1 Introduction

2 Data Source

Figure 2. Time Series of Used Data

Figure 2. Time Series of Used Data

3 Mathematical Modal

3.1 Geometric Brownian Motion (GBM) Model

  • Brewer points out in his paper that the Geometric Brownian Motion Model means the logarithm of the data follows a Brownian motion and provided the following equations(Brewer, K., Feng, Y., & Kwan, C, 2016).

  • The original differential equation is defined as \[ dN = \mu Ndt+\delta Ndz\] where \(N\) is the foreign exchange rate on that day, \(dz=\epsilon \sqrt{dt}\) and \(\epsilon\) is a random draw from the normal distribution with mean 0 and variance \(\sigma\). \(\mu\) and \(\delta\) are usually the drift parameter and the volatility parameter, respectively.

  • The equation is equivalent to \[ d\log(N) = (\mu -\frac{\delta ^2}{2})dt+\delta dz\]

  • After solving the differential equation, we get \[ N_{t+\Delta{t}}=N_{t}e^{(\mu -\frac{\delta ^2}{2})\Delta{t}+\delta \epsilon \sqrt{\Delta{t}}}\] Set \(\Delta{t}\) equal to 1, we have \[ N_{t+1}=N_{t}e^{(\mu -\frac{\delta ^2}{2})\frac{1}{n}+\frac{\delta }{\sqrt{n}}\epsilon }\] where \(n\) is the number of days in a year, which is 260 (only accounts for open market days).

3.2 Parameter Description

  • There are 3 parameters \(\mu\), \(\delta\) and \(\sigma\).

  • \(\mu\) is the drift parameter that shows the increasing or decreasing trend.

  • \(\delta\) is the volatility parameter that measures the deviations from the mean.

  • \(\sigma\) is the variance of the state parameter \(\epsilon\). By increasing sigma, it will increase the deviations from the mean.

4 POMP Model

4.1 Build POMP Model

  • The rprocess is based on the GBM model with two state variables, \(N\) and \(\epsilon\).

  • The parameters are \(\mu\), \(\sigma\) and \(\delta\).

  • The initial value of N is drawn from a random poisson distribution with mean 1.5. The initial value of \(\epsilon\) is drawn from a poisson distribution with mean 1.

  • The rmeasure is defined as Rate being drawn from a random draw from the normal distribution with mean 0 and variance \(N\), which is the state variable.

  • The detailed implementation is shown below.

dmeas <- Csnippet("lik = dnorm(Rate,0,N,give_log);")
rmeas <- Csnippet("Rate = rnorm(0,N);")
Ne_initializer <- "
 N=rpois(1.5);
 e=rpois(1);
"
stochStep <- Csnippet("
                      e = rnorm(0,sigma);
                      N = N*exp((mu-delta*delta/2)/260+delta/sqrt(260)*e);
                      ")

stopifnot(packageVersion("pomp")>="0.75-1")
pomp(data = dt2,
     times="Date",
     t0=0,
     rprocess=discrete.time.sim(step.fun=stochStep,delta.t=1),
     rmeasure = rmeas,
     dmeasure=dmeas, 
     obsnames = "Rate",
     paramnames=c("mu","delta","sigma"),
     statenames=c("N","e"),
     initializer=Csnippet(Ne_initializer)
     ) -> fx

4.2 Set Run Level

  • There are three run levels. The analysis of this report is based on level 3.

  • Detailed parameters are defined below

run_level = 3
level_Np = c(100,1000,5000)
level_Nmif = c(10,100,300)
level_Nreps_eval = c(4,10,20)
level_Nreps_local = c(10,20,20)
level_Nreps_global = c(10,20,100)

5 Likelihood Slice

Figure 3. Slicing for mu

Figure 3. Slicing for mu

Figure 4. Slicing for delta

Figure 4. Slicing for delta

Figure 5. Slicing for sigma

Figure 5. Slicing for sigma

6 Partical Filter

##                          se 
## -576.34332106    0.01696181

7 Iterated Filtering on Data

7.1 Maximization and Likelihood Evaluation

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -569.01 -568.96 -568.95 -568.91 -568.83 -568.81
  • The Local Maximum Search gives us a maximum of likelihood of -568.81 with a standard error of 0.007.

7.2 Likelihood Surface Evaluation

Figure 6. Pairwise Plot

Figure 6. Pairwise Plot

  • The pairwise plot shows us the geometry of the likelihood surface in a neighborhood (20 units around the maximum likelihood) for each parameter.

