Pricing gold options under Markov-modulated jump-diffusion processes

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Applied Financial Economics

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Pricing gold options under Markov-modulated

jump-diffusion processes

Shih-Kuei Lina, Yu-Min Liana & Szu-Lang Liaoa

a

Department of Money and Banking, National Chengchi University, Taipei City, Taiwan Published online: 29 Apr 2014.

To cite this article: Shih-Kuei Lin, Yu-Min Lian & Szu-Lang Liao (2014) Pricing gold options under Markov-modulated jump-diffusion processes, Applied Financial Economics, 24:12, 825-836, DOI: 10.1080/09603107.2014.914142

To link to this article: http://dx.doi.org/10.1080/09603107.2014.914142

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Pricing gold options under

Markov-modulated jump-diffusion

processes

Shih-Kuei Lin, Yu-Min Lian

*

and Szu-Lang Liao

Department of Money and Banking, National Chengchi University, Taipei City, Taiwan

In this study, we empirically investigate the properties of gold returns, and the European gold options are priced when the underlying gold price dynamics are driven by Markov-modulated jump-diffusion processes. Specifically, the jump events are captured by a compound Poisson process with a log-normal jump size, and the regime-switching intensity rate is governed by a continuous-time finite-state Markov chain. Under an incomplete market setting, we study the valuation of European gold options using the method of Esscher transform. The estimated results and numerical examples are provided.

Keywords: gold price; European gold option; Markov-modulated jump-diffusion process; Esscher transform

JEL Classification: C51; G12

I. Introduction

In a period where globalfinancial markets crash and the global economy is in recession, investors are seeking trusted sources of security for their portfolios. Because the price of gold does not correlate highly with changes in most mineral commodities and otherfinancial assets, this precious metal can play a saving role as a consistent portfolio diversifier – managing risk and mitigating poten-tial losses in the portfolios of investors, an imperative in the prevailing environment (Capie et al.,2005; Baur and Lucey, 2010; Baur and McDermott, 2010; Reboredo,

2013; Zagaglia and Marzo,2013).

A variety of gold-linked instruments (e.g. gold options) have been invented for hedging the fluctuating risk in the gold market. Beckers (1984) and Ball et al. (1985) study the gold options market under the Black–

Scholes framework. Ogden et al. (1990) investigate gold spot and futures options. However, the presence of jumps in the time series of commodities or equities

can have serious implications on pricing-related deriva-tives. Over the last couple of decades, the increasing number of jump events, particularly the globalfinancial crisis in the late 2000s, has created largefluctuations in the gold market. Therefore, it is critical to model the dynamic jump process adequately and price gold options corresponding to the changing gold price according to actual market developments.

Figure 1 depicts several significant jumps of the gold

price within the last four decades. This empirical evidence reveals that the geometric Brownian motion (GBM) is not completely consistent with the reality. As a consequence, incorporating the sudden random shocks into a dynamic model is necessary and significant. The bottom panel of

Fig. 1 exhibits different frequencies for jump events

through time. Specifically, it shows larger jump activities in some time periods. In the energy crisis of the 1970s and global financial crisis of 2008, for instance, gold prices had larger jumps over the past four decades (based on the daily prices of gold from 1971 to 2010). In addition, there

*Corresponding author. E-mail:[email protected]

Vol. 24, No. 12, 825–836, http://dx.doi.org/10.1080/09603107.2014.914142

© 2014 Taylor & Francis 825

exists the so-called jump and volatility clustering in the logarithmic return series of gold prices, which means that periods of large (small) changes tend to be followed by periods of continued large (small) changes. Roughly speaking, a stochastic process is said to have a long memory if it has an autocorrelation function that is not integrable. This happens, for instance, when the autocor-relation function decays asymptotically as a power law. Cheung and Lai (1993) investigate the long memory behaviour in gold returns. Nevertheless, the existing jump-diffusion processes, such as in Merton (1976) and

Kou (2002), are unable to address the phenomena of volatility clustering and long memory.

According to the changing prices of gold in the top panel of Fig. 1, we could identify two regimes of the gold market. Thefirst state is the relatively low-volatility regime and can be viewed as the ordinary state. The second state is the relatively high-volatility regime and can be regarded as the volatile state. Analysing returns from different periods allows us to investigate the poten-tial effect of different jumps over time (switching regimes). The data set used in the descriptive analysis

3 January 1968 1 January 1974 1 January 1980 1 January 1986 1 January 1992 1 January 1998 1 January 2004 31 December 2010 0 200 400 600 800 1000 1200 1400 1600 Time

Gold Prices (US$/Ounce)

Larger Jump Rate Larger Jump Rate Larger Jump Rate 4 January 1968 1 January 1974 1 January 1980 1 January 1986 1 January 1992 1 January 1998 1 January 2004 31 December 2010 –0.21 –0.18 –0.15 –0.12 –0.09 –0.06 –0.03 0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 Time Logarithmic Returns Jump Jump Jump 2008 Global Financial Crisis 1979 Oil Crisis 1973 Oil Crisis

Fig. 1. Daily data for the gold prices (top panel) and logarithmic returns (bottom panel)

