Chapter 3 Valuation Of Currency Options Under A Regime-Switching Gaussian HJM Model
3.2 Econometric analysis of spot-FX rate markets
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Black-Scholes model (BSM or GBM). In Section III, we investigate empirical data of zero-coupon yield curves and deal with stochastic forward interest rates under a regime-switching Gaussian HJM model. In Section IV, the RSJD model is accommodated with Markovian systems of stochastic interest rates. In addition, we apply the forward Esscher transform with a regime-switching condition under which currency options on a spot-FX rate can be valued. Section V provides empirical study showing that regime-switching risk exhibits a significant impact on prices of currency options. The final section concludes the results.
3.2 Econometric analysis of spot-FX rate markets
In this section, we investigate spot-FX rates to find evidence of regime-switching risk with the proposed Markov-chain system. The volatilities clustering feature of spot-FX rates shows time-inhomogeneous fluctuations in different time periods, as implied by different jump intensities that sometimes exhibit high or low intensities due to change in the state of the economy or financial crises. In addition, a model can be used to formularize autocorrelations by a Markovian system that is able to capture the volatilities clustering feature of spot-FX rates (Timmermann (2000)). Hence, we propose the RSJD model with two-state Markov chain to modulate jump intensities.
3.2.1 Descriptive statistics and time-inhomogeneous fluctuations in spot-FX rates
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We note that the time-inhomogeneous fluctuation feature in a spot-FX rate is due to regime-switching jumps. To show it, the daily data of the GBP/USD spot-FX rate and the 100JPY/USD spot-FX rates are collected form 1 January, 1990 to 30 December, 2011, and the EUR/USD spot-FX rate is collected form 3 January, 2000 to 30 December, 2011.
Panel A: The dynamics of the GBP/USD spot-FX rates.
Panel B: The returns dynamics of the GBP/USD spot-FX rate returns.
Panel C: The dynamic returns of the squared GBP/USD spot-FX rates.
12/90 12/91 12/92 12/93 12/94 12/95 12/96 12/97 12/98 12/99 12/00 12/01 12/02 12/03 12/04 12/05 12/06 12/07 12/08 12/09 12/10 12/11 1.3
12/90 12/91 12/92 12/93 12/94 12/95 12/96 12/97 12/98 12/99 12/00 12/01 12/02 12/03 12/04 12/05 12/06 12/07 12/08 12/09 12/10 12/11 -0.05
12/90 12/91 12/92 12/93 12/94 12/95 12/96 12/97 12/98 12/99 12/00 12/01 12/02 12/03 12/04 12/05 12/06 12/07 12/08 12/09 12/10 12/11 0
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Panel D: The probabilistic dynamics of low intensity
1 with spot-FX rate GBP/USD returns under the RSJD model.Figure 1: The daily data for the GBP/USD spot-FX rate starts form 1 January, 1990 to 30 December, 2011.
Figure1 exhibits evidence of jumps and the time-inhomogeneous fluctuations in the GBP/USD spot-FX rate. In Panel A of Figure 1, the spot-FX rate shows two different regions of time-inhomogeneous fluctuations during four time periods, 01/1990-12/1993, 01/1994-12/1999, 01/2000-12/2007, and 01/2008-12/2011. Panel B provides evidence of different volatilities during the four time periods; the time periods 01/1990-12/1993 and 01/2008-12/2011 show higher volatility than the other two periods. Higher volatilities in these two periods are induced, respectively, by the Oil crisis and the subprime lending crisis. The financial crises with information lag lead to alternate high and low jump intensities with high and low volatilities. In addition, the high and low volatilities reveal strong evidence of time-inhomogeneous
12/90 12/91 12/92 12/93 12/94 12/95 12/96 12/97 12/98 12/99 12/00 12/01 12/02 12/03 12/04 12/05 12/06 12/07 12/08 12/09 12/10 12/11 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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fluctuations in the spot-FX rate. Furthermore, Panel C of Figure 1 reports the returns dynamics of the squared GBP/USD spot-FX rate against time. It shows clearly that not only there are jump trends, but also there exists a different degree of jumps with high or low jump intensities in the daily spot-FX data.