  • I will then perform global search to see if the result is stable and compare the likelihood with the GARCH model.

8 Global Search with Randomized Starting Values

8.1 Iterated Filtering with randomized starting values

  • For our data, the randomized starting values could be chosen from the following box.
fx_box <- rbind(
  mu = c(-5,10),
  delta = c(0.1,1.5),
  sigma = c(0.5,3)
)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -587.61 -582.05 -575.61 -575.90 -568.98 -568.82

  • Parameter estimation requires many and random starting values. Now I draw a starting value uniformly from an interval for each parameter. If the estimation gives stable result with different starting values, then it will provide confidence that an adequate global search has been carried out (Ionizes, E., 2016).

  • It gives us a best likelihood of -568.82 with a standard error of 0.008.

  • The likelihood is similar to our local search and the global search provides a quite stable statistic.

8.2 Global Geometry of the likelihood surface

Figure 7. Pairwise Plot

Figure 7. Pairwise Plot

  • By using the pairwise plot, we can get a feeling of how does the geometry of the likelihood surface look like in the neighborhood (10 units around the maximum likelihood) for each parameter.

  • We can see that the points are clustered and end up with comparable likelihoods though the parameters are drawn from different starting values. It shows that the model gives stable maximization of likelihood.

9 GARCH Model

9.1 Model Description

  • According to notes from the class, the GARCH(p,q) model has the form \[ Y_n = \epsilon_n \sqrt{V_n},\] where \[ V_n = \alpha_0 + \sum_{j=1}^p \alpha_j Y_{n-j}^2 + \sum_{k=1}^q \beta_k V_{n-k}\] and \(\epsilon_{1:N}\) is white noise (Ionides E., 2016).

  • From comparing the GARCH(1,0), GARCH(0,1) and GARCH(1,1) model by checking their log likelihoods, I decide to use the GARCH(0,1) model which gives the largest likelihood.

9.2 Log Likelihood

## 'log Lik.' -560.0184 (df=2)

  • The GARCH(0,1) model gives the log likelihood of -560.0184. It is slightly larger than than the likelihood we get from the POMP model.

  • The likelihood does not change significantly between the POMP model and GARCH model. However, the GARCH model is difficult to interpret. So personally speaking, I favor the POMP model. But we may still need to use some preliminary results to see if it can be improved to beat the GARCH model.

9.3 Diagnose Plots

Figure 10. Diagnostics Plots

Figure 10. Diagnostics Plots

  • The QQ-plot shows the residuals are normally distributed.

  • The ACF plot shows 2 lags slightly falling outside the dashed line. Since most autocorrelation values are within the confidence lines, the do not reject the assumption that errors are independent.

10 Conclusions

11 Limitations and Further Work

12 Acknowledgment

The project is based on the course on Time Series Analysis, taught by Professor Ionides at the University of Michigan Winter, 2016. The POMP model and R coding consults examples in the class notes from http://ionides.github.io/531w16/notes13/notes13.html.

13 References

[1] Brewer, K., Feng, Y., & Kwan, C. (2012, November). Geometric Brownian Motion, Option Pricing, and Simulation. Retrieved April 21, 2016, from http://epublications.bond.edu.au/cgi/viewcontent.cgi?article=1131&context=ejsie

[2] Foreign Exchange Rates - H.10. (2016). Retrieved April 21, 2016, from https://www.federalreserve.gov/releases/h10/hist/dat00_uk.htm

[3] Geometric Brownian motion. (2016, February 18). In Wikipedia, The Free Encyclopedia. Retrieved 21:37, April 21, 2016, from https://en.wikipedia.org/w/index.php?title=Geometric_Brownian_motion&oldid=705536768

[4] Gerber, H., & Shiu, E. (2003, June). Geometric Brownian Motion Models for Assets and Liabilities: From Pension Funding to Optimal Dividends. Retrieved April 23, 2016, from https://www.researchgate.net/publication/267017769_Geometric_Brownian_Motion_Models_for_Assets_and_Liabilities_From_Pension_Funding_to_Optimal_Dividends

[5] Ionizes, E. (2016, March 22). Practical likelihood-based inference for POMP models. Retrieved April 23, 2016, from http://ionides.github.io/531w16/notes13/notes13.html

[6] Ionides, E. (2016, April 5). Case study: POMP modeling to investigate financial volatility. Retrieved April 23, 2016, from http://ionides.github.io/531w16/notes15/notes15.html