Notes: The bottom panel supposes that the daily logarithmic returns over ±3% and ±5% in magnitude are jumps. The spot data are from Datastream and cover the period from 3 January 1968 to 31 December 2010.

consists of the daily gold prices,1 and the descriptive statistics of the logarithmic return series are presented

inTable 1. The skewness and kurtosis suggest a

lepto-kurtic distribution with negatively skewed returns in the gold market. Particularly, in 2006 and 2008, we can observe extreme movements and jumps in the daily returns, respectively. These can be regarded as the volatile state (state 2). In contrast, other periods can be viewed as the ordinary state (state 1). Suppose that the gold logarithmic returns that are over
3% in magnitude are jumps. Table 1 shows that the mean frequency of the jumps in the entire period is 10.5, where the mean frequencies of the jumps in state 1 and state 2 are 3.25 and 25, respectively. State 1, there-fore, is below the long-term average jump activity, whereas state 2 is above the long-term average jump activity. As shown in Table 1, the periods 2005 and 2007 are in state 1, but the regime transitions to state 2 in the periods 2006 and 2008, and then transitions back to state 1 in the periods 2009 and 2010. A further implication of the figures in Table 1 is that during the sample period, jump events could happen over time and the economy switches between two states of jump rates. As shown in Fig. 1and the resulting statistics in Table 1, the changing gold price is likely to exert a slow shift in jump intensity from one state to another, and thus, the arrival intensity of jump events is significantly different under different regimes of the gold price. According to these previously observed characteristics in the gold market returns, such as unanticipated jump events, jump and volatility clustering, leptokurtic and asymmetric features, and the existence of regime-switching intensity rates, we are motivated to develop a Markov specification of jump intensity rates. Thus, allowing for different jump rates under different regimes, our dynamic process of jump activity is set to be a Markov-modulated Poisson process (Last and Brandt, 1995; Chang et al., 2013) instead of a single Poisson process used in the Merton-type jump-diffusion model (JDM; Merton,1976).

The jump-diffusion class of models is applied to capture the asset price dynamics and is used for option valuation. Amin (1993) develops a simple, discrete time model to price options when the underlying process follows a jump-diffusion process, but the model does not explain volatility clustering. Duan et al. (2006) evaluate options when there are jumps in the pricing kernel and correlated jumps in asset prices and volati-lities. The models capture leptokurtosis and volatility clustering but do not show the regime-switching phe-nomenon. Under the regime-switching environment, Elliott et al. (2005) and Elliott and Osakwe (2006) investigate the option prices for pure diffusion dynamics and for pure jump processes, respectively. Chan and Maheu (2002) propose a time-varying Poisson jump model to describe the jump dynamics in stock market returns. Because these models are devel-oped in the discrete time circumstance, they do not provide the closed-form solutions for option prices, and most of them do not test the empirical character-istics nor assess the models.

In this study, we incorporate both jump events and regime-switching intensity rates for the gold price by identifying a Markov-modulated jump-diffusion model (MMJM). Under such a model, the jump events are described as a compound Poisson process with the log-normal jump size setting used by Merton (1976), and the regime-switching intensity rate is captured by a continuous-time finite-state Markov chain whose states represent the hidden states of an economy. Under such gold price dynamics, which is an incomplete market, we employ the Esscher transform technique developed by Gerber and Shiu (1994) to determine the pricing kernel and the Esscher parameters (risk premiums), and then derive the pricing formulas of European gold options. Because jump arrivals and market states are hidden variables, we estimate the model parameters using the expectation maximization (EM) algorithm (Lange, 1995a, b) and obtain the SEs of parameter estimators using the supplemented expectation

Table 1. Descriptive statistics of the gold logarithmic returns

Classification 2005 2006 2007 2008 2009 2010 Total

Trading days 251 252 253 254 256 261 1527 Mean 0.0007 0.0009 0.0011 0.0001 0.0010 0.0009 0.0008 SD 0.0082 0.0152 0.0098 0.0215 0.0131 0.0103 0.0137 Skewness 0.0451 −0.4077 −0.5152 −0.0852 −0.0476 −0.3808 −0.2500 Kurtosis 3.9677 4.1217 4.4910 5.6810 3.9999 4.6083 7.7529 Days exceeding
3% 0 19 3 31 8 2 63 Days exceeding
5% 0 1 0 9 0 0 10

Notes: The descriptive statistics are reported for the gold logarithmic returns from 2005 to 2010. This table shows the number of days each year in which the gold yields a logarithmic return series over
3% and
5% in magnitude.

1 _{The tested data are from Datastream and cover the period from 4 January 2005 to 31 December 2010.}

maximization (SEM) algorithm (Meng and Rubin,

1991). From the empirically estimated parameters in the dynamic model and the derived option prices, we show that the model is more accurate than competing models in pricing European gold call options.

This study makes three major contributions. First, we analyse the behaviour of gold prices to understand the operation of the gold market and the risks involved. Our findings are valuable for the valuation of other gold derivative assets for which the gold price dynamics are expected to follow the Markov-modu-lated jump-diffusion process. Second, we derive the generalized gold option pricing formula via Esscher transform and illustrate that the derived formula can be reduced to the pricing formulas of Merton (1976) and the Black–Scholes model (BSM). Finally, we use actual market data to investigate the pricing perfor-mance of the gold option pricing. The pricing results are significant for investors and for the organization of the gold market.

