Jump Diffusion Models for the Foreign Exchange Market
Abstract
In this project, we look at the developments and limitations of some of the mainstream financial
models and build a framework of assumptions to develop a univariate and a multivariate Jump
Diffusion model to represent the returns of assets. We will implement a Double Exponential (DE)
distribution to a Gaussian model in a mixture model to account for large jumps in asset returns.
Finally, we will employ Markov chain Monte Carlo (MCMC) techniques to estimate the parameters
for the jump diffusion models using data from the Foreign Exchange (FX) market.
Keywords: Geometric Brownian motion, Jump Diffusion, Mixture model, Multivariate Double
Exponential, Markov chain Monte Carlo, Metropolis-Hastings within Gibb Sampler, Leptokurtosis,
Foreign Exchange Market
Jump Diffusion Models for the Foreign Exchange Market
1. Introduction
1.1 Financial Models
Brownian motion is an observation made by botanist Robert Brown of the erratic movements of
particles suspended in fluid. This observation was quantified and modelled as a continuous time
stochastic process known as the geometric Brownian motion (GBM) in which the increment of the
stochastic process is normal with respect to its current value1. It was French mathematician Louis
Bachelier who first conceptualise the assumption that an asset price movement is a GBM in 1900.
Many of today’s financial theories, like the Modern Portfolio Theory (Markowitz 1952), the Capital
Asset Pricing Model (Sharpe 1964) and the Black-Scholes-Merton Option Pricing model (1973), are
based on this assumption2.
Figure 1.1.1: Illustration of the Brownian motion in 1- and 2-dimensions
Prior to Harry Markowitz’s paper on portfolio selection in 1952, the conventional wisdom called for
an investor to choose a portfolio that maximises the profit. Markowitz hypothesized that the one can
maximise their profit while minimising their risk by selecting a well-diversified portfolio3. His mean-
variance portfolio theory described the assets’ returns as jointly normally distributed random variables
with the risk or volatility defined as the standard deviation. By selecting a portfolio of well-diversified
1 Sheldon M. Ross, 2007. “Introduction to Probability Models”. 7th ed. Elsevier/Academic Press 2 Eugene F. Fama, 1965. ”Random Walks in Stock Market Prices.” Financial Analyst Journals, Vol. 21, No. 5 page 55-59, Sep.-Oct.,1965 3 Harry Markowitz, 1952. “Portfolio Selection”, The Journal of Finance. Vol. 7, No. 1, page 77-91, Mar., 1952.
Jump Diffusion Models for the Foreign Exchange Market
securities, Markowitz aimed to reduce the total variance (and hence the risk) of the portfolio to the
intrinsic systematic or market risk. This became known as the Market portfolio. The investors will
then choose a weighted combination of this portfolio and a riskless asset (government bonds etc.) to
maximise their profits according to their risk appetite.
Expanding on Markowitz’s work, William Sharpe introduced the Capital Asset Pricing model
(CAPM) in 1964. CAPM formulates the expected returns of an asset as the sum of the riskless rate of
returns and a ratio of the market premium i.e. the additional expected returns of the Market portfolio
above that of the riskless asset4. The ratio is represented by the coefficient Beta, which measures the
elasticity of the asset’s return to the market’s return. Beta is usually computed through historic data of
the asset. As Sharpe was working upon the framework of Markowitz’s portfolio theory, it inherited
the same assumptions (and therefore, also its limitation)5. CAPM is use effectively to determine the
price of individual security and its widespread use forms one of the cornerstone of asset pricing
models today.
One of the most influential financial models today is the Black-Scholes-Merton Option Pricing model
(B-S model). The model was conjured up by Fischer Black and Myron Scholes (1973), and Robert C.
Merton (1973) on a separate paper and is primarily used in pricing European-style options. The option
prices are equilibrated with the prices of the underlying asset so that there can be no arbitrage in the
market and the fundamental assumption in the B-S model framework is that returns of the underlying
assets is a GBM. The B-S model formalise the process of option pricing and its widespread
acceptance saw the boom in options trading in the 1980s6.
4 S. Ross, R. Westerfield, J. Jaffe, B. Jordan, 2008, “Modern Financial Management”, 8th ed., McGraw-Hill Irwin, 2008 5 Harry Markowitz, 1999. “The early history of portfolio theory: 1600-1960”, Financial Analyst Journal , Vol.55, No.4,. Page 5-16, Jul.- Aug. 1999 6 Ajay Shah, 1997. “Black, Merton and Scholes: Their Work and its Consequences”. Economic and Political Weekly, Vol. 32, No.52, Page 3337-3342, Dec,. 1997
Jump Diffusion Models for the Foreign Exchange Market
1.2. Limitations of Financial Models
When modelling the financial markets, there are many justified assumptions made to the behaviour of
returns (normally distributed and fixed correlation between assets) and investors (price takers, rational
and risk-averse), the market efficiency (information symmetry, no arbitrage and frictionless
transaction) and the legislation pertaining to the financial markets (tax free market and short selling
laws). These assumptions are made when modelling market movements to simplify the reality and yet
derive a robust formulation that represents the real world.
