Comparison of Jump-Diffusion Parameters Using Passage Times Estimation

Abstract

The main purposes of this paper are two contributions: (1) it presents a new method, which is the first passage time generalized for all passage times (PT method), in order to estimate the parameters of stochastic jump-diffusion process. (2) It compares in a time series model, share price of gold, the empirical results of the estimation and forecasts obtained with the PT method and those obtained by the moments method applied to the MJD model.

1. Introduction

The parameters estimation is one of main dynamic models problems in many scientific fields, particularly in economics and finance. In the reference model proposed in 1973 by Black and Scholes [1] some assumptions (constant volatility, log-normality of returns, continuities of trajectories,…) are required. This process is known as the geometric Brownian. A number of empirical observations clearly contradict these assumptions [2–7]. Now, many different models are proposed (see [4, 7, 8]) to modify the Black-Scholes model. To solve the problems associated with the Black-Scholes model, Merton, in 1976, introduced a new financial model by using the discontinuities by a Poisson process with Gaussian jumps. We take Merton-Jump Diffusion (MJD) model to estimate the parameters using the method of moments. We then present and use the Passage Time (PT) method to estimate the same parameters and we finally compare the empirical results obtained by the differents methods on a time series model: gold share price. Our sample comprises observations data of gold share price (US$ Gold Fixing Price in London Bullion Market, in U.S. Dollars per Ounce) and observations started on January 2, 2007 to October 31, 2007 covering a period of 212 days.

2. Merton Jump-Diffusion Model

We consider that the asset price 𝑆_𝑡 presents log-normal jumps 𝑉₁,…,𝑉_𝑗 at random times 𝜏₁,…,𝜏_𝑗, which represent the moments of jumping of a Poisson process [3, 4, 9–11]. Between two instants, we assume that the dynamics of the model follows the Black-Sholes process model. It is a continuous time model. We suppose that the behavior of the stock price is determined by the stochastic differential equation: 𝑑𝑆_𝑡=𝜇𝑆_𝑡𝑑𝑡+𝜎𝑆_𝑡𝑑𝑊_𝑡,𝑆₀=𝑥₀, (1) where 𝑊 is a standard Brownian motion and 𝜇 and 𝜎 are, respectively, the drift and diffuse coefficient.

In the MJD model, the price process 𝑆_𝑡 is assumed to follow the stochastic differential equation 𝑑𝑆_𝑡𝑆_𝑡−=𝜇𝑑𝑡+𝜎𝑑𝑊_𝑡+𝑑^𝑁𝑡_𝑗=0𝑉_𝑗−1, (2) where log𝑉_𝑗∼i.i.d.𝑁(𝛼,𝛿²), normally distributed. The last term models the jumps. A jump is modelled by the random variable 𝑉 which transforms the price 𝑆_𝑡 to 𝑉𝑆_𝑡. The difference (𝑉−1) is the relative change in price when a Poisson jump occurs. Using Ito’ lemma, the solution of (2) is 𝑆_𝑡=𝑆₀1exp𝜇−2𝜎²𝑡+𝜎𝑊_𝑡^𝑁𝑡_𝑗=0𝑉_𝑗, (3) where 𝑆₀ is the asset price at time zero and with 𝑌_𝑗=log𝑉_𝑗 we can write 𝑋_𝑡𝑆=log^𝑡𝑆₀=1𝜇−2𝜎²𝑡+𝜎𝑊_𝑡+^𝑁𝑡_𝑗=0𝑌_𝑗. (4)

The processes 𝑊, 𝑁 and the random variables 𝑌_𝑗∼𝑁(𝛼,𝛿²) are supposed to be independent.

The discontinuities of the price process are described by the Poisson process 𝑁 with intensity 𝜆 (mean arrival rate of jumps per unit time) and jump 𝑉_𝑗. Introduction of the MJD model adds three extra parameters (𝛼,𝛿²,𝜆) to the Black-Sholes process model which contains two parameters (𝜇,𝜎²). We consider that the jumps, so Ball and Torous [12] and Beckers [13], are symmetrical and with mean value null. More sophistical asymmetric models are proposed [14–16].

