnS~t). However, the existing jump-estimators are restricted to the BPV case of the Realised Multi-Power Variation (RMPV) processes, and existing models do not accommodate these features. Therefore, this study was designed to construct unrestricted Particular Higher-Order Cases (PHOC) jump-estimators, and build suitable models that can accommodate the jumps and non-normal features found in (∆(lnS~t). The limits in probability and distribution were used to derive the Jump Test Models (JTM) in the PHOC of the RMPV processes. The JTM were used to test for jumps under the null hypothesis (H0) of no jump in (∆(lnS~t) at a 5% level of significance in three stock markets, namely: Nigerian, UK, and Japan. These were used to build the dynamics of (∆(lnS~t). The convolution of densities and the L´ evy-It^ o decomposition methods were used to derive the Probability Density Functions (PDF) and the L´ evy-Khintchine (LK) formulae of two novel skewed models. The maximum likelihood estimation method was used to estimate the optimal values of the parameters in the models. The Kolmogorov-Smirnov, Anderson-Darling statistics, and the basic moments were used to test the suitability of these models to the empirical stock market data and compared with three existing models viz: Black-Scholes (BS), normal and double-exponential jump-diffusion models. The JTM derived for the PHOC of the RMPV processes were: ^ Z m = ∆−0:5 µ −m 2=mfXg[r1;:::;rm] ∆;t fXg[2] ∆;t − 1! ’RMP V rmax 1; p^q^2 −1, for m = 2 · · · 10, where, fXg[r1;:::;rm] ∆;t ; fXg[2] ∆;t; ’RMP V ; p^and ^ q are the RMPV, realised variance, asympvtotic variance, estimators of bi-power and quad-power variation, respectively. Jumps in ∆(lnS~t) were observed and H0 was rejected. The dynamics of ∆(lnS~t) was derived as: ∆(lnS~t) = (µ− 1 2σ2)∆+σ∆Wt +J(Qu j )∆Ntu +J(Qd j)∆Ntd, where,µ; σ; Wt, J(Qu j ); J(Qd j); Ntu and Ntd are respectively drift and volatility parameters, standard Brownian motion, upward and downward jump measures with intensities λu j and λd j, respectively. The PDF of the Asymmetric-Laplace (AL) and the Modified DoubleRayleigh (MDR) were models derived were: f∆(ln S~t)(x) = (1−λ∆t) σp∆t ’ x−(µ− 1 2 σ2)∆t σp∆t + ∆t pκα2λu j exp 2α1µj+α1σ2∆t 2 exp − x − µ − 12σ2 ∆t α1 Φ aµj + qkα2λd jexp 2α2µj+α2σ2∆t 2 exp − x − µ − 12σ2 ∆t α2 Φb − µj and f∆(ln S~t)x = (1−λ∆t) σp∆t ’ x−(µ− 1 2 σ2)∆t σp∆t + pηexp θ2−ρ # 2θexp −(µj − θ)2 # +θpπ#Φa(µj)−µjpπ#Φa(µj) −qηexp ^ θ^2−ρ^ #^ #2^exp −(µj − θ^)2 #^ !+ θpπ#^Φb(µj) − µjpπ#^Φb(−µj) ∆t, where, θ = σju(µd∆t)+µjσ2∆t (σju+σ2∆t) , ρ = µ2 j σ2∆t (σju+σ2∆t), # = 2σ2∆tσju (σju+σ2∆t), θ^ = σjd(µd∆t)+µjσ2∆t (σjd+σ2∆t) , ^ ρ = µ2 j σ2∆t (σjd+σ2∆t), #^ = 2σ2∆tσjd (σjd+σ2∆t), η = λu j (σju)’ x−(µ− 1 2 σ2)∆t σp∆t and ^ η = λd j (σjd)’ x−(µ− 1 2 σ2)∆t σp∆t . The derived LK formulae of the novel models were: (u) = iuµ − 1 2σ2u2 − λu j pkα1 α1−iu + λd j qkα2 α2+iu eiuµj + λd jqk + λu j pk and (u) = iuµ − 1 2σ2µ2 − pλu j σu j + qλd j σd j + pλu j σu j + qλd j σd j eiuµj, respectively.The optimal values of the parameters: (µd; σ; α1; α2; pk; qk; λu j ; λd j; µj) and (µd; σ; σju; σjd; p; qλu j ; λd j; µj) in the models were obtained. The AL and MDR were models fit the empirical distributions better than the existing models, having the BS model in the worst-case scenario. The jump test models of the particular higher-order cases were found to be better jump-estimators. The asymmetric-Laplace and modified double-Rayleigh jumpdiffusion models proved more suitable for capturing jumps and non-normal features in the stock market indices log-returns.