However, despite the elegance of these models, recent events in the financial market have forced us to
re-evaluate some of these assumptions. In this report, we shall investigate the limitation of the
Gaussian assumption made earlier. Over the bubble and bust cycle in the financial market, jumps of
magnitude over 5 standard deviations (σ) are observed more frequently that predicted under the
Gaussian assumption (1 in 3 million event). Empirical data showed the FX market, by far the most
volatile and liquid financial market, saw jumps of 7σ in the DEM-GBP rates (a 10-12 event; Black
Wednesday 1992), 9σ in the USD-THB rates (Asian financial crisis 1997), and rebounds of 8σ in the
EUR-ISK (Icelandic financial crisis 2008). These huge deviations created the fat-tail phenomena,
known as leptokurtosis7, in the distribution of the drifts. Probability of such events occurring are
almost negligible under the Gaussian model but ignoring this statistic, as history has shown, is
disastrous. The inability of the Gaussian model to accommodate these frequent large jumps in the
financial markets opposes the idea that assets return is a GBM. Therefore a more accurate model will
be required if we were to improve our predictions in the financial markets.
7 Peter Verhoeven and Michael McAleer, 2004. “Fat tails and asymmetry in financial volatility models”, Mathematics and Computers in Simulation, Vol.64 No.3-4 , Pages 351-361, Feb., 2004
Jump Diffusion Models for the Foreign Exchange Market
1.3. The Foreign Exchange Market
As mentioned earlier, the FX market is by far the largest financial market8. It boasted a $4 trillion
daily turnover worldwide and is traded round the clock all year long. Currencies are traded in pairs,
with a country’s currency exchanging for another’s. Therefore, a currency trade may be considered as
a structure of long and short trades. Due to this long-short arrangement, the exchange rates across the
world’s currencies are intricately linked. Large quantity of speculative trades in the FX market gives
the market additional liquidity, which in turn ensures that all arbitrage opportunities are eliminated
quickly after an exogenous shock. The fast changing and volatile nature of the FX market may hinder
the effectiveness of a predictive model for currency exchange. As such, there are few financial models
developed for the FX market. However, we are precisely looking to develop a model for an eccentric
and interrelated class of asset. Therefore, we will use the market data from the FX market to evaluate
our model.
1.4. Jump Diffusion
One possible improvement is to employ a jump diffusion (JD) model to represent the returns of assets
(Merton (1976); Kou (2002)). Jump diffusion is a stochastic process that separates the diffusion
(drifts) component from the spontaneous jumps. While the diffusions follow a GBM, which is a
continuous time stochastic process, market data we obtain are usually discretised in time. Therefore,
we will simplify the model by using a discrete time space for greater practicality and develop a
probability distribution function (pdf) for the model. We will use a Gaussian distribution to represent
the drifts while a Bernoulli model will separate the jumps and the drifts from occurring at the same
time. In reality, the Bernoulli random variable will mimic the occurrence of a shock (both positive and
negative) amidst the “well-behaved” drifts in the financial market. The Double Exponential (DE)
8 Bank of International Settlement, 2007. “Foreign Exchange and Derivative market activity in 2007”, Triennial Central Bank Survey,19 Dec.2007
Jump Diffusion Models for the Foreign Exchange Market
Distribution (also known as the Laplace distribution) will model the resultant jump9. The DE will give
us the leptokurtosis feature for our model.
We will extend our investigation of the jump diffusion model to look at a multivariate model. Due to
the complex intricate nature of the securities in the market, price movements are rarely only security-
specific. They usually have a corresponding effect on related assets. This was the motivation behind
Markowitz mean-variance model in 1952. Furthermore, the recent proliferation of structured financial
products, like exotic options, Mortgage backed securities and Collateral Debt Obligations, which are
themselves composed of multiple securities, exemplify the need to developing a multivariate version
of the jump diffusion model. The multivariate model will be useful in representing the movement of a
portfolio of securities or the market in general in view of the jumps and may be instrumental in
finding optimal portfolio allocations, identifying the market portfolio and pricing options. These
applications are however not within the scope of this report but are worth further investigations.
