Multivariate Jump Diffusion Models for the Foreign Exchange Market

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


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.


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, 337-3342, Dec,. 1997


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


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


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


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


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 .


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


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.


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


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


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.


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.


, 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.


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


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

.


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


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


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.


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


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.


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.


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


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.


References

Bank of International Settlement, 2007. “Foreign Exchange and Derivative market activity in 2007”,

Triennial Central Bank Survey,19 Dec.2007

Browne, W. J., 2006, “MCMC algorithms for constrained variance matrices”, Comuptational

Statistics & Data Analysis, Vol. 50. pages 1655-1677, 2006

Browne, W.J., Draper, D., 2000, “Implementation and performance issues in Bayesian and likelihood

fitting of multilevel models”, Computational Statistics”, Vol.15, page 391-420, 2000

Eltoft, T., Kim, T, Lee, T-W, 2006, “On the Multivariate Laplace Distribution”, IEEE Signal

Processing Letters, Vol. 13 No.5 May., 2006.

Fama, E. F., 1965. ”Random Walks in Stock Market Prices.” Financial Analyst Journals, Vol. 21, No.

5, page 55-59, Sep.-Oct.,1965.

Gelman, A., Carlin, J. B., Stern, H., 2004, “Baysian Data Analysis”, 2nd ed. Boca Raton, 2004

Kou, S.G., 2002, “A Jump-Diffusion Model for Option Pricing”, Management Science, Vol.48, No. 8

page1086-1101, Aug. 2002

Markowitz, H., 1952. “Portfolio Selection”, The Journal of Finance. Vol. 7, No. 1, page 77-91, Mar.,

1952.

Markowitz, H., 1999. “The early history of portfolio theory: 1600-1960”, Financial Analyst Journal ,

Vol.55, No.4,. Page 5-16, Jul.- Aug. 1999

Ramezani, C.A., Zeng Y., 1998. “Maximum Likelihood Estimate of the double exponential jump-

diffusion process”, Annals of Finance, Vol. 3, Issue 4, Page 487-507. Oct.2007

Ross, S. M., 2007. “Introduction to Probability Models”. 7th ed. Elsevier/Academic Press.

Ross, S., Westerfield, R., Jaffe, J., Jordan, B., 2008, “Modern Financial Management”, 8th ed.,

McGraw-Hill Irwin, 2008

Shah, A., 1997. “Black, Merton and Scholes: Their Work and its Consequences”. Economic and

Political Weekly, Vol. 32, No.52, Page 3337-3342, Dec,. 1997

Verhoeven, P., McAleer, M., 2004. “Fat tails and asymmetry in financial volatility models”,

Mathematics and Computers in Simulation, Vol.64 No.3-4 , Pages 351-361, Feb., 2004


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)


Consider

Similarly,

Observe that , where ,

, where

For ,

(2)


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.


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));


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


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);


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

Multivariate Jump Diffusion Models for the Foreign Exchange Market

Documents

Transcript of Multivariate Jump Diffusion Models for the Foreign Exchange Market