The remainder of this article is organized as follows. The next section illustrates the continuous-time dynamic model. Section III presents the change of measures and the pricing formulas of gold options. Section IV discusses the empirical and numerical results. Thefinal section presents the conclusions of this study.

II. Model Setting

LetðΩ; F; PÞ be a complete probability space, where P is the physical probability measure. For each t2 0; T½ , the MMJM is used to model the gold price dynamics as follows: SðtÞ ¼ Sð0Þ exp μ 1 2σ 2_{ Λκ}
 tþ σWðtÞ þX ΦðtÞ k¼1 Yk ( ) (1) where the appreciation rate μ and the volatility σ are constants and WðtÞ is a Wiener process under P.

Yk: k ¼ 1; 2; :::

f g are the jump sizes, which are assumed to be independently identically distributed nonnegative random variables with the density function fYðyÞ. If a jump event occurs at time k, the jump size Ykis normally distributed with meanμ_J and varianceσ2

J. Therefore, the mean percentage jump size of the gold price is κ ¼ E exp Y½ ð Þ  1k  ¼ exp μJþ12σ2J

 

 1. The Markov-modulated Poisson process ΦðtÞ is a Poisson process whose stochastic jump intensityλXðtÞ changes according to a hidden Markov chain XðtÞ, with the transition func-tion PijðtÞ on the finite state space X ¼ 1; 2; :::; If g. For

i; j 2 X, we denote the transition rate Ψði; jÞ from state Xð0Þ ¼ i to state X ðtÞ ¼ j of ΦðtÞ as follows:

Ψði; jÞ ¼ Pψði; jÞ; iÞj j; jÞiψði; jÞ; otherwise (

(2)

The notation Ψ ¼ ðΨði; jÞÞ_II represents the I I matrix of the transition rate with diagonal elements ψii¼ 

PI

j¼1; jÞiψij¼ vi. vi is the departure rate at which the process leaves state i. Since the Markov chain has a finite number of states, the Poisson arrival intensity takes discrete values corresponding to each state. Last and Brandt (1995) give the moment-gener-ating function for the joint distribution function of XðtÞ and ΦðtÞ via the Laplace inverse transform as follows:

PΘðζ ; tÞ ¼X 1

n¼0

Pðn; tÞζn_{; 0  ζ  1 (3)} where Pðn; tÞ ¼ ðPijðn; tÞÞII represents the I I transition probability matrix and Pijðn; tÞ ¼ PðX ð0Þ ¼ i; X ðtÞ ¼ j; ΦðtÞ ¼ nÞ denotes the transition probability with n jump times from state Xð0Þ ¼ i to state XðtÞ ¼ j. Here, Pðn; 0Þ ¼ ð1fn¼0gDijÞ, where Dij¼ 1 if i ¼ j and 0, otherwise. Using the Kolmogorov forward equation, the derivative of Pðn; tÞ becomes:

d

dtPðn; tÞ ¼ ðΨ  ΛÞPðn; tÞ þ 1fn1gΛ Pðn  1;tÞ (4) where Λ ¼ ðλXðtÞÞII denotes the I I diagonal matrix of the intensity rate with diagonal elements λi. Thus, the unique solution of PΘðζ ; tÞ can be obtained as

PΘðζ ; tÞ ¼ exp Ψ  ð1  ζ ÞΛðð ÞtÞ (5) where the exponential power series are given as eA_¼P1

n¼0A

n

n! for any I I matrix A and A0¼ ðDijÞ. Applying the Laplace inverse transform of Equation 3

and the unique solution ofEquation 5, we have the joint distribution of XðtÞ and ΦðtÞ at time t with the following equation:

Pðn; tÞ ¼_{@ ζ}@nn_n!PΘðζ ; tÞ_{ζ ¼0} 

 (6)

In Equation 1, we also assume that all random shock

processes WðtÞ, ΦðtÞ, X ðtÞ and Yk are mutually independent.

III. Measure Change and Option Pricing Esscher transform for Markov-modulated jump-diffusion model

There are infinitely equivalent martingale measures to price options since the security economy described by the MMJM is incomplete. In this circumstance, we relax the assumption of a diversifiable jump risk made by Merton (1976) and then apply the Esscher transform used by Elliott et al. (2005) and Elliott and Osakwe (2006) for the MMJM to determine a risk-neutral pricing measure. By decomposing ZðtÞ ¼ log SðtÞ=Sð0Þð Þ ¼ CðtÞ þ JðtÞ, we then get a continuous diffusion part CðtÞ ¼ μ 1

2σ2 Λκ

 

tþ σWðtÞ and a jump part JðtÞ ¼PΦðtÞ_k¼1Yk, for all t2 0; T½ . Now, write FtZand FtX for the P-augmentation of the naturalfiltrations generated by ZðtÞ and X ðtÞ, respectively. For each t 2 0; T½ , we define Ft¼ FtZ_ FtX as the σ- algebra. Then, the Esscher measure Qθh equivalent to P on Ft, with respect toθh2 R for h ¼ C; Jf g, is given by the following:

θh t ð Þ ¼dQθ h dP    Ft ¼ exp θ C_{σW tð Þ}   EP _exp_θC_{σW tð Þ} FZ 0 h i  exp θJ P Φ tð Þ k¼1 Yk ! EP _{exp θ}JΦ tPð Þ k¼1Yk !  _FX 0 2 4 3 5 ¼ exp θC_{σW tð Þ }1 2 θ C_σ  2 t
  exp θJX Φ tð Þ k¼1 Yk Λκθ J t ! (7) whereθC andθJ are the Esscher parameters of the con-tinuous diffusion part and the jump part, respectively. The mean percentage jump size of the gold price becomes κθJ ¼ E exp θJ_Y k    1   ¼ exp θJ_μ Jþ12ðθ J_σ JÞ 2  1. In addition, the concrete form of the Esscher transform densityθhðtÞ is an exponential Ft-martingale.

To facilitate the no-arbitrage pricing with respect to the risk-neutral pricing measure, it is important to con-struct the gold price process so as to satisfy the mar-tingale condition, that is, the existence of a risk-neutral pricing measure, under which the Markov-modulated jump-diffusion process for the gold price is an Ft-martingale. Let the Esscher transform be defined by

Equation 7. Then, the martingale condition is satisfied if

and only if:

θC_¼r μ þ Λκ σ2 (8) and θJ _¼μJ12σ 2 J σ2 J (9)

Appendix A presents the detailed proof.

An equivalent martingale measure can be treated as the Esscher measure Qθh with respect to the measure P. We begin with identifying the gold price dynamics under the risk-neutral pricing measure Qθh. Let θC and θJ _{be the Esscher parameters of the risk-neutral} Esscher measure. Then, under Qθh and conditional on FZ

t;

WθhðtÞ ¼ W ðtÞ  θCσt (10) is a Wiener process. Furthermore, under Qθh, the transi-tion probability matrix Pθhðn; tÞ, the stochastic jump intensity Λθh of the Markov-modulated Poisson process Φθh

ðtÞ, and the jump size Yθh

k are, respectively, given by the following: Pθhðn; tÞ ¼ Pðn; tÞ κθJ þ 1 n exp ΛκθJt (11) Λθh ¼ Λ κθJ þ 1 ¼ Λ exp θJ_μ Jþ 1 2 θ J_σ J  2
 (12) and Y_kθh i:i:d:, N μ _Jþ θJσ_J2; σ2_J (13) which means that the investors receive the premiums  θC_{σ and κ}θJ

þ 1

for the continuous diffusion risk and jump risk at time t, respectively. Therefore, the Wiener process and jump intensity are affected by the measure change. Under Qθh, the risk-neutral transition probability matrix becomes Pθhðn; tÞ with transition rate matrix Ψ and stochastic jump intensity Λθh. Through the change of measures, the jump size Y_kθhi:i:d:,Nðμ_Jþ θJ_σ2

J; σ2JÞ. Appendix B shows the detailed proof. By determining the Esscher parameters, we then get:

WθhðtÞ ¼ W ðtÞ þ μ r  Λκ σ
 t (14) Λθh ¼ Λ exp  μ2J 2σ2 J þ σ2J 8
 (15)

and Y_kθh i:i:d:, N 1 2σ 2 J; σ 2 J
 (16)

whereμrΛκ_σ and exp μ2J

2σ2 J þ σ2 J 8

are the market prices of the continuous diffusion risk and jump risk at time t, respectively. In addition, under Qθh, the jump size Y_kθh is normally distributed with mean 1₂σ2_J and varianceσ2_J. Consequently, the gold price dynamics under Qθh is the following: SðtÞ ¼ Sð0Þ exp r1 2σ 2
 tþ σWθhðtÞ þ X Φθh_ðtÞ k¼1 Y_kθh 8 < : 9 = ; (17) where r¼ μ  Λκ þ θCσ2_{þ Λ}θh_κθJ_{, for all t2 0; T½ .} Pricing European gold options

Under the assumption that there are no arbitrage opportu-nities in the market, we price gold options under the risk-neutral pricing measure Qθh. For the European gold call options with strike price K and time to expiration T , its price at time zero is given by the following:

where Λθh denotes the stochastic jump intensity of the Markov-modulated Poisson process ΦθhðTÞ, πi denotes the stationary distribution in state i, Pθhðn; TÞ denotes the transition probability matrix and n denotes the number of jumps in the time interval 0; T½ . In addition, NðÞ denotes the cumulative distribution function of a standard normal random variable, and

d1;n¼lnð Sð0Þ K Þ þ ðr þ_{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi}12σ2ÞT þ12σ2Jn σ2_T þ σ2 Jn p (19) and d2;n¼ d1;n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2_Tþ σ2 Jn q (20)

Equation 18 can be viewed as a weighted sum of the

expected European gold call option with weights being the transition probabilities. Appendix C gives the detailed proof.