In both our univariate JD (UVJD) and multivariate JD (MVJD) model, we did not separate the
occurrence and the magnitude of the positive and negative jumps (as oppose to Ramezani and Zeng
(1998)) as we seek not to complicate our model by introducing too many parameters and lose the
degrees of freedom. Therefore, we will assume that the jumps are centralised at zero, and for the
multiple assets in our MV model, with the common magnitude. The drifts in the MVJD are modelled
as a multivariate Gaussian distribution with positive semi-definite variance-covariance matrix to
demonstrate the correlated nature of the price movements. We assume that the jumps of the multiple
assets occur at the same time (when our Bernoulli random variable is 1). From an economic
standpoint, this assumption is a practical one as in a liquid market, within a feasible unit of discrete
time, jumps occur simultaneously.
9 S.G. Kou, 2002, “A Jump-Diffusion Model for Option Pricing”, Management Science, Vol.48, No. 8 page1086-1101, Aug. 2002
Jump Diffusion Models for the Foreign Exchange Market
1.5. Markov chain Monte Carlo
For our analysis, we opted for the Bayesian approach to inference over the Frequentists’ technique of
Maximum Likelihood Estimate (MLE)10. We chose this method as it allows us to incorporate our
knowledge and the information we have about the data into the analysis by specifying a prior
distribution. To compute the marginal posterior density, we used the standard Metropolis-Hasting
within a Gibbs Sampling MCMC algorithm. The algorithm gives us flexibility in implementing a
suitable proposal distribution for the Markov chain. We will obtain a series of values for each
parameter estimate, which will allow us to analyse the range and the precision of the estimate. In
addition, the derivation of the MLE for our mixture (JD) model, and especially the multivariate form,
will be extremely complicated.
This report will first describe the structure of the UVJD and MVJD model and illustrate the derivation
process for their probability density function. We will then adopt these two models for the four most
liquid currency-pair in the FX market. Finally, we will analyse and conclude our findings and discuss
the limitations of the models and possible avenues for further research.
10 Cyrus A. Ramezani and Yong Zeng, 1998. “Maximum Likelihood Estimate of the double exponential jump-diffusion process”, Annals of Finance, Vol. 3, Issue 4, Page 487-507. Oct.2007
Jump Diffusion Models for the Foreign Exchange Market
2. Model Development
2.1 Univariate Jump Diffusion (UVJD) Model
In this section, we describe the construction of our model through a mixture of the Gaussian model
and the DE model. We first consider the day-to-day returns of assets as the random variable of
interest.
, where denotes the price on day i.
If drifts in the returns are GBM i.e. the process is in diffusion, the returns should be normally
distributed with mean and volatility .
In a case of a jump, we have modelled the returns as DE random variable (Ramezani and Zeng 1998;
Kou 2002). The JD model has mean and volatility . As we do not separate the positive and
negative jumps, we will set .
Jump Diffusion Models for the Foreign Exchange Market
Both the Normal and DE distributions are members of the Generalised Normal (GN) distribution
family.
where , , is the Gamma function
The Normal distribution has , while the DE distribution has and . The
family has kurtosis (“peakiness”) of . Therefore, the Normal distribution (zero
kurtosis) has a smoother peak than the DE distribution (kurtosis of 3). Consequently, the DE
distribution has a fatter tail than Normal.
Figure 2.1: Difference between the Gaussian distribution and DE distribution with
the same mean and variance
The occasion switch from the diffusion model to the jump model (when a jump occurs) give rise to
the leptokurtic property we were looking for.
Standard Gaussian model
DE model with mean 0 and variance 1
Jump Diffusion Models for the Foreign Exchange Market
The switch will be controlled by a Bernoulli random variable, . is an Indicator
function where
Therefore we have the random variable
, where , ,
Upon derivation (refer to Appendix A and B), we find that has conditional posterior density
,
where is the cumulative distribution function of a standard Normal.
The pdf is a complicated equation consisting of the special function and 4 distinct parameters.
As such, finding the MLE is not feasible as the derivative process would be difficult and a point
estimate provides little analytical value. On the other hand, the MCMC technique for parameter
estimation offers a reasonable alternative as the simulation process generates a series of values, with
which we can evaluate the robustness of our assumptions and the model, and to better understand the
meaning of the parameters.
Jump Diffusion Models for the Foreign Exchange Market
The Metropolis-Hastings (MH) algorithm will be used within a Gibbs sampler. In the Gibbs sampler,
we will simulate the parameters as random variables separately. This simplifies the process of
evaluating the marginal density and the likelihood of a parameter through multiple integrations of the
conditional density function. Instead, by looking at a specified parameter at each step, we compute the
full conditional density and in turn, update the parameter.
We will adopt the MH algorithm for proposal acceptance in our model. The algorithm will be
performed using the MATLAB package (refer to Appendix C and D for code)
For ith iteration
Do Propose where is the parameter of interest and is the
hyperparameter(s) of the proposal density.
