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Volatility Forecasting

Candidate number:

Abstract
This paper constructs a hedged portfolio with a long positon in S&P 500 index and a short position in FTSE 100 index. To calculate the time-varying hedge ratio, we use four methods, rolling window, EWMA, GARCH model and B-S model. Firstly, we explain the methods we used, including the assumptions, formulas and implications. Also, we implement the methods in the Excel to get the value of hedge ratios. Finally, we show the advantages and disadvantages of every method by comparing between every two methods. We evaluate the methods both in theory and practical application.

Introduction:
We want to hold a long position in S&P 500 index, at the same time we want to minimum-variance. As this reason, we introduce to hedge the long position in S&P 500 index with a short position in the FTSE 100 index. We suppose that the return on the hedged portfolio is: rp,t=rS&P-htrFTSE Then the variance of hedged profile will be: δF2=δS&P2+h2δFTSE2-2hδS&P,FTSE
If we want to minimum the variance of the hedged portfolio, we must derivate of h in this function and let the equal to zero. Where ht is the time-varying hedge ratio, given by:

ht=δS&P,FTSE,tδS&P2
Because we want to get the result only focus on the two-index, we ignore the currency flotation by assume that the currency rate is perfectly hedged.
At very beginning, risk manager have assumed that the volatility is constant over time, which allow us to estimate it use sample variance of past time: δ2=1T-1t=1T(rt-r)2 By estimate the historical data, our find the variance tend to be clustered, it is obvious to see all the index not only have high volatility during some period, but also have some low volatility at some other periods. So an approach to estimating time-varying volatility is to use the data from the recent past. This leading us introduce rolling window model: δ2=1M-1i=t-M+1t(ri-rt)2
We use the most recent days, says M, as the windows length to evaluate the new variance. This approach show longer periods more stable, but this approach leave wholly unanswered the choice of the rolling windows. Long precision of the estimate but could miss undying variation in volatility (Value at risk, chapter 9.2). As this reason, the limitations of this model is quit obvious, if we think the most recent past data are more valuable than the pervious data, why yesterday’s data have equal weight to the 100 days ago data? And dose the data just passed window length really mean nothing? For solve this limitation of RW (rolling windows) model, we introducing EWMA model: δt+12=λδt2+(1-λ)rt2 Which states that today’s variance is a weighted average of yesterday’s variance and yesterday’s squared return, with the determined by λ. This approach give us the biggest weight for the most recent data, and a smoothly decline for the past.
EWMA model is the simplified model from GARCH model. The conditional variance depends on the latest innovation but also on the previous conditional variance. We usually use the GARCH(1,1) model, the simplest model to forecast the volatility, σt+12 = а0+ а1σt2+ ß1rt2.
We decided to use the multivariate Rolling Window, EWMA and GARCH approaches, and to evaluate each model over a five-year period at first. Show the effectiveness of each model by compare the reduction of each hedge variance depend on RW, EWMA and GARCH model.

Method to using 1. Classic method
To obtain a control group, we computed an unconditional volatility. The constant variance and covariance are based on the first year simple return. The reduction, is (the unhedged variance – the hedged portfolio variance)/the unhedged variance, as a yard stick to measure the hedging.
Suppose we only use the data in the first year 2007 to compute the constant variance and covariance. We do not add new information set into our forecast. Then we used the variance and covariance to calculate the hedge ratio in the next 5 years. Variance of FTSE 100 | 0.000188 | Covariance (S&P 500-FTSE 100) | 0.000090 | Hedge ratio | 0.482044 | Unhedged variance | 0.000271 | Hedged variance | 0.000173 | Reduction | 36.01% |

2. Rolling window method
Rolling Window method is a very crude method, but one that is employed widely, is the use a rolling windows of fixed length for estimating volatility. (Value at risk, 3rd Ed, chapter 9.3) Assuming that we use data over T days, the volatility estimate is analysis from rolling windows: δ2=1T-1t=1T(rt-r)2 Rolling window model estimates the time-varying volatility based on returns from the recent historical data. It uses the most recent data from M days ago, M known as a window. We observe returns rt over M days, the volatility estimate is constructed from a moving average. Each day, the forecast is updated by adding information from the preceding day and dropping information from (M+1) days ago. The information set is updated daily, we can get the conditional volatility based on the information we already have. All weights on past returns in the window are equal to 1/(M-1).

