Estimation for a Second-Order Jump Diffusion Model from Discrete Observations: Application to Stock Market Returns

This paper proposes a second-order jump diffusion model to study the jump dynamics of stock market returns via adding a jump term to traditional diffusion model. We develop an appropriate maximum likelihood approach to estimate model parameters. A simulation study is conducted to evaluate the performance of the estimation method in finite samples. Furthermore, we consider a likelihood ratio test to identify the statistically significant presence of jump factor.The empirical analysis of stock market data from North America, Asia, and Europe is provided for illustration.


Introduction
Continuous-time stochastic processes have been widely used to model securities prices for option valuation.Nicolau considered a second-order diffusion process which is defined by where (  ) and (  ) are the drift and diffusion functions, respectively [1].  is a standard one-dimensional Brownian motion.  is directly observable and differentiable.In this model,   can also be expressed as the integrated process For model (1), a nonparametric approach which is based on the infinitesimal generator and Taylor series expansion has been developed to estimate the drift and diffusion functions [1].Thereafter, Wang and Lin [2] presented the local linear estimation of the diffusion and drift functions and proved that the estimators are weakly consistent.Wang et al. [3] proposed the reweighted estimation of the diffusion function and investigated the consistency of the estimator.Furthermore, Hanif [4] studied the nonparametric estimation of the drift and diffusion functions using an asymmetric kernel and proved that the estimators are consistent and asymptotically normal.
As pointed out by Nicolau [1,5], model ( 1) is especially useful in empirical finance.First, the model accommodates nonstationary integrated stochastic process   that can be made stationary by differencing.Second, in the context of stock prices,   represents stock return and   indicates the cumulation of   .The model suggests directly modeling return rather than stock price and meets many general properties of stock returns such as stationarity in the mean, nonnormality of the distribution and weak autocorrelation.
In financial markets, the correct specification of drift and volatility is essential and instructive among practitioners in obtaining valid conclusions.Unfortunately, the existing economic theory generally provides little guidance about the precise specification of them.Model misspecification may lead to misleading conclusions in estimation and hypothesis testing.Therefore, much attention has been paid to the issue of specifying the functions forms for continuous-time diffusion models.On the other hand, it has the advantage that the problem of its estimation can be reduced to the determination of some low-dimensional parameters by applying more efficient statistical methods (see [6,7] for details).In particular, Nicolau [1,5] pointed out that is a promising model for stock returns and possesses some interesting properties.In Nicolau's empirical studies on the American stock markets, a regular pattern in all estimators of drift and diffusion can be observed: the drift is clearly linear, the volatility is a quadratic function with a minimum in the neighborhood of zero, and the specification  2 (  ) =  + (  − ) 2 fits the nonparametric estimators very well.Furthermore, Yan and Mei [8] developed the generalized likelihood ratio test to check the empirical finding of Nicolau.
The empirical analysis of real-world data sets demonstrates that it is in general reasonable to suppose that the volatility is a quadratic function in stock markets.
The empirical distribution of stock returns typically exhibits skewness and excess kurtosis, which could be induced by various macroeconomic shocks, such as the unemployment announcement, the Gulf war, and the oil crisis.Unfortunately, the standard second-order diffusion framework (3) is not tailored to capture these stylized facts.A more appropriate specification is to modify the aforementioned standard second-order diffusion model to allow for discontinuity.This is easily obtained by combining the general diffusion model with a jump factor.The popularity of the jump diffusion models stems from at least two facts.First, as distinguished from pure diffusion processes, the jump processes can affect and match high levels of kurtosis and skewness.Second, they are economically attractive because they admit that stock prices change by sudden jumps in a short time, which is a reasonable assumption for an efficient stock market.Recently, jump noises have been also applied in population models to describe the abrupt changes of population sizes (see [9,10], etc.).For example, Zhou et al. [9] introduced a two-population mortality model with transitory jump effects and applied it to pricing catastrophic mortality securitizations.
In view of the abovementioned facts, we extend the work of Nicolau [1,5] and then concentrate on a new second-order jump diffusion model: where the arrival of jumps   is governed by the continuoustime Poisson process   with frequency parameter , which denotes the average number of jumps per year.The jump size may be a constant or may be drawn from a probability distribution.The diffusion and Poisson process are independent of each other, and each of them is independent of jump   as well.Consequently, the stock return is the sum of three components.The component ( 0 +  1   ) represents the instantaneous expected return on the stock.
The √  + (  − ) 2   part describes the instantaneous variance of the stock return due to the arrival of "normal" information, and the     () part describes the total instantaneous stock return owing to the arrival of "abnormal" information.Next, we develop parameter estimation methodology to estimate the coefficients in the drift, diffusion, and jump factor.