Compute logarithm of the MH acceptance ratio
Generate
If then else
Return
Jump Diffusion Models for the Foreign Exchange Market
We decided on the following priori and proposal densities (at the ith iteration) that gives most stable
set of results and the best mixing of proposals.
Parameters
Priors
Proposals
Table 2.1.1 Table of Prior and Proposal distributions used for the parameters
Remarks: t1, t2, t3 and t4 are tuning parameters which we adjust to achieve better mixing.
and have independent proposal densities.
and have random walk proposal mechanism centered at previous iteration.
We noted the relationship between the standard deviation of the proposal densities and the rate of
proposal acceptance11, which increases with a decreasing proposal variance. However, by increasing
the variance, we are proposing small differences between the iterations, and therefore a convergence
towards the most probably parameter value will be slower. We chose a tuning factor that stabilizes the
proposals (i.e. converging series of iterations) that has good mixing (acceptance rate of > 0.23).
11 William J. Browne, David Draper, 2000, “Implementation and performance issues in Bayesian and likelihood fitting of multilevel models”, Computational Statistics”, Vol.15, page 391-420, 2000
Jump Diffusion Models for the Foreign Exchange Market
2.1 Multivariate Jump Diffusion (MVJD) Model
For our multivariate version, we consider the returns of k assets as a random vector .
The diffusion drifts of follow a multivariate Normal distribution with k x 1 mean vector and k x
k variance-covariance matrix , where is positive semi-definite.
The jumps of follow the Multivariate Double Exponential (MDE) distribution. The Multivariate
Double Exponential distribution can be considered as a multivariate normal variance mixture model
comprising of a Gaussian and an exponential variable12. The model has the form
, where ,
and is a positive definite constant matrix. has a determinant of 1.
We assume that while the jumps occur at the same time, the sizes of the jumps are comparable but
independent. Therefore, we let to be the identity matrix.
The resultant conditional posterior density for the jumps is
12 T.Eltoft, T. Kim, T-W. Lee 2006, “On the Multivariate Laplace Distribution”, IEEE Signal Processing Letters, Vol. 13 No.5 May., 2006.
Jump Diffusion Models for the Foreign Exchange Market
As MCMC is itself a numerical approximation process, we can avoid evaluating the analytical form of
the double integral by finding approximation for the equation. Suppose we generate a series of
and from the distributions and respectively. By the strong
law of large numbers, we have
1Ni N j
1(2πwi )
k / 2 ⋅ e−
12wi
(y−zk )T (
y−zk )
i=1
Ni
∑j=1
N j
∑ N j ,Ni →∞⎯ →⎯⎯⎯ EW [EZ [1
(2πW )k / 2 ⋅ e−
12W
(y−Zk )T (
y−Zk )
]] ,
which we will use to estimate the likelihood for the jump component. We can tolerate variance in the
approximation if it is sufficiently small i.e. less than the variance of the likelihoods in the MCMC.
Figure 2.1.1 Illustration of the variance of the simulation and the size of simulated sample
We can see that the variance of the approximation decreases with the size of the simulated and
. However, the approximation, when used in an MCMC process is very computationally intensive.
This is due to the long chain of simulations and the generations of the and , and evaluations of
the double summations at each iteration.
Jump Diffusion Models for the Foreign Exchange Market
, 40 80 120 160 200 240 280
Variance 95.38 59.33 33.04 25.44 19.26 16.90 18.87
Computation time (s) 0.113 0.258 0.347 0.560 0.858 1.337 1.780
Table 2.1.1 Table of size of simulated sample, variance of estimation and computation time per iteration
Therefore we will use a suitable size that gives us sufficiently precise likelihood
estimate in the quickest time.
We use the same Bernoulli variable to regulate the jumps. i.e.
The random vector in our MV JD model is
, where , , ,
The conditional density function has the form
which we will estimate via
for a sufficiently large N.
Jump Diffusion Models for the Foreign Exchange Market
Again, we have chosen the following priori and proposal densities (at the ith iteration) that promotes
stability and mixing. As far as possible, we have also tried to use the same distributions in the MV
model as our UV model.
Parameters
Priors
Proposals
Table 2.1.2: Table of Prior and Proposal Distributions used for the parameters
Remarks: t1, t2, t3 and t4 are tuning parameters which we adjust to achieve better mixing.
and have independent proposal densities.
and have random walk proposal mechanism centered at previous iteration.
The Wishart distribution13 is as both the prior and proposal distribution for . The Wishart
distribution is a generalization of the distribution to a multiple dimension vector space.