Assumptions:
We can forecast the future volatility on the information set that investors have now, namely, the historical data can be used to predict the future.
A moving window of fixed length for estimating volatility, we assume the window length M is constant.
Observations in the window have the same fixed positive weight 1/M, and zero weight if they are outside the window. It means recent information receives the same importance as older observations in the window, and we reject the older observations outside the window.

The implementation in Excel * Historical data selection
We download six years of data for the S&P 500 index and FTSE 100 index from YAHOO! FINANCE. We evaluated each model over a five year period, so we need one more year data to get the information set for rolling window estimator.
The Standard & Poor's 500, is a stock market index based on the market capitalizations of 500 leading companies publicly traded in the U.S. stock market, as determined by Standard & Poor's. The index value is updated every 15 seconds during trading sessions, and starts at 09:30 and ends at 16:00. The FTSE 100 Index, is a share index of the 100 companies listed on the London Stock Exchange with the highest market capitalization. The index is maintained by the FTSE Group, an independent company which originated as a joint venture between the Financial Times and the London Stock Exchange. It is calculated in real time and published every 15 seconds. Continuous trading on the London Stock Exchange starts at 08:00 and ends at 16:30 (when the closing auction starts), and closing values are taken at 16:35.
As a result of the difference of trading sessions between U.S. stock market and London Stock Exchange, the transaction date is not exactly the same with each other. We need to delete the dates that are not on the same trading day. Otherwise, it will have data error on the covariance. We used the ‘VLOOKUP’ function to screen the data are not on the same trading day for the two index, then deleted them. To obtain a series of correct data, we have to abandon the noisy data.

* Rolling window estimator
We chose 1 year as the window length. Actually, the first year 2007 only have 249 simple returns, M = 249. We compute the time-varying variance and covariance from 04/24/2008 to 04/19/2013, everyone chose the preceding 249 days’ simple returns as the information set. Unhedged variance | 0.000271 | Hedged variance | 0.000165 | Reduction | 39.06% |

* The optimal rolling window estimator
In the last step, we assumed a window length. But this is unlikely to be optimal, we decided to estimate an optimal window length to compute hedge ratio. We already computed the result when M=249, then we used the “table” in “what if analysis” to list the reduction on M= 2,3,4…249. Then the match function told us the optimal window length. Maximum reduction | 39.26% | Optimal window length | 124 |

3. EWMA model
“Formally, the forecast for time t is a weighted average of the previous forecast, using weight λ, and of the latest squared innovation, using weight(1-λ), that is,” (Value at risk, 3rd Ed, chapter 9.2.4). Today’s variance is a weighted average of yesterday’s variance and yesterday’s squared return, with the weight determined by λ. δt+12=λδt2+(1-λ)rt2 We call the λas decay factor, and it must less than 1. If we recursively replace δt2

into the equation, we can get δt+12 = (1-λ)i=0∞λirt-i2.
It places less and less weight on the observations as they move further into the past, by using the decay factor. The larger the value of decay factor, the more weight is placed on past observations and so the smoother the series becomes.
EWMA model is from GARCH model, the GARCH(1,1) process is σt+12 = а0+ а1σt2+ ß1rt2 when а0=0 and ß1 = 1-а1, the GARCH(1, 1) model reduces to the EWMA model.
In the application of finance, we also need to estimate covariance or correlation. Multivariate EWMA model can be applied to the covariance between two series of returns, σij,t = λσij,t + (1-λ)ri,trj,t.

Assumptions:
The mean return is equal to zero.
Decay factor λ is a constant parameter, and less than unity.
The recent information is more important than the past information. EWMA weight observations in the sample in a such way that their importance declines smoothly into the past.