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Table 1: Statistics Of The GBP/USD Spot-FX Rate Returns Starts Form 1 January, 1990 to 30 December, 2011.
Period 01/1990 Kurtosis 1.2674 2.3290 4.2818 1.4145 0.5305 2.2411 0.2464 0.2955 0.8179 2.2246 0.6426
Number of returns more
than 2 (0.0164) 22
Number of returns less
than -2 (-0.0163)23
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The descriptive statistics of GBP/USD spot-FX returns rate are reported for every two-year period in Table 1. Jumps are identified as follows: returns that are more than two standard deviations (2 ) or less than 2 from the mean are identified as jumps. Notice that a high number of jumps is associated with a high jump intensity, and vice versa, which implies that the spot-FX dynamics has two jump intensities (high and low) due to change in the state of the economy. The above result for the GBP/USD spot-FX rate carries over to the 100JPY/USD spot-FX rate and the EUR/USD spot-FX rate.
Based on the above observed phenomena of the spot-FX rate data, a hidden Markovian process is employed to formulate regime switching as an important feature for modeling the spot-FX rate dynamics, which also specifically considers jump sizes and their intensities. Hence, in the next subsections, the spot-FX rate dynamics is formulated with the RSJD model modulated by Markov chain. We then provide the estimated parameters and the test statistics for the BSM, the JDM, and the RSJD models based on the GBP/USD, the 100JPY/USD and the EUR/USD spot-FX rates.
In addition, a regime-switching jump diffusion model is also established for spot-FX rates.
3.2.2 Regime-switching jump diffusion model for the spot-FX rate
A complete probability space is denoted by
, ,
with a real probability measure ,and a finite time is specified by 0, T in the continuous-time setting.
( )
t TY Y t is a continuous-time, finite-state Markov chain defined on the actual probability space
, ,
. We take the state space of Y to be a finite state set
1, 2
(1, 0), (0, 1)
2IY e e , which specifies a boom or a recession (good or
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bad time) for the state of the economy. Hence, the RSJD model of the spot-FX rate S t( ) under the actual measure is proposed and given by: a sequence of mutually independent jump variables. The jump variable Zn has a normal distribution with mean J and variance J2 . Moreover, the MMPP ( )t modulated by Markov chain Y has low and high jump intensity of the Poisson process denoted, respectively, by , i i1, 2. The ’s are governed by change in i the state ei of the economy modeled by Markov chain Y at time t. Specifically, we have the i
1, 2
ei , 1, i 2, where the dot ( ) denotes the scalar product, and the MMPP is governed by continuous-time Markov chain Y with a transition matrix given below:
Ψ denotes a transition-rate matrix of Markov chain Y t( )
whose element ai, 1, i 2, represents the transition rate of the process leaving from state eiIY to the other state e , andj ai is the transition rate of staying at state ei.
Let Y t( ) and ( )t be the joint probability, ( ( ) t n Y, (0)e Y ti, ( )ej)
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Proposition 1 Under the Kolmogorov's forward equation, the moment-generating
function of the joint probability has a unique solution
well-known BSM given as follows:( ) ( ), (0) 0
( ) S
dS t dt dW t S
S t
. (3.2.4)
3.2.3 Empirical analysis: the GBP/USD Foreign Exchange rate
We estimate the parameters of the discrete-time BSM, the JDM, and the RSJD models, and test the models empirically using the likelihood ratio test (LRT). The RSJD
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parameters are estimated using the maximum likelihood method with the Expectation Maximization algorithm (EM, Dempster et al. (1977)) and a gradient algorithm (Lange, (1995)). [ 4 ] Their standard errors are estimated using Supplemented Expectation Maximization algorithm (SEM, Meng et al. (1991)).