Discretized over [𝑡,𝑡+Δ[, the MJD model takes the form Δ𝑋_𝑡=1𝜇−2𝜎²Δ+𝜎Δ𝑊_𝑡+^Δ𝑁𝑡_𝑗=0𝑌_𝑗, (5) where Δ𝑊_𝑡=𝑊_𝑡+Δ−𝑊_𝑡∼𝑁(0,Δ) and Δ𝑁_𝑡=𝑁_𝑡+Δ−𝑁_𝑡 is the number of jumps occuring during the time interval [𝑡,𝑡+Δ[.

The probability density of Δ𝑋_𝑡 can be expressed [17] as 𝑓(𝑥)=^∞_𝑛=0(𝜆Δ)^𝑛𝑒𝑛!^−𝜆Δ×12𝜋𝜎²Δ+𝑛𝛿²×exp−𝑥−𝜇−(1/2)𝜎²Δ−𝑛𝛼²2𝜎²Δ+𝑛𝛿², (6) where 𝑛=0,1,2,… are the possible number of jumps.

Putting Δ=1, that is, the interval [𝑡,𝑡+1[ and 𝛼=0 (i.e., symmetrical jumps) the density function for Δ𝑋_𝑡=𝑥_𝑡 becomes 𝑓(𝑥)=^∞_𝑛=0𝑒^−𝜆𝜆^𝑛×1𝑛!2𝜋𝜎²+𝑛𝛿²−𝜎×exp𝑥−𝜇−²/2²2𝜎²+𝑛𝛿². (7)

From this density we can deduce 𝐸(𝑋)=^∞_𝑛=0𝑒^−𝜆𝜆^𝑛×𝑛!_∞−∞𝑥2𝜋𝜎²+𝑛𝛿²−𝜎×exp𝑥−𝜇−²/2²2𝜎²+𝑛𝛿²𝐸((𝑋𝑑𝑥𝑋−𝐸))^𝑘=^∞_𝑛=0𝑒^−𝜆𝜆^𝑛×𝑛!_∞−∞𝑥−𝜇−𝜎²/2^𝑘2𝜋𝜎²+𝑛𝛿²−𝜎×exp𝑥−𝜇−²/2²2𝜎²+𝑛𝛿²𝑑𝑥, (8) for 𝑘≥1. We note that the improper integral is the central moment of order 𝑘 of the normal law 𝑁(𝜇−(𝜎²/2),𝜎²+𝑛𝛿²).

3. Parameter Estimation

3.1. Method of Moments

Our model is described by four parameters 𝜇, 𝜎², 𝜆, and 𝛿² and we will calculate the estimators by the moments method which was used by Beckers [13] and Press [18]. The idea was based on the equalization of four empirical moments with the corresponding theorical central moments. This leads to solve a system of four equations that allowed determination of estimators [17]. Given the law of returns 𝑋, the central moments of odd order are null and the central moments of even order can be written 𝐸(𝑋−𝐸(𝑋))^2𝑘=(2𝑘)!2^𝑘𝑘!^∞_𝑛=0𝑒^−𝜆𝜆^𝑛𝜎𝑛!²+𝑛𝛿^2𝑘. (9)

One way to determine the values of the parameters 𝜇, 𝜎², 𝜆, and 𝛿² is to fit the market data, is to solve the following system of equations 𝐸(𝑋)𝜎=𝜇−²2,𝐸(𝑋−𝐸(𝑋))²=𝜎²+𝜆𝛿²,𝐸(𝑋−𝐸(𝑋))⁴=3^∞_𝑛=0𝑒^−𝜆𝜆^𝑛𝜎𝑛!²+𝑛𝛿²²=3𝜎²+𝜆𝛿²²+𝜆𝛿⁴,𝐸(𝑋−𝐸(𝑋))⁶=15^∞_𝑛=0𝑒^−𝜆𝜆^𝑛𝜎𝑛!²+𝑛𝛿²³=15𝜎²+𝜆𝛿²³+3𝜆𝛿⁴𝜎²+𝜆𝛿²+𝜆𝛿⁶. (10)