The implementation in Excel * Historical data selection
We chose the simple return data the same as rolling window, the first year 2007 was used to initialize the EWMA estimator. We assume the variance and covariance on 04/24/2007 are zero. * EWMA(0.94)
The decay factor is assumed as 0.94. We calculate the variance and covariance from 04/25/2007 to 04/19/2013, then we got hedge ratio from 04/24/2008 to 04/19/2013.

Unhedged variance | 0.000271 | Hedged variance | 0.000170 | Reduction | 37.15% | * EWMA(optimal)
Decay factor λ= 0.94 is unlikely to be optimal, we decided to test a third model, which is multivariate EWMA model with a decay factor that is estimated. The optimal decay factor was obtained by solver. Solver changed the value of decay factor to get the maximum reduction.

Unhedged variance | 0.000271 | Hedged variance | 0.000165 | Optimal λ | 0.995177 | Reduction | 39.11% |

4. GARCH model
A more sophisticated conditional volatility model is known as GARCH. “Volatility estimation has moved toward models that put more weight on recent information. The first such model was generalized autoregressive conditional heteroskedastic (GARCH) model. The GARCH model assumes that the variance of returns follows a predictable process. The conditional variance depends on the latest innovation but also on the previous conditional variance.”(Value at risk, 3rd Ed, chapter 9.2.2). The simplest such model is the GARCH(1,1) process, that is rt=c+ εt σt+12 = а0+ а1σt2+ ß1εt2 where rt is the daily return and σt denotes the conditional variance of εt. In the ‘plain vanilla’ GARCH model the conditional variance is assumed to be normal distribution with mean zero.
So the GARCH(1,1) process is, σt+12 = а0+ а1σt2+ ß1rt2. Parameters а0, а1and ß1 are estimated by maximum likelihood estimation. We assume that returns rt are drawn from a normal distribution with zero mean. The probability density of rt is f(rt,σt2) = 12πσt2exp(-rt22σt2).
The sum of the log-likelihood function is t=1Tln L(а0, а1, ß1; rt)= t=1T(-lnπ2-12lnσt2-12rt22σt2).
We can get the estimators а0, а1and ß1 by setting the sum maximum.
As for the covariance between S&P 500 and FTSE 100, we used the Harris, Stoja and Tucker(2007) approach to estimate the implied covariance. “Consider two assets, i and j, whose per-period abnormal returns are given by εit= rit- υit εjt= rjt- υjt where rit and rjtare actual returns and υit and υjt are conditional mean returns. Given the time t −1 information set, an estimate of the conditional covariance matrix requires estimates of the conditional variances σit2 and σjt2, and an estimate of the conditional covariance. σij,t. The multivariate GARCH model that we propose involves firstly estimating the conditional variances, σit2 and σjt2, using a univariate GARCH model. We then construct the new series ε+,t= εit+εit ε-,t= εit-εit and use a univariate GARCH model to estimate σ+,t2 and σ-,t2. An estimate of the conditional covariance σij,t can then be obtained using the following identities.” (A Simplified Approach to Modeling the Comovement of Asset Returns, 2004) σ+,t2= σit2 + σjt2 + 2σij,t σ-,t2= σit2 + σjt2 - 2σij,t
In particular, combining the two functions, we have σij,t= 14(σ+,t2- σ-,t2)

Assumptions
The mean return is zero.
Returns rt are drawn from a normal distribution with zero mean: rt ~ N(0, σt2.

The implementation in Excel
To implement GARCH model in the Excel, we proceed as follows: * Historical data selection
We chose the simple return data the same as EWMA, the first year 2007 was used to initialize the EWMA estimator. We chose the initial value for the conditional variance σ02 = 0 on 04/24/2007. With the passing of one year, variance tends to be the true value. * Parameter estimation
Arbitrarily chose starting values for the parameters, а0=0.5, а1=0.3and ß1=0.4., and constructed the conditional volatility series σt2=а0+ а1σt-12+ ß1rt-12. Then calculated the log-likelihood for each return rt, and used Solver function to maxing the sum t=1Tln L(а0, а1, ß1; rt)= t=1T(-lnπ2-12lnσt2-12rt22σt2) by changing the parameters а0, а1and ß1. We got the estimators of pararmeters.
Then we constructed two new series equal to the sum and difference of S&P 500 and FTSE 100 returns, and used the same method above to estimate the two sets parameters.