, where is a p x p positive definite matrix and denotes the degree of freedom has
mean .
13 William J. Browne, 2006, “MCMC algorithms for constrained variance matrices”, Computational Statistics & Data Analysis Vol. 50. pages 1655-1677, 2006
Jump Diffusion Models for the Foreign Exchange Market
3 Data Analysis
3.1 Preliminary Analysis
We used the day-to-day returns of the Sterling pounds (GBP) to Euros (EUR) currency pair
(GBPEUR) from 1st January 2008 to 31th December 2009 as our data for the (UVJD) model.
Figure 3.1.1: Day-to-day time series plot of the GBP-EUR currency pair
Figure 3.1.2: Day-to-day time series plot of the returns of t
The returns of the GBPEUR have mean -0.0002753 and volatility ( ) 0.0051. Currency pairs are
normally traded in high precision due to the large volume of trade, hence, a small change in the
exchange rate will relate to a large profit or loss. We can observe that returns of GBPEUR are with +
3% of the rate. Out of the 730 days, we have 39 days with returns > 2 and 2 days of returns of > 5
.
Jump Diffusion Models for the Foreign Exchange Market
We will use the MH within a Gibbs sampling algorithm to estimate the parameters. The fitted model
will then be used for inference. We start by undertaking some preliminary analysis of the likelihood
function so that we can propose appropriate priori values to start the Markov chain.
As the model is a mixture of two separate models (with possibly two distinct peak), we need to find
out that the likelihood is unimodal in the parameter space. If there is more than one maximum
likelihood value, then the likelihood function in the Markov chain may converge to any of the peaks,
giving rise to inconsistent results.
=0.005 =0.0002
Figure 3.1.3: Surface plot of the Log-likelihood against the and for a fixed .
We can see that given the same µ (fixed at 0), when we set =0.005, the likelihood is higher when
and µ are both large. On the other hand, when we set =0.0002, the likelihood is higher when
is small but η is large.
Due to the high precision of the returns, tuning the MCMC becomes a very delicate process. While
the scale of the tuning for the proposal densities parameters µ, and η can be scaled
correspondingly (i.e. smaller proposal variances, proportionate to the proposal mean), lies within
Jump Diffusion Models for the Foreign Exchange Market
the range [0,1]. Since the value of would affect how the rest of the parameters are updated, we
must be careful in using a reasonable priori and tuning factor to propose s.
3.2 UVJD Analysis
With the suitable configuration for the tuning, we ran the MCMC algorithm for 10000 iterations.
Figure 3.2.1: Plot of the log-likelihood against the iterations (1000 and 10000).
We observed that there are two large jumps in the chain before the chain arrives at a stable state. We
will remove the first 1000 iterations (burn-in) for our analysis.
Acceptance rate Mean Variance Mean (after
burn in) Variance
(after burn in) 2.96x103 343.74 2.96x103 3.2329
0.3112 -1.81x10-5 3.29x10-9 -1.79x10-5 2.03x10-10
0.7249 2.08x10-4 3.54x10-7 9.27x10-5 2.86x10-10
0.2273 3.9x10-3 1.01x10-7 3.9x10-3 2.58x10-8
0.2539 0.8527 0.0112 0.8714 2.52x10-4
Table 3.2.1 Table of results (acceptance rate, mean and variance) of the MCMC
Jump Diffusion Models for the Foreign Exchange Market
Figure 3.2.2: Fitting the histogram of data with the UVJD (red)
and Gaussian (green) posterior density function
A peculiar observation was that mean ( ) from the MCMC process is 0.8527. In other words, there is
a shock in the FX market for GBPEUR 85% of the trading days. Intuitively, this meant that the DE
distribution is controlling the drifts more often than the Normal distribution giving rise to the
leptokurtic effect. This implies that jumps are recognized as the usual trading pattern for GBPEUR.
As the DE distribution has a greater effect for our model, its is smaller that what it would be if the
is small. If the is small, when an occasional jump occurs, the size of the jump will have to be
sufficiently big in order to give the leptokurtic characteristic. Conversely, when jumps occur
frequently, the size of each jump (and hence, η) will need to be small so there would not be excessive
leptokurtosis.
Jump Diffusion Models for the Foreign Exchange Market
The prior distribution we used for was centered at a small value at as we expected jumps to occur
sporadically. However, the MCMC converges within the first 500 iterations to a stable state.
Figure 3.2.3 Plot of the jumps of the MCMC for
3.3. MVJD Analysis
For our multivariate model, we included the Euro to US Dollars (USD), GBP to USD and USD to
Japanese Yen (JPY) currency pairs. Together, they made up the four most traded currency pairs in the
FX market.