3.2. PT Method

This method proposes the parameter estimation of the process 𝑋 by the observation of the first passage time 𝑇, [14, 19, 20], by the line of equation 𝑋=𝑆. Janssen et al. used the first passage time method to estimate the two parameters of the stochastic differential equation (Black-Scholes equation). This method consists of determining a constant boundary limit 𝑆 after having decomposed the original time-series in several (𝑘) independent series having the same initial point 𝑥₀=𝑥_1,0=⋯=𝑥_𝑘,0, where 𝑥_𝑖,0 (𝑖=1,…,𝑘) denotes the initial point of the series number 𝑖. This constant terminal 𝑆 intersects all the 𝑘 trajectories. Each first intersection (with the trajectory 𝑖) determines a moment 𝑡_𝑖 (called first passage time). The random variable 𝑇 is the first passage time of the process 𝑋(𝑡)=exp(𝑌(𝑡)) by the point 𝑆. We have 𝑇=inf{𝑡∣𝑋(𝑡)=𝑆,𝑡>0}. The first passage time 𝑇 follows, as proved by Chhikara and Folks [21] and Folks and Chhikara [22], an inverse Gaussian distribution, and its density function, with 𝑋(𝑡₀)=𝑥₀, is 𝑓𝑡𝑆,𝑥₀,𝑡₀=log𝑆/𝑥₀√2𝜋𝜎𝑡−𝑡₀^3/2×exp−log𝑆/𝑥₀−(𝜇−(1/2)𝜎²)𝑡−𝑡₀²2𝜎²𝑡−𝑡₀. (11)

In finance, the trajectory which is composed of 𝑛 observations of the stock price is unique and to apply this method we consider several 𝑘 trajectories from only one. The random variables 𝑇₁,…,𝑇_𝑘 are the moments of the first passage time by the constant 𝑆 in the 𝑘 trajectories (Figure 1), with observed values 𝑡_𝑖: 𝑖=1,…,𝑘.

Figure 1 

First passage time.

Solving the maximum likelihood equations (11) gives the following estimators: 𝜇=log𝑆/𝑥₀𝑡+12𝜎²,𝜎²=log𝑆/𝑥₀²𝑘^𝑘_𝑖=11𝑡_𝑖−1𝑡, (12) where 𝑡=(1/𝑘)∑^𝑘_𝑖=1𝑡_𝑖.

In our work we study, in the first step, the quality of the estimations. We solve this problem by adopting an algorithm (from the simulation of uniform law on the variation intervals of 𝑆) to determine the best estimators 𝜇 and 𝜎², according to certain criteria (RMSE, %RMSE, and %RME) by comparison with the initial series. The chosen couple is the one which minimizes the optimization criteria.

(i) We consider the 𝑘 independent trajectories, each with 𝑛_𝑖 observations (𝑖=1,…,𝑘) and we define 𝑛_𝑖: observations number of the trajectory 𝑖, with ∑^𝑘_𝑖=1𝑛_𝑖=𝑛. 𝑥_𝑖𝑗: observation number 𝑗 of the trajectory number 𝑖, (𝑖=1,…,𝑘; 𝑗=1,…,𝑛_𝑖). 𝑡_𝑖𝑗: observation number instant 𝑗 of the trajectory number 𝑖.

(ii) We find the value 𝑆 by random simulation of the uniform law on the interval [max_{1≤𝑖≤𝑘}min_{1≤𝑗≤𝑛𝑖}𝑥_𝑖𝑗,min_{1≤𝑖≤𝑘}max_{1≤𝑗𝑛𝑖}𝑥_𝑖𝑗].

We note and distinguish two cases: when (𝜇−(1/2)𝜎²)>0(resp.<0), 𝑆>𝑥₀(resp.<𝑥₀).

In the second step, we generalize this method. Figure 2 shows many passage times for each trajectory. The difference is that we use all passage times crossed at the value 𝑆 so that this value is reached by all possible trajectories. This method principle is similar to that of the first method. The random variable 𝑇 follows an inverse Gaussian law. Its density function is written in the form (11).

Figure 2 

All passage times.