| FTSE 100 | The sum of S&P 500 and FTSE 100 | The difference of S&P 500 and FTSE 100 | а0 | 0.00000179 | 0.00000838 | 0.00000211 | а1 | 0.90939119 | 0.89603092 | 0.80962323 | ß1 | 0.81728052 | 0.09095506 | 0.19614655 |

* The calculation of time-varying variance and covariance
We applied the estimators into the formula σt2=а0+ а1σt-12+ ß1rt-12, to get σFTSE 100,t2, σ+,t2 and σ-,t2. Then computed covariance between S&P 500 and FTSE 100, σij,t= 14(σ+,t2- σ-,t2).

Hedged variance | 0.00017522 | Reduction | 35.34% |

5. Option method
An alternative way to estimates variance of those index from analysis the option price. As we already know the index option price from today, option method is infer from the option price, and the implied volatility is determined by the option price. Because of rest of parameter we are already know. And the formulas of B-S model is:

C=S*Nd1-X*exp-r*T*N(d2) d1= lnSX+r+σ22*TσT d2= d1- σ*T

Assumptions
The log normality of stock return
Constant risk free interest rate
No dividend payment (even the FTSE 100 index do not have any dividend)

The implementation in Excel * Historical data selection
All the data are coming from the yahoo finance and NYSE EURONEXT in 24/04/2013. Consider about the exchange market at US, we choose the US Treasuries as the risk free rate.

FTSE 100 index | Strike Price | Risk free rate | Maturity time (year) | Option price | 6411.83 | 6400 | 3% | 0.4 | 132 |

* Calculation of the implied volatility of FTSE 100
Set to an initial guess of the unknown δ and using the guess sigma to valued a computed call option price by using the B-S model formulas. Then write down the actual call option price and calculated the different of those two call option price, actual C and computed C. At last, set the different between actual C and computed C to zero using Solver by changing sigma. Then we get the implied volatility of the underlying asset is equal to 0.2503% per annum. Which means 0.01597% per day. (δm2=mδ12) The parameter m is the period and δm2and δ12 are the volatility during m days and single day, separately.

* Calculation of the covariance between FTSE 100 and S&P 500
Using the single day volatility of FTSE 100 from option method and the covariance between FTSE 100 index and S&P 500 index from other method (GARCH, Rolling Windows and EMWA) to copulated the hedge ratio and the reduction of volatility of each hedged portfolio, since 2013/1/24. (Because we using a four months call option to evaluated the volatility.)

| Option (COV-GARCH) | Option (Cov-RW) | Option (Cov-EWMA) | GARCH | RW | EWMA | Unhedged Volatility | 5.29E-05 | 5.29E-05 | 5.29E-05 | 5.29E-05 | 5.29E-05 | 5.29E-05 | Hedged Volatility | 4.44E-05 | 4.14E-05 | 4.64E-05 | 4.26E-05 | 4.09E-05 | 4.21E-05 | %Reeducation | 16% | 21.06% | 12% | 19% | 23% | 20% |

It is obvious to see by using different covariance basic on different methods, the option method have different performance. But the option method at least close and some time batter than other methods.

Results and Analysis 1. Classic method VS rolling window
The unconditional volatility only used the first year data to compute the constant volatility. It has serious drawbacks. It ignores the new information totally, and applied the constant volatility to the next 5 years. If we add the new observations, the weights of the data will be smaller and smaller. Weight 1/M decline as the growth of the denominator As time goes on, the new information plays insignificant role in the estimation, for the weight is too small.
Rolling window solve the problem. It specify the scope of the data, window length M, ensure the weight is constant after the after the mew observations join in. The forecast is updated by adding information from the preceding day and dropping information from (M+1) days ago. The results below shows rolling window model has a 3.05% improvement than classic method. | Classic method | Rolling window | Hedged variance | 0.000173 | 0.000165 | Reduction | 36.01% | 39.06% | 2. Rolling window VS EWMA
Limitations of rolling window * Ghost features
Due to the large one day return, large increase in volatility the following day. If the influential observation leaves the sample, a large reduction in volatility occurred. This is the ‘ghost features’ in the estimation of volatility.
We can reject the ‘ghost features’ data on purpose. Using the excel functions to screen the absolute value of simple returns greater than 4% and deleted them. We can get the result below, the reduction decreased. It is useless to remove the ghost data. Because we cannot distinguish between the random unusual data and the data depend on market issue. It would make the result worse if we delete the date reflect the real market situation.

