Sensitivity analysis of market and stock returns by considering positive and negative jumps
Authors: Theodosiadou Ourania , Polimenis Vassilis , Tsaklidis George 
Published: October 2016

Purpose
This paper aims to present the results of further investigating the Polimenis (2012) stochastic model, which aims to decompose the stock return evolution into positive and negative jumps, and a Brownian noise (white noise), by taking into account different noise levels. This paper provides a sensitivity analysis of the model (through the analysis of its parameters) and applies this analysis to Google and Yahoo returns during the periods 2006-2008 and 2008-2010, by means of the third central moment of Nasdaq index. Moreover, the paper studies the behavior of the calibrated jump sensitivities of a single stock as market skew changes. Finally, simulations are provided for the estimation of the jump betas coefficients, assuming that the jumps follow Gamma distributions.

Design/methodology/approach
In the present paper, the model proposed in Polimenis (2012) is considered and further investigated. The sensitivity of the parameters for the Google and Yahoo stock during 2006-2008 estimated by means of the third (central) moment of Nasdaq index is examined, and consequently, the calibration of the model to the returns is studied. The associated robustness is examined also for the period 2008-2010. A similar sensitivity analysis has been studied in Polimenis and Papantonis (2014), but unlike the latter reference, where the analysis is done while market skew is kept constant with an emphasis in jointly estimating jump sensitivities for many stocks, here, the authors study the behavior of the calibrated jump sensitivities of a single stock as market skew changes. Finally, simulations are taken place for the estimation of the jump betas coefficients, assuming that the jumps follow Gamma distributions.

Findings
A sensitivity analysis of the model proposed in Polimenis (2012) is illustrated above. In Section 2, the paper ascertains the sensitivity of the calibrated parameters related to Google and Yahoo returns, as it varies the third (central) market moment. The authors demonstrate the limits of the third moment of the stock and its mixed third moment with the market so as to get real solutions from (S1). In addition, the authors conclude that (S1) cannot have real solutions in the case where the stock return time series appears to have highly positive third moment, while the third moment of the market is significantly negative. Generally, the positive value of the third moment of the stock combined with the negative value of the third moment of the market can only be explained by assuming an adequate degree of asymmetry of the values of the beta coefficients. In such situations, the model may be expanded to include a correction for idiosyncratic third moment in the fourth equation of (S1). Finally, in Section 4, it is noticed that the distribution of the error estimation of the coefficients cannot be considered to be normal, and the variance of these errors increases as the variance of the noise increases.

Originality/value
As mentioned in the Findings, the paper demonstrates the limits of the third moment of the stock and its mixed third moment with the market so as to get real solutions from the main system of equations (S1). It is concluded that (S1) cannot have real solutions when the stock return time series appears to have highly positive third moment, while the third moment of the market is significantly negative. Generally, the positive value of the third moment of the stock combined with the negative value of the third moment of the market can only be explained by assuming an adequate degree of asymmetry of the values of the beta coefficients. In such situations, the model proposed should be expanded to include a correction for idiosyncratic third moment in the fourth equation of (S1). Finally, it is noticed that the distribution of the error estimation of the coefficients cannot be considered to be normal, and the variance of these errors increases as the variance of the noise increases.