4 For details, see the text book: McLachlan and Krishnan 2008, The EM algorithm and extensions.
John Wiley & Sons, Inc, Hoboken, New Jersey.
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Table2: Estimated Parameters of The BSM, The JDM And The RSJD In The GBP/USD Spot-FX , The 100JPY/USD and The EUR/USD Spot- FX RSJD 0.9976 0.9916 0.0003 -0.0003 0.0041 0.0054 0.2152 2.4741 533.77*
(0.0008) (0.0030) (0.0001) (0.0001) (0.0001) (0.0003) (0.0273) (0.2762) --
BSM -- -- 0.0001 -- 0.0071 -- -- -- -- RSJD 0.9930 0.9911 -0.0002 0.0005 0.0043 0.0071 0.2451 1.4889 260.39*
(0.0029) (0.0047) (0.0001) (0.0002) (0.0001) (0.0002) (0.0321) (0.1305) --
BSM -- -- 0.0001 -- 0.0067 -- -- -- -- RSJD 0.9956 0.9937 0.0003 -0.0003 0.0045 0.0052 0.2265 1.7791 180.95*
(0.0043) (0.0042) (0.000.) (0.0002) (0.0004) (0.0000) (0.4010) (0.4407) -- Note: 1. The data are the daily GBP/USD and 100JPY/USD spot FX rates (01/01/1990 to 30/12/2011.)
2. The data are the daily data for EUR/USD spot FX rates (03/01/2000 to 30/12/2011.) 3. The denotes the mean of the logarithmic return at discrete time . t
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4. (
) denotes the standard errors.5. The BSM represents Black-Scholes Model, the JD denotes jump diffusion and the RSJD is regime-switching jump diffusion.
6. The LRT of the JDM represents the likelihood ratio test for the null hypothesis of the BSM against the alternative hypothesis of the JDM. The LRT of the RSJD model is for the null hypothesis of the JDM against the alternative hypothesis of the RSJD model.
7. Asterisk * denotes ‘significant at the 5% level’.
8.The parameters are rounded off to the fourth decimal place.
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The parametric estimates and the statistical tests based on the GBP/USD, the 100JPY/USD, and the EUR/USD spot-FX rates are listed in Table 2 for the three models: the BSM, the JDM, and the RSJD models. For example, the mean
andthe standard deviation
for the GBP/USD spot-FX rate under the BSM are, Srespectively, 0.0000 and 0.0062. The LRT in Table 2 shows rejecting the BSM at the 95% significance level. In addition, the jump frequency
, the mean 1 (J) and the volatility
J of the logarithmic jump variable of the JDM are, respectively, 0.2633, -0.0007 and 0.0085. In Table 2, based on the LRTs of the three models, the JDM is found to have a better fit than the BSM (with the LRT 631.90 significant at the 95% level), and the RSJD model is better than the JDM (with the LRT 533.77 significant at the 95% level). In addition, the estimated transition probabilities p11Yand p22Y under the RSJD model are, respectively, 0.9976 and 0.9916, both of which imply that the probability of regime switching from a low frequency (0.2152) to a high frequency (2.4741) is very small, and vice versa.
Panel D of Figure 1 reports the probabilistic dynamics of the GBP/USD spot-FX rate returns with the low intensity (or the smoothing probability[5]) under the RSJD model, which shows that the low level of squared returns coincides with the low probabilistic dynamics of the low intensity. Specifically, the (smoothing) probability of the low jump intensity changes frequently during the period 01/1990-12/1993, and the corresponding squared returns in Panel C exhibit low and high frequency in an alternating feature. In addition, when the (smoothing) probability of the low jump intensity in Panel D is more stable during the period 01/1994-12/2007, the corresponding squared returns in Panel C are lower on average than the other period
5 See Hamilton (1989).
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The above results clearly show that not only a regime-switching feature of the low jump intensity but also the corresponding lower volatilities occur in the period of the low jump intensities. The result for the GBP/USD spot-FX rate also carries over to the 100JPY/USD spot-FX and the EUR/USD spot-FX rates.