Model Estimation
For model ( 4), the integrated process   is generally observable at the time points {  ,  = 0, 1, ⋅ ⋅ ⋅ ,  + 1}, while   is a nonobservable process.In fact, for the fixed sampling interval Δ =   −  −1 , the exact distribution of {   } +1 =0 is generally not explicit.An exception is the case where   follows an Ornstein-Uhlenbeck process [11].In practical applications, the time points are equally spaced.For example, when the time unit is a year, weekly data are sampled at   =  0 + (/52) ( = 0, 1, ⋅ ⋅ ⋅ ,  + 1) for a given initial time point  0 .Based on   =  0 +∫  0    and the discrete-time observations {   } +1 =0 , we have Then    can be approximated by Naturally, the smaller the time span Δ is, the closer    is to    .In fact, stock prices are usually observed daily or higher frequency.
Let Δ   =   +1 −   and    =  +1 −   .According to the independent increment property of the standard Brownian motion,    ( = 1, 2, ⋅ ⋅ ⋅ , ) are independently and normally distributed with mean zero and variance Δ.Therefore, we can write are independently and identically distributed as the standard normal distribution.Then the Euler discretization of model ( 4) can be expressed as where    is normally distributed with mean   and variance  2  .Δ   () is the discrete-time Poisson increment.We set the mean jump size equal to zero,   = 0, guaranteeing a symmetric stock return distribution according to [12,13].
It is well recognized that discretization of continuous-time diffusion models for estimation does introduce an estimation bias, but this is relatively small (see [14]).In this case the discrete-time approach (7) allows us to estimate the proposed model where the jump is normally distributed.In this model, the density functions, a mixture of Poisson−Gaussian distribution is generally used to define the jump diffusion model.Here, we employ a Bernoulli approximation, first introduced in [8], to approximates Poisson−Gaussian distribution underlying the discretized (7).We assume that in each time interval either only one jump occurs or no jump occurs.No other information arrivals over this period of time are allowed.This is tenable for short frequency data, e.g., daily stock returns, and may be debatable for data at higher frequencies.As Ball and Torous demonstrate, it provides an approximation procedure which is highly tractable, stable, and convergent [8].Hence, the discretized (7) can be rewritten as where which approximates the true Poisson−Gaussian density with a mixture of normal distributions.Then Φ can be estimated via maximizing the profile pseudo likelihood function which yields estimators of Φ.The maximization problem can be carried out in many scientific computing packages.Note that the likelihood function ( 10) is complex nonlinear function based on transition probability density; hence the analytical formula for the estimators could not derived and the large sample property of estimators becomes challenging.We will not discuss this aspect any further.For more details, we refer the reader to Xu and Wang [15] or Zheng and Lin [16].
The numerical technique, sequential quadratic programming algorithm (see [17,18]) is considered to solve this maximization problem.In our implementation, the likelihood function ( 10) is numerically maximized with the "fmincon" routine embedded in the "Optimization Toolbox" of MAT-LAB using the sequential quadratic programming algorithm.Furthermore, in [19], maximum likelihood estimation allows the construction of approximate confidence intervals for the parameters of interest and these confidence intervals are asymptotically optimal.
Remark 1.The sequential quadratic programming algorithm has been proved to be an excellent nonlinear programming method for solving constrained optimization problems in a variety of statistical models, such as linear models with longitudinal data under inequality restrictions [15], semiparametric regression models with censored data [16], autologistic models and exponential family models for dependent data [20], Cox-Ingersoll-Ross model for the interest rate [21], integer-valued GARCH models [22], and Gaussian stochastic process (GASP) models [23].Our experience shows that the algorithm performs very well for moderate sample sizes.To gain more confidence in the estimates, we also try "Global Optimization Toolbox" of MATLAB based on the "fmincon" routine.We find that choosing different initial values yields identical results.
Furthermore, we also provide a formal statistical test for the presence of jumps in the stock and stock index returns.Since a pure diffusion model is nested within a combined diffusion and jump model, a likelihood ratio test can be used to test the null hypothesis  0 : stock and stock index returns are normally distributed.The corresponding likelihood ratio test statistic is where Φ * denotes the maximum likelihood estimates under jump diffusion model and Φ 0 is the maximum likelihood estimates corresponding to the situation when no jump structure occurs (i.e., =0).Under the null hypothesis, the stock returns are consistent with a log-normal diffusion process without jump factor and Ω is asymptotically distributed  2 with −5 degrees of freedom, where  denotes the number of parameters to be estimated.More details about asymptotically distribution of the likelihood ratio test statistic can be found in ( [24,25]).

