Hi all,I'm new here and I must say this site is excellent and very helpful. I'm currently writing my masters thesis in financial economics and I have one general question about simulating currency's. In my thesis I'm building a multistage stochastic programming optimization model for asset liability management for a Icelandic pension fund. I have background in engineering (optimization in particular) and only recently begun studying the financial markets, so to speak. What I need to do is to simulate stock prices x years into the future. To convert the stock price returns (which are generally quoted in USD) to returns in ISK (Icelandic krona). Then I have to simulate the USD/ISK currency 'kross'. My question is:"what kind of time series model is suitable for simulating the USD/ISK currency (price) dynamics or currency dynamics in general". I use monthly data, and I have mild garch effects (first 3 lags statistically significant in the acf for square monthly returns) and also the first lag in the acf for raw returns is significant. My thoughts so far is to model the series with GARCH(1,1) or (AR(1)-GARCH(1,1)) time series model, calibrated with the historical data, and then simulate the price dynamics. I have also considered GBM. But GBM assumes log-normal distribution and according to the first four moments of the historical data series. If I use GBM i will not capture the high kurtosis and skewness. I have also considered GARCH(1,1) model other error terms than gaussian, for example students-t or skewed t distributions. All comments and thoughts more than welcomed and appreciated.

The same GARCH models might make sense for your currency pair. Another suggestion is to make sure youunderstand the market mechanics well; my suggestion is have that pension fund introduce you to somecurrency traders at their custody bank and take them out to lunch.

Lunch is a good idea - make sure the traders pay Welcome, glad you are finding the site useful - it's been enormously helpful for me.If you have the time (which I realise may be limited when writing a thesis), I would suggest using the GARCH or AR-GARCH model and the GBM. Reason being: as you have written, the (AR-)GARCH models are calibrated to historical data, and relying solely on these models implicitly assumes that past USD/ISK movements will be repeated in the future - including the massive devaluation of 2008-9 (http://www.xe.com/currencycharts/?from= ... K&view=10Y). In contrast, by adjusting the volatility in the GBM model you can obtain a range of plausible future outcomes. Have you considered simulating using a mixture of 2-Normal distributions (both having the same mean)? For little extra effort, these will give you some extra kurtosis. On a final note, regardless of which simulation process(es) used, you may wish to run your optimisation over a number of scenarioes to check the robustness of the optimised portfolio. Example scenarioes that I would use: 1) the Great Moderation period = low FX vol / low US equity vol, 2) turbulence in Iceland = high FX vol / low US equity vol, 3) the Great Crisis = high vol in FX and US equities, 4) US credit rating downgrade = even higher vols than in 3).Hope this helps.

Quick follow-up: I recall from the literature review for my research, that several researchers found that skewness and kurtosis are present in daily currency returns, but decay to (almost) 0 and 3 respectively over 1- to 6-months (depending on the currency pair). Consequently one can justify using GBM based on this empirical research.

Thank you Alan and ahew for quick and helpful replies. Alan: Both of my contacts are on summer holiday at the moment and I will definitely meet them over coffee when they return (and make sure they pay ). Since I have limited time I'm eager to get the ball rolling on the subject. ahew: I remember reading somewhere that skewness and high kurtosis are both greatly effected by few outlaws (extreme values) like the devaluation of 2008-9. As you pointed out monthly currency time series have relative little skewness and excess kurtosis and therefore GBM might by better option. Also the longest USD/ISK time series that I have managed to get my hands on is roughly 13 years, which equals roughly 150 data points. Furthermore the period around 2008 and after can not be considered 'normal' (I hate using the word normal in this content) due to the capital control. I have not used mixed gaussian models. "Example scenarios that I would use: 1) the Great Moderation period = low FX vol / low US equity vol, 2) turbulence in Iceland = high FX vol / low US equity vol, 3) the Great Crisis = high vol in FX and US equities, 4) US credit rating downgrade = even higher vols than in 3)." <- I will keep this in mind.Maybe one quick off topic question. Have you ever used multivariate garch models to simulate stocks price dynamics? Models like DCC-GARCH and CCC-GARCH. I took a look at the free available codes form Kevin Sheppard (http://www.kevinsheppard.com/wiki/MFE_Toolbox). However a quick google search gave me reason to doubt that the coding is right. There is also a package in R for multivariate garch models (ccgarch) but I haven't had the time to try it out myself.Thnx again.

Pleasure.Yes - the ISK devaluation will affect your sample skewness and kurtosis estimates. In my experience, there is no one right or wrong way to handle such circumstances. The data is what it is, but how representative it is of the future is unknown. You might wish to calculate the skew/kurtosis when the outliers are excluded (eg, those more than 6 standard deviations away from the mean) as this will quickly show you what effect the devaluation had.Presumably the capital controls acts as a constraint in your optimisation, and limit the value of US equities you can buy/sell in any period?I haven't used multivariate GARCH models so can't comment, but I'm sure other Wilmott'ers will gladly share their experiences.

You can incorporate some Dummy variable sort of stuff into your model to capture those uncommon phenomena. And regarding GARCH fitting, I dont think you should look into it first hand. In general estimation of model parameters should not be affected asymptotically whether you have high skewness or kurtosis. Asymptotic distribution of model parameters doesnt depend on the distribution of the innovations. Therefore you should fix the goal first of your analysis; If your goal is just to simulate futures values, construct confidence band around future values then only you need detailed distributional assumption of innovation. You should incorporate GARCH stuff only if there are legitimate ARCH effect in your data, beware of over-fitting.Regards,

Thank you again for your repliesAhew: I will estimate the the skewness and kurtosis when outliers are excluded and see the difference. The capital controls will be taken into account via constraints in the stochastic optimization problem. ronm: I'm currently using monthly data from four MSCI stock indexes as a proxy for foreign stocks in the model (the pension fund invested in those indexes before the capital controls). They all have ARCH effects and furthermore not statistically significant auto correlation or partial correlation in the raw return data series. This is the reason I turned to GARCH(1,1) in the first place. Since the stock returns have excess kurtosis and are all skew to the left I'm currently thinking of using GARCH(1,1) with t-distributed (or skewed t) innovations. Since I would like to model the movements of all indexes at the same time I have turned to multivariate GARCH. The reason for DCC/CCC-GARCH is the intuitive 'expansion' from univariate GARCH to multivariate and it is also relatively simple to estimate the model parameters (two-stage estimation method introduced by Engle and Sheppard). Also In their paper the construct a hypothesis test to test for dynamic conditional correlation. Since I reject the null of constant conditional correlation for my data I have turned my attention to DCC-GARCH.About the over-fitting. How can I make sure that it is not the case when I have estimated the model parameters. I have also considered multivariate GBM, but due to the log-normal assumption I will 'loose' the third and the fourth moments in the simulation. Any suggestions of other models to simulate stock price dynamics are of course welcomed.Thnx, again

QuoteAbout the over-fitting. How can I make sure that it is not the case when I have estimated the model parameters.I dont think any statistical test exists which tells you, your model is over-fitted! It all depends on Experience. Just keep your model simple and intuitive, so that it can explain the environment in fundamental level. Never bring unnecessary complexity in the model. In most of the time, a simple model works far better than any fancy models and simple model is most easy to implement, in terms of speed and automation.Regards,