Figure 3.3.1 Time series plot for EURUSD, GBPUSD and USDJPY rates
Jump Diffusion Models for the Foreign Exchange Market
Figure 3.3.2 Time series plot for EURUSD, GBPUSD and USDJPY returns
The returns of the multiple FX rates have mean (-0.18, -2.62, -2.89, -2.47)T and variance-covariance
matrix
From the plots of the return, we can see that some jumps of the exchange rates do occur
simultaneously across the markets. These shocks affect the different rates in different ways.
Due to the computationally intensive process of likelihood estimation for each iteration, we ran the
model for 3000 iterations. We have obtained from the univariate model about the mean of the
currency pairs and used it for our prior and proposal densities.
Jump Diffusion Models for the Foreign Exchange Market
Figure 3.3.3: Plot of log-likelihood against the first 1000 iterations
Acceptance rate Mean (after burn in) Variance (after burn in)
9.4345 x 103 5.4923 x 106
0.9997 (1.11 2.88 2.35 6.80)T x10-3
(4.76 4.17 5.22 6.54)T
x10-3
0.4430
x10-2
1.20 -0.18 0.99 0.14 -0.18 1.24 1.02 0.31 0.99 1.02 2.11 0.53 0.14 0.31 0.53 1.78
13.9 0.39 11.0 1.36 0.39 18.0 14.8 0.91 11.0 14.8 52.3 4.89 13.6 0.91 4.89 28.2
x 10-8
0.5433 0.0036 1.9216e-06
0.7807 0.0188 0.0036
Table 3.3.1 Table of results (acceptance rate, mean and variance) from the MCMC
Notice that the for the multivariate model has = 0.0188, which relates to approximate one jump
in every 53 days. This is very different from our univariate model. In our multivariate model, the
definition of a jump is stricter. Jumps in returns must occur at the same time for multiple returns.
Suppose the returns for USDJPY recorded a large drift at time t but the other currency pairs do not
display corresponding jumps at time t, then the large drift would not be considered as a jump. This
filtered out many of the “supposed” jumps from our univariate model and the actual jumps are results
of exogenous shocks that affect the returns market wide. As a result of the small , the is larger.
Jump Diffusion Models for the Foreign Exchange Market
4 Conclusion
While JD models are useful in describing movements in the financial markets, there are some
limitations and considerations when using it. In both our models, we have modelled the JD process
using to four parameters compared to the two parameters used in the Gaussian model. In order to
relax our assumptions and accommodate more information, we need to introduce additional
parameters. However, these parameters used in the process will reduce the effectiveness. Most model
comparison analyses do penalize additional parameters14, as the reduction in deviance may not
compensate the loss of degree of freedom. While the model may be a better fit, the reduced number of
effective parameters makes it less appealing.
In addition, the process of piecewise tuning of the each parameters is extremely time consuming,
especially on a precise scale such as that of currency pairs returns. For the ∑ proposals, we have
adopted the Wishart proposal mechanism. The tuning process requires us to increase the degrees of
freedom and at the same time, dividing the random variable by the same factor (recall that
. However, during the MCMC, most standard computers are unable to
evaluate the proposal density the large degree of freedom. Therefore, there is a lower bound to which
we can tune the variance. Due to the small values of the asset returns, it requires a proposal
mechanism with a much smaller variance in order to promote mixing. We are unable to do this for the
multivariate Gaussian model for model comparison.
One interesting feature of the DE distribution within the mixture model is that it may not be unimodal.
With a varying λ value, either the DE model or the Gaussian model would dominate the other,
14 Andrew Gelman, John B. Carlin, Hal Stern, 2004, “Baysian Data Analysis”, 2nd ed. Boca Raton, 2004
Jump Diffusion Models for the Foreign Exchange Market
leading the MCMC to converge to a different set of parameters. As such, we need some prior
knowledge about the data before we can choose a prior and proposal models.
The derivation of the JD model is not a simple process, and in particular, the multivariate version of
the model is very complicated. The approximation process of our MVJD model is computationally
intensive as it involves the estimation of a double integrand via two summation loops. Such time
consuming process makes this model not feasible for the fast moving FX market.
The some of assumptions we have made may not hold in the FX market. For example, we did not
separate positive and the negative jumps. We hypothesize that the impact that the exogenous shocks
have each currency pair are conditionally independent on the others, hence we set Γ to be the identity
matrix. This however, may not be true as the we know that the currency pairs are intricately linked to
each other and the fact that the jumps occurs at the same time meant that some jumps by a particular
currency pair will have corresponding effect on the others.
The FX market displayed some unique features. The overtly volatile nature makes jumps in the
markets more common than drifts. This volume of liquidity may be the cause of the volatility. On the
other hands, shocks that move the entire market, although present, are less frequent.