Simulation Study
In this section, we conduct a simulation study for secondorder jump diffusion model (4) aimed at examining the performance of the estimation methodology.We consider a special case of (4): where   ∼ (0, 0.1 2 ), probability of jump =0.5, and initial values  0 =0.01 and  0 =0.The values of the parameters except jump term are given in [1].We generate 100 replications (paths) of   and   according to the above design with each replication consisting of  (=1260, 2520, 3780, 5040) daily observations and a time step between two consecutive observation equal to Δ=1/252.Then the coefficients  0 and  1 in the drift functions, , , and , in the diffusion function, and   and , in the jump factor, can be estimated by the proposed estimation methodology.The empirical sample mean (MEAN), empirical standard deviation (SD), and root mean square error (RMSE) of the coefficient estimators α0 , α1 , β, τ, θ, σ , and q are reported (Table 1).We can see the mean of the estimators becomes more accurate; meanwhile the empirical standard deviation and root mean square error become smaller as the observations  increases from 1260 to 5040.SENG), Australia (SPASX200), Switzerland (SMI), and Taiwan (TWII).The datasets are available at the Yahoo Finance website (https://finance.yahoo.com/).Following the conventional practice in stock market research, we take the logarithmic transformation of the selected data.2).The change in stock prices is clear, positively skewed, and has excess kurtosis indicating that leptokurtosis is undeniable, which also predicates the use of a jump diffusion model.The Jarque-Bera test of both dataset rejects the hypothesis of normality.Other stock prices and indexes have similar statistical characterization.We do not enumerate them here.

Parameter Estimation and Testing
Results.We shall summarize and compare the parameter estimates between second-order diffusion model (3) and second-order jump diffusion model (4) for the stocks and stock indexes (Tables 3 and 4).In addition to the 7 parameters to be estimated ( 0 and  1 in instantaneous mean and , , and  in the   volatility component, the variance  2  and probability  of jump factor), the standard error, the log-likelihood value (ln L), and the likelihood ratio test statistic Ω are also reported.For each sampled stock and stock index, the first row and the second row present maximum likelihood estimates and corresponding standard errors of model ( 4), respectively.The third row and the fourth row present maximum likelihood estimates and corresponding standard errors of model (3).
Note that the estimates of standard errors of Φ * and Φ 0 for the maximum likelihood estimation are obtained from the main diagonal of the inverse of the Hessian matrix, respectively.
It can be seen that the standard errors of parameter estimates for jump diffusion model (4) are extremely small, confirming the preferability of the estimation methodology, and are mostly smaller than that of pure diffusion model (3) (Tables 3 and 4).For all the stocks and stock indexes, the corresponding log-likelihood values, ln L, of model ( 4) are also higher than that of model (3).Furthermore, the likelihood ratio test demonstrates the presence of jumps at the 1 percent significance level for each stock and stock index.These empirical results show that the second-order jump diffusion model ( 4) is more suitable for modeling the stock market returns from North America, Asia, and Europe than second-order pure diffusion model (3).

Conclusion and Discussion
In this paper, a new second-order jump diffusion model is proposed to investigate the dynamic characteristic of stock market returns from North America, Asia, and Europe.The maximum likelihood approach is employed to estimate parameters of the model.The simulation study demonstrates that the estimation approach works satisfactorily.By analyzing stock market data from North America, Asia, and Europe, we find that the second-order jump diffusion model can capture high levels of kurtosis and skewness of stock return distributions, while the second-order diffusion model that ignores the jump factor is misspecified.The likelihood ratio test also confirms the statistically significant presence of jump factor in the selected stock market returns.
It should be pointed out that there are some topics that can be further considered.This paper mainly focuses on the Poisson jump.In fact, there are some random perturbations, such as the telephone noise which can be further studied [26].Besides, stochastic differential equations have been recently applied into the SIS and SIRS epidemic models with Markovswitching [27][28][29], the budworm growth model with regime switching [30], and stochastic regime switching predatorprey model [31,32].It is very promising to incorporate the jump factors into these stochastic models.

4. 2 .
Descriptive Statistics.Microsoft Corporation's daily stock prices from January 25, 2000 to January 25, 2005 are plotted in Figure 1.The most striking feature in the figure is the existence of conspicuous spikes in the stock prices and returns.The descriptive statistics are reported (Table

Figure 1 :
Figure 1: Daily closing stock price of Microsoft Corporation from January 25, 2000 to January 25, 2005.
* * * * * * Indicates significance at 1% level.Note: The statistics reported are for log-price (  ) and change (Δ  ).The Jarque-Bera statistic tests for normality distribution.If the data are normally distributed, the skewness is 0, whereas the kurtosis is equal to 3.

Table 3 :
Parameter estimates for stocks.

Table 4 :
Parameter estimates for stock indexes.Standard errors of maximum likelihood estimates are in parentheses.