| Rolling window(1 year) | Rolling window(removing the ghost data) | Unhedged variance | 0.000271 | 0.000213 | Hedged variance | 0.000165 | 0.000142 | Reduction | 39.06% | 33.39% |

* The weights on the past observations
The rolling window estimator gives past observations either full weight( if they are recent enough to lie in the window) or zero weight( once they become so old that they fall out of the sample). Do the observations in the window have the same importance? Are the observations outside the window useless? Obviously not, the simple selection of data limits the effectiveness of rolling window.
If we increase the window length, the precision with the return variance will be increased. It will reduce the weight 1/(M-1) on the observations, and decrease the sensitivity of rolling variance estimator to the market. This is the built-in problem in the model. So we need a new method, GARCH model.
EWMA model is the simplification of GARCH model, and the advantages are essentially the same over rolling window. Later we will analyses the difference between them.
To correct the ‘ghost features’ problem of equal weights of rolling window model, volatility estimation has moved toward models that put more weight on recent information. EWMA model responds more quickly to the market change. It weight observations in the sample in a such way that their importance declines smoothly into the past. According to δ_(t+1)^2 = (1-λ)∑_(i=0)^∞▒〖λ^i r_(t-i)^2 〗., the weight of simple return of day t is (1-λ), day t-1 is (1-λ)λ, day t-2 is (1-λ)λ^2. We can see the weight from recent to past declines in geometric sequence. Compared with rolling window, EWMA not only solve the problem of weight is 1/(M-1) or zero, but also promise the sensitivity of the recent information.
EWMA do not suffer from “ghost feature”. The ghost date can be declined in a smooth path. The ghost date in rolling window receives weight 1/(M-1) or zero, which may cause the abnormal fluctuations of data. Obviously, EWMA can solve the problem by giving the observations the exponentially weights. | Rolling window (1 year) | Rolling window (optimal) | EWMA (0.94) | EWMA (optimal) | Hedged variance | 0.000165 | 0.000165 | 0.000170 | 0.000165 | Reduction | 39.06% | 39.11% | 37.15% | 39.11% |
In theory, EWMA progress rolling window. However, the reduction of rolling window is larger or equal than EWMA. We suppose there is calculation error in the process. The Excel may can not handle such a large amount of data. Obviously, EWMA is a more accurate model than rolling window.
In principal, of course the EWMA give us a more reasonable assumption. But in the real world market, it have a large number of fluctuation from nowhere, and have less effective to the future, we call this random variable. For example, rumour will affect the market at some time, but when the lies were exposed, the effect of it will disappear very quickly. But our model cannot distinguish which is the rumour and which information will have long-term effect. EWMA will accumulate this huge number of very small but have amazing amount size random variable than RW model. But in more efficiency market, the investors have ability to distinguish some information, have rumour will less effective. And the EWMA will have a better performance in these kinds of market.

3. GARCH VS EWMA
We know the GARCH(1,1) model is σt+12 = а0+ а1σt2+ ß1rt2. The volatility of next day is related with volatility and return today. To simplify the model, we assume а0=0 and а1+ ß1=1, is EWMA model.
In principal, GARCH nests the EWMA model, it should unequivocally offer an improvement over the EWMA model. “But the true conditional variance is not integrated introduces estimation error that is detrimental to its performance.”(Value at risk, 3rd Ed, chapter 9.2.4) EWMA doing well than GARCH is the real world sometime, and easier to using, so EWMA are much popular than GARCH. “The longer-period forecasts, however, are markedly different because the EWMA process does not revert to the mean.” (Value at risk, 3rd Ed, chapter 9.2.4) But we only need to do is to forcast the volatility of the next day.

