# Comparison of Jump-Diffusion Parameters Using Passage  Times Estimation

Authors: K. Khaldi; K. Djeddour; S. Meddahi
Journal: Journal of Applied Mathematics
 (2014)

---

## 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.


---

## Body

## 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 priceSt presents log-normal jumps V1,…,Vj at random times τ1,…,τj, 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:
(1)dSt=μStdt+σStdWt,S0=x0,
where W is a standard Brownian motion and μ and σ are, respectively, the drift and diffuse coefficient.In the MJD model, the price processSt is assumed to follow the stochastic differential equation
(2)dStSt-=μdt+σdWt+d(∑j=0Nt‍(Vj-1)),
where log⁡Vj~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 St to VSt. The difference (V-1) is the relative change in price when a Poisson jump occurs. Using Ito’ lemma, the solution of (2) is
(3)St=S0exp⁡{(μ-12σ2)t+σWt}∏j=0Nt‍Vj,
where S0 is the asset price at time zero and with Yj=log⁡Vj we can write
(4)Xt=log⁡StS0=(μ-12σ2)t+σWt+∑j=0Nt‍Yj.The processesW, N and the random variables Yj~N(α,δ2) are supposed to be independent.The discontinuities of the price process are described by the Poisson processN with intensity λ (mean arrival rate of jumps per unit time) and jump Vj. 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
(5)ΔXt=(μ-12σ2)Δ+σΔWt+∑j=0ΔNt‍Yj,
where ΔWt=Wt+Δ-Wt~N(0,Δ) and ΔNt=Nt+Δ-Nt is the number of jumps occuring during the time interval [t,t+Δ[.The probability density ofΔXt can be expressed [17] as
(6)f(x)=∑n=0∞‍(λΔ)nn!e-λΔ×[12π(σ2Δ+nδ2)×exp⁡(-(x-(μ-(1/2)σ2)Δ-nα)22(σ2Δ+nδ2))12π(σ2Δ+nδ2)],
where n=0,1,2,… are the possible number of jumps.PuttingΔ=1, that is, the interval [t,t+1[ and α=0 (i.e., symmetrical jumps) the density function for ΔXt=xt becomes
(7)f(x)=∑n=0∞‍e-λλnn!×[12π(σ2+nδ2)×exp⁡(-(x-(μ-(σ2/2)))22(σ2+nδ2))12π(σ2+nδ2)].From this density we can deduce(8)E(X)=∑n=0∞‍e-λλnn!×∫-∞∞‍x2π(σ2+nδ2)×exp⁡(-(x-(μ-(σ2/2)))22(σ2+nδ2))dxE((X-E(X))k)=∑n=0∞‍e-λλnn!×∫-∞∞‍(x-(μ-(σ2/2)))k2π(σ2+nδ2)×exp⁡(-(x-(μ-(σ2/2)))22(σ2+nδ2))dx,
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 parametersμ, 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
(9)E((X-E(X))2k)=(2k)!2kk!∑n=0∞‍e-λλnn!(σ2+nδ2)k.One way to determine the values of the parametersμ, σ2, λ, and δ2 is to fit the market data, is to solve the following system of equations
(10)E(X)=μ-σ22,E((X-E(X))2)=σ2+λδ2,E((X-E(X))4)=3∑n=0∞‍e-λλnn!(σ2+nδ2)2=3((σ2+λδ2)2+λδ4),E((X-E(X))6)=15∑n=0∞‍e-λλnn!(σ2+nδ2)3=15((σ2+λδ2)3+3λδ4(σ2+λδ2)+λδ6).


### 3.2. PT Method

This method proposes the parameter estimation of the processX 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=x1,0=⋯=xk,0, where xi,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 ti (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
(11)f(S,tx0,t0)=log⁡(S/x0)2πσ(t-t0)3/2×exp⁡{-[log⁡(S/x0)-(μ-(1/2)σ2)(t-t0)]22σ2(t-t0)}.In finance, the trajectory which is composed ofn observations of the stock price is unique and to apply this method we consider several k trajectories from only one. The random variables T1,…,Tk are the moments of the first passage time by the constant S in the k trajectories (Figure 1), with observed values ti: i=1,…,k.Figure 1
First passage time.Solving the maximum likelihood equations (11) gives the following estimators:
(12)μ^=(log⁡⁡(S/x0))t¯+12σ^2,σ^2=[log⁡(S/x0)]2k∑i=1k((1ti)-(1t¯)),
where t¯=(1/k)∑i=1kti.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 ofS) to determine the best estimators μ^ and σ^2, 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 thek independent trajectories, each with ni observations (i=1,…,k) and we defineni: observations number of the trajectory i, with ∑i=1k‍ni=n.xij: observation number j of the trajectory number i, (i=1,…,k; j=1,…,ni).tij: observation number instant j of the trajectory number i.(ii) We find the valueS by random simulation of the uniform law on the interval [max⁡1≤i≤k⁡min⁡1≤j≤ni⁡xij,min⁡1≤i≤k⁡max⁡1≤jni⁡xij].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. Figure2 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:
(13)μ^=log⁡(S/x0)t¯+12σ^2,σ^2=[log⁡(S/x0)]2m∑i=1m‍(1til-1t¯),
with t¯=(1/m)∑i=1k∑l=1mktiland where m=∑l=1kml is the total number of passage times, ml is the number of passage times in the trajectory number l, and til is the passage time number l in the trajectory number i.The random variablesT1,…,Tm are the instants of the passage time by the line X=S in the k trajectories, with observed values til: i=1,…,k; l=1,…,mk.