Solving the maximum likelihood equations (11) gives the following estimations: 𝜇=log𝑆/𝑥₀𝑡+12𝜎²,𝜎²=log𝑆/𝑥₀²𝑚^𝑚_𝑖=11𝑡^𝑙_𝑖−1𝑡, (13) with 𝑡=(1/𝑚)∑^𝑘_𝑖=1∑^𝑚𝑘_𝑙=1𝑡^𝑙_𝑖and where 𝑚=∑^𝑘_𝑙=1𝑚_𝑙 is the total number of passage times, 𝑚_𝑙 is the number of passage times in the trajectory number 𝑙, and 𝑡^𝑙_𝑖 is the passage time number 𝑙 in the trajectory number 𝑖.

The random variables 𝑇₁,…,𝑇_𝑚 are the instants of the passage time by the line 𝑋=𝑆 in the 𝑘 trajectories, with observed values 𝑡^𝑙_𝑖: 𝑖=1,…,𝑘; 𝑙=1,…,𝑚_𝑘.

4. Empirical Results

4.1. Observations

We used the 212 daily observations of share price of gold to calculate the returns 𝑋_𝑡=log(𝑆_𝑡+1/𝑆_𝑡), 𝑡=0,1,…,211. We obtain an i.i.d. sample with which we calculate the empirical estimators of the necessary four moments: the empirical expectation and the first moments. These observations are represented in Figure 3. We note that our data are not Gaussian. Otherwise, the Black-Scholes model is sufficiently robust. The normality test of Jarque and Bera [23] gives a 𝑃-value equal to 46.0063<𝜒²_1−𝛼=5.99 for 𝛼=0.05. This leads to reject the null hypothesis; that is, the law of return can not be a normal distribution (see [10]).

Figure 3 

Returns evolution.

4.2. Estimations

4.2.1. Method of Moments

According to the MJD model, the parametrization, solving the system (10), gives 𝜇=1.0288×10⁻³, 𝜎²=4.8731×10⁻⁵, 𝜆=0.244, 𝛿²=1.2361×10⁻⁴.

4.2.2. PT Method

This study is based on certain criteria to chose the best method, in particular the criterion wich minimizes the RMSE (root mean squared error loss function), %RMSE (relative root mean squared error loss function), and %RME (relative mean error): RMSE=(1/𝑁)∑^𝑁_𝑖=1𝑒²_𝑖(Θ), %RMSE=(1/𝑁)∑^𝑁_𝑖=1(𝑒_𝑖(Θ)/𝑥_𝑖)², and %RME where 𝑒𝑖(Θ)=𝑥_𝑖−𝑥_𝑖(Θ) are the error terms, such that 𝑥_𝑖and 𝑥_𝑖(Θ) are, respectively, the observed and simulated values. In our time-series the initial value 𝑥₀=608.4 divides the trajectory in two (𝑘=2) trajectories and by random simulation we find 𝑆=640.75, such that all the 2 trajectories are reached by this value 𝑆. The numerical results are 𝜇=0.0021 and 𝜎²=6.5117×10⁻⁴.

4.3. Simulation

We take the parameters 𝜇 and 𝜎² estimated by the two methods. Concerning the other parameters 𝜆 and 𝛿², we used the estimated values by the method of moments (the PT method gives only 𝜇 and 𝜎²). Figure 4 represents the simulation graph and the real values.

Figure 4 

Simulation and true values.

According to the criteria (see Table 1), the PT method minimizises the errors relatively to the method of moments.

Table 1 

Simulation errors.

4.4. Forecast

The forecast days are 10 days. The graphs of the forecast errors for each method are represented in Figure 5. The result analysis shows that the PT method 𝑚𝑎𝑥 error does not exceed 7%. Table 2 represents the forecast errors for the two methods.

Table 2 

Forecast errors.

Figure 5 

Relative errors.

5. Conclusion

In this paper, we evaluate the performance of a new method in the parameters estimation problem of MJD model. According to the simulation and forecast results, we deduce that this PT method gives better results than the method of moments and can be used in other problems based on the stochastic differential equations with jumps.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.