| EWMA(optimal) | GARCH | Hedged variance | 0.000165 | 0.00017522 | Reduction | 39.11% | 35.34% | The GARCH model our using is a simplify model. In the real world our using this model, we can analysis the parameters day by day to get the dynamic GARCH model, which will report different decay factor bias on each dataset impact. But as the limitation of our knowledge, our team do not know how to use excel analysis the dynamically model automatically. But even the dynamic model can realization, as a coin have to side, such many parameters tend this model to be computationally burdensome (A Simplified Approach to Modeling the Comovement of Asset Returns, page 2). And as the true conditional variance is not integrated in the real world, even it is close to being integrated, such many computationally method introduces estimation error that will decline the accurate of GRACH model. That is the reason why the EWMA model has a better performance than the GRACH model sometime.
But even the model as complicate as GRACH model, still have some defective should be improved. More specifically, depend on analysis a widely range of historical data from all around the world, we find the expected stock returns will decline, volatility tends to increase; when the good news, that the expected stock returns will rise, volatility tends to decrease. GARCH model cannot explain this asymmetric phenomenon, because this model using rt2 as the residual factor which we lend a no different reflection of positive or negative return. So Nelson (1991) introduce E-GARCH model: lnht2=α0+i=1pαiϵt-1ht-1+i=1priϵt-1ht-1+i=1qβlnht-i2 In this model, ε introduce by two different ways, the observed value and absolutely value. This method help us analysis the asymmetric phenomenon. 4. Option method VS other methods
To analysis the option method, what we actually do is using the B-S method and the formula, through we already know the price of call option the investor give to us calculated what is the variance him think about this index, or we can say it is a forecast of the professional investor.
And compare the different between option methods; GARCH, RW and EWMA we can figure out all the method can decline the risk of the portfolio very effectively. We should make decision to use different method in different situation. But the option gives us a professional advisor and we should not ignore it. We should better to use option method to be a reference value or a testing value.

Conclusion
To determine the optimal hedge ratios based on the minimum risk framework. We used rolling window method, EWMA method, GRACH model and implied volatility method to estimate the minimum-variance hedge ratio for S&P 500 index portfolio, hedged using FTSE 100 index. We evaluated the performance of each model economically, by considering the variance of the hedged portfolio. We found that the four methods all reduced the portfolio variance effectively.
The comparison between methods is as follows:

| Rolling window (1 year) | EWMA(0.94) | Reduction | 39.06% | 37.15% |

Rolling window has a better performance than EWMA both under a certain value of parameter and the optimal parameter. It maybe not accord with theory. We think this may be caused by data deviation and contingency. In principal, EWMA is a more mature theory.

| Rolling window(optimal) | EWMA(optimal) | GARCH | Reduction | 39.26% | 39.11% | 35.34% |

We can see from the result above, GARCH model suffer from a number of problems in practice owing to the complexity of their specification. Due to the large number of parameters that must be estimated, it is too complicated for excel to compute. We can not guarantee that maximum likelihood estimation will converge to the correct parameter values. As to implied volatility, it base on the totally different theoretical basis with the three methods. The three methods make predictions on the historical data, while implied volatility depends on the option price that has the function of “discovery of volatility”. It contains the option writers’ expectation of volatility, so we only treat the implied volatility as a reference.

Reference:
Jorion, P., 2000. Value at Risk. 3rd ed. New York: McGraw Hill. Ch.9.
Alexander, C., and Leigh, C.,1997. On the covariance matrices used in Value at Risk. Journal of Derivatives, 4:3(1997), 50-62.
Harris, R.D.F., Stoja, E. and Tucker, J., 2007. A simplified approach to modeling the comovement of asset returns, Journal of Futures Markets. [online] Available at: <http://onlinelibrary.wiley.com/doi/10.1002/fut.20262/pdf> [Accessed 19 April 2013]
Harris, R.D.F., 2013, Volatility forecasting, ECONM2029_2012_13. Bristol University, unpublished.
Benninga, S., 2008. Financial Modeling. 3rd ed. Cambridge: The MIT Press. Ch.19. http://finance.yahoo.com/ https://globalderivatives.nyx.com/

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