In order to improve on our model, some further research will be needed. A possible extension will be
to derive the analytical form for the multivariate posterior density. This will eliminate reduce the time
taken, and the error due to the numerical approximation process. We can also consider the full model
by relaxing the assumptions made to Γ and κ.
Jump Diffusion Models for the Foreign Exchange Market
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fitting of multilevel models”, Computational Statistics”, Vol.15, page 391-420, 2000
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page1086-1101, Aug. 2002
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Jump Diffusion Models for the Foreign Exchange Market
Appendix A
Derivation of the UVJD posterior function
Let denotes price of asset at time t
, , where , , ,
We have
We want
Now,
Consider ,
(1)
(1)
, (2)
Jump Diffusion Models for the Foreign Exchange Market
Consider
Similarly,
Observe that , where ,
, where
For ,
(2)
Jump Diffusion Models for the Foreign Exchange Market
Appendix B
Derivation of the MVJD posterior density and estimate
Let , where denotes return of asset i at time t.
For , where , ,
, , , ,
Now, ,
Now, where
, are generated values of random variables W and Z respectively.
Jump Diffusion Models for the Foreign Exchange Market
Appendix C
MCMC algorithm for UVJD model function [lliks,mus,sigmas,etas,lambdas] = jdmcmc(m0,s0,e0,a,b,t1,t2,t3,t4,Y,N) lliks = zeros(1,N); mus = zeros(1,N); sigmas = zeros(1,N); etas = zeros(1,N); lambdas = zeros(1,N); mup = zeros(1,N); sp = zeros(1,N); ep = zeros(1,N); lap = betarnd(a,b,1,N); mus(1) = normrnd(m0,s0,1,N); sigmas(1) = gamrnd(t1,s0/t2); etas(1) = gamrnd(t2,e0/t3); lambdas(1) = lap(1); lliks(1) = sum(log(jdlik(lambdas(1),sigmas(1),mus(1),etas(1),Y))); for(i = 2:N) mup(i) = normrnd(m0,s0*t1); llc = sum(log(jdlik(lambdas(i-1),sigmas(i-1),mus(i-1),etas(i-1),Y))); llp = sum(log(jdlik(lambdas(i-1),sigmas(i-1),mup(i),etas(i-1),Y))); lpp = log(normpdf(mup(i),m0,s0)); lpc = log(normpdf(mus(i-1),m0,s0)); ar = (log(rand) <= llp-llc+lpp-lpc); % Metropolis acceptance ratio mus(i) = mus(i-1)*(1-ar)+mup(i)*ar; sp(i) = chi2rnd(t2*sigmas(i-1))/t2; llc = sum(log(jdlik(lambdas(i-1),sigmas(i-1),mus(i),etas(i-1),Y))); llp = sum(log(jdlik(lambdas(i-1),sp(i),mus(i),etas(i-1),Y))); lpp = log(exppdf(sp(i),s0)); lpc = log(exppdf(sigmas(i-1),s0)); lprop = log(chi2pdf(t1*sp(i),t2*sigmas(i-1))); lproc = log(chi2pdf(t1*sigmas(i-1),t2*sp(i))); ar = (log(rand) <= llp-llc+lpp-lpc+lproc-lprop); % Metropolis acceptance ratio sigmas(i) = sigmas(i-1)*(1-ar)+sp(i)*ar; llc = sum(log(jdlik(lambdas(i-1),sigmas(i),mus(i),etas(i-1),Y))); llp = sum(log(jdlik(lambdas(i-1),sigmas(i),mus(i),ep(i),Y))); lpp = log(exppdf(ep(i),e0)); lpc = log(exppdf(etas(i-1),e0));
Jump Diffusion Models for the Foreign Exchange Market