To further illustrate the property of the generalized gold option pricing formula, we consider several special cases and show their specific formulas in the following exam-ples. In a general setting with I states, if λ1 ¼ λ2¼ . . . ¼ λI ¼ λ, then the Markov-modulated Poisson process reduces to the single Poisson process with intensity λ. Therefore, the solution of European gold call options can be derived as follows:

Cλ_MMJMθh ð0Þ ¼ X 1 n¼0 expðλθhTÞðλθhTÞn n! CBSM Sð0Þ; K; r; T; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2_Tþ σ2 Jn T r ! (21) whereλθh ¼ λ κθJ þ 1

denotes the jump intensity of the Poisson process under Qθh. If κθJ ¼ 0, this equation reduces to the pricing formula of Merton (1976), and the jump intensity and distribution are not altered by the measure change. Ifλ ¼ 0, σJ¼ 0 and n ¼ 0, this equation

reduces to the standard Black–Scholes option pricing formula: CBSMðSð0Þ; K; r; T; σÞ ¼ Sð0ÞNðd1Þ  K expðrTÞNðd2Þ (22) where d1¼ lnð ÞSð0ÞK þ rþð 12σ2ÞT σpffiffiffiT and d2¼ d1 σ ffiffiffiffi T p .

IV. Empirical and Numerical Analyses Empirical results

The previous discussion of Table 1 sheds doubts on the validity of the GBM assumption. Motivated by these findings, we examine the ability of various C_MMJMΛθh ð0Þ ¼X 1 n¼0 XI i¼1 XI j¼1 πiPθ h ijðn; TÞ Sð0ÞNðd 1;nÞ  K expðrTÞNðd2;nÞ ¼X1 n¼0 Pθhðn; TÞCBSM Sð0Þ; K; r; T; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2_{Tþ σ}2 Jn T r ! (18)

continuous-time processes in capturing the gold price dynamics. Uncovering the underlying gold price is a necessary step for choosing the appropriate option pricing model. In the empirical analyses, we employ the BSM as a benchmark for the actual data analysed. For simplicity, we consider that the economy shifts between the ordinary state (state 1) and volatile state (state 2). The transition probability matrix of the two-state Markov chain XðtÞ is given by P11 P12

P21 P22  ¼ P11 1 P11 1 P22 P22 

. Because the market state is hidden at time zero, the stationary distribution can be evalu-ated by the transition probability. Thus, the stationary distributions of state 1 and state 2 are π1¼ v2=v₁þ v₂ and π2¼ v1=v₁þ v₂, respectively. Given the gold price dynamics defined in Equation 1, we express the loga-rithmic return in discrete time as follows:

RðtÞ ¼ μ,þ σ,Zþ P N1ðΔtÞ k¼1 Yk if XðtÞ ¼ 1 P N2ðΔtÞ k¼1 Yk if XðtÞ ¼ 2 8 > > > < > > > : (23) where μ~ ¼ μ 1 2σ2 Λκ   Δt with Λκ ¼ pð 11λ1þ p22λ2Þ fð Yð1Þ1ÞÞ, σ~ ¼ σ ffiffiffiffiffi Δt p , ZeNð0; 1Þ, YkeNðμJ; σ2JÞ and NiðΔtÞ is a Poisson process with the intensity rate λiin the interval timeΔt when the Markov chain X ðtÞ remains in state i. The states and the jump arrivals are unobserved. We apply the EM algorithm (Lange,1995a, b) to calculate

the maximum likelihood (ML) estimations. In the first step, given the observed return data R and the former one-period parametersΘðk1Þ, we compute the conditional expectation of the log complete-data likelihood fun-ction asΓ Θ; Θ ðk1Þ¼ E log PrðR; X; NjΘÞjR; Θ ðk1Þ. Then, for the second step, we maximize theΓ-function to use the parameter set asΘðkÞ¼ arg max Γ Θ; Θ ðk1Þ. By the iteration and recursive computation of these two steps, the parameters converge the Γ-function to the local maximum in the incomplete-data likelihood function. Applying the code of the EM algorithm and the com-plete-data information matrix, we get the SEs of parameter estimators by the SEM algorithm (Meng and Rubin,

1991). Khalaf et al. (2003) combine bounds and Monte Carlo simulation techniques to test the generalized auto-regressive conditional heteroscedasticity (GARCH) class of models with nuisance parameters. To determine whether the JDM outperforms the BSM, and whether the data fit the MMJM better than the JDM, we apply the likelihood ratio test as follows:

¡ ¼ 2 ln Lð 1ðΘÞ  ln L0ðΘÞÞ ! asy

χ2

d;1α (24)

where LmðΘÞ represents the likelihood function under the hypothesis Hmfor m¼ 0; 1f g and d denotes the difference of the parameters between the H0 and H1 constraints. If ¡ > χ2

d;1α, H0is rejected. The respective null hypotheses are that the BSM and JDM hold.