## 3.1. Method of Moments

Our model is described by four parametersμ, 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
(9)E((X-E(X))2k)=(2k)!2kk!∑n=0∞‍e-λλnn!(σ2+nδ2)k.One way to determine the values of the parametersμ, σ2, λ, and δ2 is to fit the market data, is to solve the following system of equations
(10)E(X)=μ-σ22,E((X-E(X))2)=σ2+λδ2,E((X-E(X))4)=3∑n=0∞‍e-λλnn!(σ2+nδ2)2=3((σ2+λδ2)2+λδ4),E((X-E(X))6)=15∑n=0∞‍e-λλnn!(σ2+nδ2)3=15((σ2+λδ2)3+3λδ4(σ2+λδ2)+λδ6).


## 3.2. PT Method

This method proposes the parameter estimation of the processX 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=x1,0=⋯=xk,0, where xi,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 ti (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
(11)f(S,tx0,t0)=log⁡(S/x0)2πσ(t-t0)3/2×exp⁡{-[log⁡(S/x0)-(μ-(1/2)σ2)(t-t0)]22σ2(t-t0)}.In finance, the trajectory which is composed ofn observations of the stock price is unique and to apply this method we consider several k trajectories from only one. The random variables T1,…,Tk are the moments of the first passage time by the constant S in the k trajectories (Figure 1), with observed values ti: i=1,…,k.Figure 1
First passage time.Solving the maximum likelihood equations (11) gives the following estimators:
(12)μ^=(log⁡⁡(S/x0))t¯+12σ^2,σ^2=[log⁡(S/x0)]2k∑i=1k((1ti)-(1t¯)),
where t¯=(1/k)∑i=1kti.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 ofS) to determine the best estimators μ^ and σ^2, 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 thek independent trajectories, each with ni observations (i=1,…,k) and we defineni: observations number of the trajectory i, with ∑i=1k‍ni=n.xij: observation number j of the trajectory number i, (i=1,…,k; j=1,…,ni).tij: observation number instant j of the trajectory number i.(ii) We find the valueS by random simulation of the uniform law on the interval [max⁡1≤i≤k⁡min⁡1≤j≤ni⁡xij,min⁡1≤i≤k⁡max⁡1≤jni⁡xij].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. Figure2 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:
(13)μ^=log⁡(S/x0)t¯+12σ^2,σ^2=[log⁡(S/x0)]2m∑i=1m‍(1til-1t¯),
with t¯=(1/m)∑i=1k∑l=1mktiland where m=∑l=1kml is the total number of passage times, ml is the number of passage times in the trajectory number l, and til is the passage time number l in the trajectory number i.The random variablesT1,…,Tm are the instants of the passage time by the line X=S in the k trajectories, with observed values til: i=1,…,k; l=1,…,mk.


## 4. Empirical Results

### 4.1. Observations

We used the 212 daily observations of share price of gold to calculate the returnsXt=log⁡(St+1/St), 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<χ1-α2=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, σ^2=4.8731×10-5, λ^=0.244, δ^2=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)∑i=1Nei2(Θ), %RMSE=(1/N)∑i=1N(ei(Θ)/xi)2, and %RME where ei(Θ)=xi-xi(Θ) are the error terms, such that xiand xi(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 σ^2=6.5117×10-4.


### 4.3. Simulation

We take the parametersμ^ and σ^2 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 σ^2). Figure 4 represents the simulation graph and the real values.Figure 4
Simulation and true values.According to the criteria (see Table1), the PT method minimizises the errors relatively to the method of moments.Table 1
Simulation errors.
RMSE
%RMSE
%RME
Method of moments
22.891
0.0327
2.743
PT Method
17.959
0.0271
2.171


### 4.4. Forecast

The forecast days are 10 days. The graphs of the forecast errors for each method are represented in Figure5. 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.
%RME
Method of moments
10.836
PT Method
2.751Figure 5
Relative errors.


## 4.1. Observations

We used the 212 daily observations of share price of gold to calculate the returnsXt=log⁡(St+1/St), 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<χ1-α2=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, σ^2=4.8731×10-5, λ^=0.244, δ^2=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)∑i=1Nei2(Θ), %RMSE=(1/N)∑i=1N(ei(Θ)/xi)2, and %RME where ei(Θ)=xi-xi(Θ) are the error terms, such that xiand xi(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 σ^2=6.5117×10-4.


## 4.2.1. Method of Moments

According to the MJD model, the parametrization, solving the system (10), gives μ^=1.0288×10-3, σ^2=4.8731×10-5, λ^=0.244, δ^2=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)∑i=1Nei2(Θ), %RMSE=(1/N)∑i=1N(ei(Θ)/xi)2, and %RME where ei(Θ)=xi-xi(Θ) are the error terms, such that xiand xi(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 σ^2=6.5117×10-4.


## 4.3. Simulation

We take the parametersμ^ and σ^2 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 σ^2). Figure 4 represents the simulation graph and the real values.Figure 4
Simulation and true values.According to the criteria (see Table1), the PT method minimizises the errors relatively to the method of moments.Table 1
Simulation errors.
RMSE
%RMSE
%RME
Method of moments
22.891
0.0327
2.743
PT Method
17.959
0.0271
2.171


## 4.4. Forecast

The forecast days are 10 days. The graphs of the forecast errors for each method are represented in Figure5. 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.
%RME
Method of moments
10.836
PT Method
2.751Figure 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.



---

*Source: 975418-2014-05-21.xml*