lprop = log(gampdf(ep(i),t2,e0/t3)); lproc = log(gampdf(etas(i-1),t2,e0/t3)); ar = (log(rand) <= llp-llc+lpp-lpc+lproc-lprop); % Metropolis acceptance ratio etas(i) = ep(i)*ar+etas(i-1)*(1-ar); llc = sum(log(jdlik(lambdas(i-1),sigmas(i),mus(i),etas(i),Y))); d = t4*(1-lambdas(i-1))/lambdas(i-1); lap(i) = betarnd(t4,d); lpp = log(betapdf(lap(i),a,b)); lpc = log(betapdf(lambdas(i-1),a,b)); lprop = log(betapdf(lap(i),c,d)); d = t4*(1-lap(i))/lap(i); lproc = log(betapdf(lambdas(i-1),t4,d)); llp = sum(log(jdlik(lap(i),sigmas(i),mus(i),etas(i),Y))); lambdas(i) = lambdas(i-1)*(1-ar)+lap(i)*ar; lliks(i) = sum(log(jdlik2(lambdas(i),sigmas(i),mus(i),etas(i),Y))); end
Jump Diffusion Models for the Foreign Exchange Market
Appendix D
MCMC Algorithm for MVJD model
function [lliks,mus,sigmas,etas,lambdas] = jdmcmcMV(M0,S0,e0,a,b,t1,t2,t3,t4,Y,N,J) K = size(Y,2); lliks = zeros(1,N); mus = zeros(N,K); sigmas = zeros(K,K,N); etas = ones(1,N); lambdas = 0.1.*ones(1,N); % Independent proposals at the beginning mup = mvnrnd(repmat(M0,N,1),S0.*t1); ep = gamrnd(t3,e0/t3,1,N); % RW proposals start with series of zeros Sp = zeros(K,K,N); lap = zeros(1,N); mus(1,:) = mup(1,:); sigmas(:,:,1) = wishrnd(S0,t2)./t1; % 1st RW proposal lambdas(1) = betarnd(a,b,1,n); % 1st RW proposal etas(1) = ep(1); lliks(1) = mvjdlik(Y,mus(1,:),etas(1),squeeze(sigmas(:,:,1)),lambdas(1),J); for i = 2:N % for each iteration, chain moves from mu -> sigma -> eta -> lambda llc = mvjdlik(Y,mus(i-1,:),etas(i-1),squeeze(sigmas(:,:,i-1)),lambdas(i-1),J); llp = mvjdlik(Y,mus(i,:),etas(i-1),squeeze(sigmas(:,:,i-1)),lambdas(i-1),J); lprp = log(mvnpdf(mus(i,:),M0,S0)); lprc = log(mvnpdf(mus(i-1,:),M0,S0)); ar = (log(rand) <= llp-llc+lprp-lprc); %ar = MH acceptance ratio mus(i,:) = mus(i-1,:).*(1-ar)+mup(i,:).*ar; Sp(:,:,i)= wishrnd(sigmas(:,:,i-1),t2)./t2; llc = mvjdlik(Y,mus(i,:),etas(i-1),squeeze(sigmas(:,:,i-1)),lambdas(i-1),J); llp = mvjdlik(Y,mus(i,:),etas(i-1),squeeze(Sp(:,:,i)),lambdas(i-1),J); lprop = logwishpdf(t1.*Sp(:,:,i),sigmas(:,:,i-1),t2); lproc = logwishpdf(t1.*sigmas(:,:,i-1),Sp(:,:,i),t2); lprp = logwishpdf(K.*squeeze(Sp(:,:,i)),S0,K); lprc = logwishpdf(K.*squeeze(sigmas(:,:,i-1)),S0,K); ar = (log(rand) <= llp-llc+lproc-lprop+lprp-lprc);
Jump Diffusion Models for the Foreign Exchange Market
sigmas(:,:,i) = sigmas(:,:,i-1).*(1-ar)+Sp(:,:,i).*ar; llc = mvjdlik(Y,mus(i,:),etas(i-1),squeeze(sigmas(:,:,i)),lambdas(i-1),J); llp = mvjdlik(Y,mus(i,:),ep(i),squeeze(sigmas(:,:,i)),lambdas(i-1),J); lprp = log(exppdf(ep(i),e0)); lprc = log(exppdf(etas(i-1),e0)); lprop = log(gampdf(ep(i),t3,e0/t3)); lproc = log(gampdf(etas(i-1),t3,e0/t3)); ar = (log(rand) <= llp-llc+lprp-lprc+lproc-lprop); etas(i) = ep(i)*ar+etas(i-1)*(1-ar); lap(i) = betarnd(c,c*(1-lambdas(i-1))/lambdas(i-1)); llp = mvjdlik(Y,mus(i,:),etas(i),squeeze(sigmas(:,:,i)),lap(i),J); llc = mvjdlik(Y,mus(i,:),etas(i),squeeze(sigmas(:,:,i)),lambdas(i- 1),J); lprp = log(betapdf(lap(i),a,b)); lprc = log(betapdf(lambdas(i-1),a,b)); lprop = log(betapdf(lap(i),t4,t4*(1-lambdas(i-1))/lambdas(i-1))); lproc = log(betapdf(lambdas(i-1),t4,t4*(1-lap(i))/lap(i))); ar = (log(rand) <= llp-llc+lprp-lprc-lprop+lproc); lambdas(i) = lambdas(i-1)*(1-ar)+lap(i)*ar; lliks(i) = mvjdlik(Y,mus(i,:),etas(i),squeeze(sigmas(:,:,i)),lambdas(i),J); end