Table 2 presents several interesting results. First, in

the case of the JDM versus MMJM, we can determine the value of d by the difference in the number of

Table 2. Estimated results of continuous-time models

Parameter BSM JDM MMJM p11 0.9939 (0.0057) p22 0.9885 (0.0080) μ 0.0008 (0.0004) 0.0014 (0.0004) 0.0013 (0.0003) μJ −0.0009 (0.0007) −0.0004 (0.0004) σ 0.0137 (0.0002) 0.0068 (0.0002) 0.0066 (0.0006) σJ 0.0144 (0.0003) 0.0107 (0.0012) λ1 0.6593 (0.1766) 0.3942 (0.1991) λ2 2.7658 (0.7103) ¡1 244.20 ¡2 173.66

Notes: This table presents the empirical results of dynamic models, reporting the estimated para-meters and corresponding SEs. The estimation settings for the BSM and JDM/MMJM are deter-mined via the maximum likelihood (ML) approach and EM algorithm, respectively. The SEs of the parameter estimators for the BSM and JDM/MMJM obtained by the ML approach and SEM algorithm, respectively, are reported in parentheses.¡1represents the likelihood ratio test for ML

functions with the null hypothesis that there is no jump event; that is, the dynamic model is a BSM. ¡2 shows the likelihood ratio test for ML functions with the null hypothesis that there is no

switching regime, that is, the dynamic model is a JDM. Performance is evaluated in terms of both the likelihood ratio test and statistical accuracy.

parameters between these two models. At the con fi-dence level of 1 α ¼ 0:95, the critical value for the aforementioned test is χ2

4;0:95 ¼ 9:49. In the gold mar-ket, we can observe that the MMJM is better than the JDM by the likelihood ratio ¡2 of 173.66 > 9.49 at the 0.05 statistical significance level, as shown in Table 2, which implies that the addition of the Markov-modu-lated Poisson process clearly dominates the single Poisson process. Through a similar procedure, we find that the JDM clearly dominates the BSM by the like-lihood ratio ¡1 of 244:20 > χ2_3;0:95¼ 7:82 at the 0.05 statistical significance level, as shown in Table 2. Second, these two transition probabilities of the gold returns for the MMJM are almost 1, implying that the economy stays in each state for a period of time and then transitions to the other. The mean and SD of the gold logarithmic returns for the MMJM are 0.0013 and 0.0066, respectively. The additional jump component prescribes a drift of −0.0004 and a volatility of 0.0107. The jump intensity is found to be 0.3942 in state 1 and 2.7658 in state 2, which clearly shows different intensity rates in different states. The findings indicate that the gold price has a GBM structure with Markov-modulated Poisson processes; that is, they are subject to regime-switching movements that cannot be explained by standard jump-diffusion processes. Furthermore, they are consistent with the findings in the descriptive analysis of Table 1, namely the non-normality of returns and the existence of different jumps over time.

Table 3 shows the same mean and variance for the

original data and continuous-time models. Comparing

with the original data, our MMJM characterizes the skew-ness and kurtosis of the logarithmic return series.

Figure 2clearly shows the different frequencies of price

jumps over the sample period. The time-varying volatility is captured by the MMJM in the form of regime-switching behaviour, that is, the oscillating periods of higher and lower intensity rates. As shown in Fig. 2, our sample period ends with two relatively consistent periods of higher intensity rates, corresponding to the Iran nuclear crisis of 2006 and US subprime mortgagefinancial crisis of 2008.

Concerning the volatility clustering,Fig. 3exhibits a substantially positive autocorrelation in the squared loga-rithmic returns of gold in which the trend steadily declines as the lag length increases. The MMJM then captures not only the existence of volatility clustering but also the magnitude and decay of this phenomenon. Overall, these empirical results suggest that the MMJM provides an adequate description for the logarithmic returns of gold. It overcomes the shortcomings of the GBM and

jump-Table 3. Distributional statistics for data and continuous-time models

Classification Data BSM JDM MMJM Mean 0.0008 0.0008 0.0008 0.0008 Variance 0.0002 0.0002 0.0002 0.0002 Skewness −0.2500 0 −0.1461 −0.1462 Kurtosis 7.7529 3 5.5581 5.9224 Notes: This table reports the distributional statistics for the ori-ginal data and continuous-time models.

4 January 2005 2 January 2007 2 January 2009 31 December 2010 400 600 800 1000 1200 1400 1600 Time

Gold Prices (US$/Ounce)

5 January 2005 2 January 2007 2 January 2009 31 December 2010 0 0.2 0.4 0.6 0.8 1 Time

The Probability of State 1

–0.1 –0.05 0 0.05 0.1 Time Logarithmic Returns 5 January 2005 2 January 2007 2 January 2009 31 December 2010 5 January 2005 2 January 2007 2 January 2009 31 December 2010 0.2 0.4 0.6 0.8 1 Time

The Probability of Jumps

Fig. 2. Time series and jump characteristics for gold

diffusion processes (Merton,1976; Kou, 2002) in asset price modelling.

Pricing performance

In order to assess the empirical validity of the MMJM, we evaluate the out-of-sample pricing performance using actual option market data2from the European Exchange (Eurex). As a benchmark, we employ the BSM for pricing European gold options. The 1-year US treasury bill rate is used as a proxy for the risk-free rate. We apply the relative mean square errors (RMSEs) for model evaluation. An in-sample analysis and an analysis across moneyness levels are not possible because of the limited actual option price data. Table 4 reports the RMSE (pricing error) of each model using the out-of-sample data. With different strike prices K, it shows that pricing errors under the MMJM are all smaller than those of competing models in terms of RMSEs. Taking the strike price K¼ 1400 as an example, the largest improvement offered by the MMJM over the benchmark model and JDM varies between 0.0848 and 0.0538 in the reduction of RMSEs. The reduction of the RMSEs between the JDM and MMJM is more substantial than that of the RMSEs between the BSM and JDM. One can infer that for this reason, the Markov component contributes more to the superior pricing performance rather than the pure jump process. The numerical results show that the MMJM is more accurate than the BSM and JDM in pricing gold call options. As a consequence, the evidence presented suggests that it is worth accounting for regime-switching jump risks when pricing European gold call options.

V. Conclusion

By analysing the gold logarithmic returns, this study finds different jump behaviours of gold prices in different time periods. The empirical results indicate that the gold price is better approximated by a GBM with Markov-modulated Poisson processes. This dynamic model also perfectly characterizes the leptokurtosis and volatility clustering of the logarithmic return series. Because the gold market described by the MMJM is incomplete, this study uses the Esscher transform to identify a risk-neutral pricing measure. After determining the risk premiums, we further derive the generalized pricing formula for

0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Lag Autocorrelation of MMJM

Fig. 3. Model autocorrelation of the squared logarithmic returns for gold

Table 4. Out-of-sample pricing errors of European gold call option pricing models

K BSM JDM MMJM 1300 0.0970 0.0906 0.0810 1320 0.1309 0.1217 0.1064 1340 0.1776 0.1649 0.1422 1360 0.2447 0.2274 0.1957 1380 0.3335 0.3104 0.2683 1400 0.4530 0.4220 0.3682

Notes: This table presents the RMSEs of various option pricing models. The parameters of each model are calibrated in the period from 5 January 2005 to 31 December 2010 and then the estimated parameters are used to evaluate the out-of-sample performance in the period from 24 January 2011 to 15 April 2011. The RMSEs are estimated by minimizing the sum of the squared pricing errors between model-determined prices and market prices (divided by the market prices). Pricing performance is evaluated for the aggregate sample on the basis of the RMSEs.

2 _{The option data are from Bloomberg. These data correspond to the gold prices and cover the period between 24 January 2011 and 15}

April 2011 (expiration date). There are a total of 60 observations for each call option contract.

gold options and demonstrate that the pricing formulas of the BSM and JDM are special cases of the generalized pricing formula. This study goes further to investigate the pricing performance of gold options under the BSM, JDM and MMJM in our sample data. The results show that the MMJM generates lower pricing errors than competing models, and pricing errors can be reduced with different strike prices depending on the RMSEs. In other words, we find that jump risks implied by our MMJM have a more significant impact on the gold option prices. As a consequence, the ability of market stakeholders to speculate and hedge their positions in the gold market is of utmost importance for considering regime-switching jump risks and for ensuring market efficiency.

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Appendix A

Proof of the martingale condition: Let Eθh denote the mathematical expectation operator with respect to the Esscher measure Qθh equivalent to P. UsingEquation 7, we obtain:

From the mutual independence of random shocks WðtÞ, ΦðtÞ and Yk, and then the martingale condition Eθh½expðrtÞSðtÞ Fj 0 ¼ Sð0Þ holds if and only if the Esscher parameters θC and θJ satisfy:

μ  r  Λκ þ θC_σ2_{¼ 0 (A2)} and μJþ 1 2σ 2 Jþ θJσ2J ¼ 0 (A3)

for all t2 0; T½ . Therefore, we can define a pair of solu-tions of Esscher parameters for the martingale condition

byEquations 8and9.

Appendix B

Proof of the Esscher transforms under the MMJM: Based onEquation 7, we apply the Girsanov theorem, and by the mutual independence of random shocks WðtÞ, ΦðtÞ and Yk, we find that Wθ

h ðtÞ ¼ WðtÞ  θCσt is a Wiener process under Qθh. Next we denote the moment-generating function of the random variable Yk byfYðθJÞ ¼ E exp θJYk

 

 

¼ κθJ

þ 1. This

does not depend on the index k because

Yk: k ¼ 1; 2; :::

f g all have the same distribution.

Then, we have:

where πi denotes the stationary distribution in state i. This limiting distribution can be computed by ψ_jjπi¼ PI

k¼1;kÞjψkjπj along with the constraint PI

j¼1πj¼ 1. Specifically, we note that θJPΦðtÞ

k¼1Yk Λκθ

J t is a martingale at time t. Given ΦðtÞ ¼ n, the Radon–

Nikodym derivative of the transition probability can be set as follows: dQθ_prob:h dPprob: ΦðtÞ¼n   ¼ κθJ þ 1 n exp ΛκθJt (B2) Then, we get dPθhðn; tÞ ¼ dPðn; tÞ κθJ þ 1 n exp ΛκθJ t

, where Pθhðn; tÞ denotes the transition prob-ability matrix under Qθh. Also, we use Equation 3 and its unique solution given by Equation 5. Letting Pθhðn; tÞ ¼ Pðn; tÞ κθJ þ 1 n exp ΛκθJt , we get: PθhΘðζ ; tÞ ¼ X 1 n¼0 Pðn; tÞ κθJ þ 1 n exp ΛκθJt ζn ¼X1 n¼0 Pðn; tÞ ζ κθJ þ 1 n exp ΛκθJt ¼ exp Ψ  ð1  ζ ÞΛ κθJ þ 1 t (B3) Therefore, under Qθh, the jump risk can be formulated by the Esscher transform intensity of the Markov-modulated Poisson process. The stochastic jump intensity Λθh ¼ Λ κθJ þ 1 ¼ Λ exp θJ_μ Jþ12 θ J_σ J  2 is Sð0Þ ¼ exp rtð ÞEθh SðtÞ F0    ¼ exp rtð ÞE dQθ h dP SðtÞ F0   " # ¼ Sð0ÞE exp μ  r 1 2σ 2_{ Λκ}
 tþ σWðtÞ þX ΦðtÞ k¼1 Yk ! dQθh dP " # ¼ Sð0Þ exp μ  r 1 2σ 2_{ Λκ}
 tþ1 2 1þ θ C   σ  2 t1 2 θ C_σ  2 t
  exp Λ κð1þθJ Þ  κθJ t (A1) E exp θJX ΦðtÞ k¼1 Yk ! " # ¼ P ΦðtÞ ¼ 0ð Þ þX1 n¼1 E exp θJX n k¼1 Yk ! ΦðtÞ ¼ n   " # PðΦðtÞ ¼ nÞ ¼X1 n¼0 E exp θJYk      n Pðn; tÞ ¼X1 n¼0 XI i¼1 XI j¼1 πi fYðθJÞ  n Pijðn; tÞ ¼ exp Λκθ J t (B1)

altered by the measure change. Finally, we investigate the jump size, where Yf 1; Y2; . . . ; Yng are independently iden-tically distributed random variables. Hence the Radon– Nikodym derivative of each specific jump size can be written as: dQθ_Yh dPY FX t   ¼ exp θJYk   E exp θJYk   FX 0    " # (B4) Then, we obtain dQθ_Yh ¼ ffiffiffiffiffiffiffi1 2πσ2 J p exp ðYk μðJþθJσ2JÞÞ 2 2σ2 J
 . Furthermore, under the physical probability measure P, the density function of each specific jump size Yk is fYðyÞ. Through the change of measures, under Qθ

h , the density function of each specific jump size Yθh

k is f_YθhðyÞ ¼ fYðyÞ dQ θh Y dPY _FX t   . Appendix C

Proof of the derivation of the generalized gold option pricing formula: Let CΛ_MMJMθh ð0Þ represent the value of the option at time zero with strike price K and matured at time T , and we have the following equation:

CΛ_MMJMθh ð0Þ ¼ exp rTð ÞEθh ðSðTÞ  KÞþjF0

 

¼ A  B (C1) First, we calculate B as follows:

where d_2;n¼lnð Sð0Þ KÞþðrffiffiffiffiffiffiffiffiffiffiffiffiffiffi12σ2ÞT12σ2Jn σ2_Tþσ2 Jn p . Second, we calculate A: A¼ exp rTð ÞEθhSðTÞ1fSðTÞKg (C3)

We denote the Radon–Nikodym derivative for the continuous diffusion part by the following formula:

dQð1þθhÞ dQθh Ft   ¼ exp σWθh ðtÞ 1 2σ 2_t
 (C4)

For each t2 0; T½ , under the risk-neutral pricing measure Qð1þθhÞ, we have the gold price dynamics as: SðtÞ ¼ Sð0Þ exp rþ1 2σ 2
 tþ σWð1þθhÞðtÞ þ X Φð1þθhÞ_ðtÞ k¼1 Y_kð1þθhÞ 8 < : 9 = ; (C5) where Wð1þθhÞðtÞ is a Wiener process, Φð1þθhÞðtÞ is a Markov-modulated Poisson process with the transition probability matrix Pð1þθhÞðn; tÞ ¼ Pθhðn; tÞ, and the jump size Y_kð1þθhÞi:i:d:,N 1

2σ2J; σ2J

 

. Hence, Equation C3 can be rewritten as: where d1;n¼lnð Sð0Þ KÞþðrþ 1 2σ 2_ÞTþ1 2σ 2 Jn ffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2_Tþσ2 Jn p . Combining Equations

C2 and C6, the generalized gold option pricing for-mula is obtained. Sð0ÞEð1þθhÞ1fSðTÞKg ¼ Sð0ÞX1 n¼0 XI i¼1 XI j¼1 πiPð1þθ h_Þ ij ðn; TÞQð1þθ h_Þ Nð0; σ_{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi}2Tþ σ2JnÞ σ2_T þ σ2 Jn p  d1;n Φð1þθ h_Þ ðTÞ ¼ n   ! ¼ Sð0ÞX1 n¼0 XI i¼1 XI j¼1 πiPð1þθ h_Þ ij ðn; TÞNðd1;nÞ (C6) B¼ K exp rTð ÞEθh1fSðTÞKg ¼ K exp rTð ÞX1 n¼0 XI i¼1 XI j¼1 πiPθ h ijðn; TÞQθ h Nð0; σ_{ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi}2Tþ σ2JnÞ σ2_Tþ σ2 Jn p  d2;nΦθ h ðTÞ ¼ n   ! ¼ K exp rTð ÞX1 n¼0 XI i¼1 XI j¼1 πiPθ h ijðn; TÞNðd2;nÞ (C2)