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Time Series technical analysis via new fast estimation
methods
Time Series technical analysis via new fast estimation
methods
Yan Jungang A0075380EHuang Zhaokun A0075386UBai Ning A0075461E
Yan Jungang A0075380EHuang Zhaokun A0075386UBai Ning A0075461E
Fundamental Approach based on a wide range of data regarded as
fundamental economic variables that determine exchange rates
Forecast foreign exchange rates
Steps of fundamental approach1. starts with a model
2. collects data to estimate the forecasting equation
3. generation of forecasts
4. evaluation of the forecast
Fundamental Approach
Trading Signal 1. significant difference between the expected
foreign exchange rate and the actual rate
2. a mispricing or a heightened risk premium
3. a buy or sell signal is generated
Fundamental Approach
Technical Approach does not rely on a fundamental analysis of
the underlying economic determinants of exchange rates or asset prices, but only on extrapolations of past price trends
Forecast foreign exchange rates
Steps of technical approach1. recognize the type of trend the market is
2. a level of support
3. form trend lines
Technical Approach
in-sample: works within the sample at hand
out-of-sample works outside the sample
Two kinds of forecasts:
where is the trendline which satisfies the above linear equation
is the mismatch between the real data and the trendline
Linear difference equations
1( ) ( 1) ( ) 0 (1)
( ) ( ) ( ) (2)n
trendline
x t n a x t n a x t
x t x t t
( )trendlinex t
( )t
Thus
we only assume that
Linear difference equations
1
1
( ) ( 1) ( ) ( ) (3)
( ) ( ) ( 1) ( ) (4)n
n
x t n a x t n a x t t
t t n a t n a t
,
(0) (1) ( )lim 0
1,
( ) ( 1) ( )( )
10 large enough
N
N
v N
Nt N the moving average
t v t t NMA t
Nis close to if N is
From equation (4) that also satisfies (5) and (6).
Hence, the finite linear combinations of i.i.d. zero-mean process, do satisfy almost surely such a weak assumption.
Þ Our analysis Does not make any difference between non-
stationary and stationary time series
Linear difference equations
( )t
Consider again Equation (1). The Z-transform X of x satisfies
where
Rational generating functions
1 ( 1)
1
[ (0) (1) ( 1) ]
[ (0)] 0 (7)
n n
n n
z X x x z x n z
a z X x a X
1 20 1 1
1
( )
( )
( ), ( ) ( )
n nn
nn n
b z b z b P zX
z a z a Q z
P z Q z R z
We introduce the Wronskian matrix
Parameter identifiability
1 2
1 21
( 1) ( 1) ( 1
1
)1 2
1
For real or complex valued functions , ..., , which are
1
, ..., as a function on is def
( ) ( ) ( )
( ) ( ) ( )( , , )( ) , .
( ) ( )
ined b
( )
y
n
nn
n n nn
n
n
f x f x f x
f x f x f xW f f x x I
f x
n f f
n
W f
f
f
f
I
x x
times differentiable on an interval I, the Wronskian
The unknown linearly identificable parameters can be solved by the matrix linear equation
Parameter identifiability
1 1 2
1 1 2
2 2 1 2 2 1 2 2 2
2 2 2 2 2 2
1
( ) ( ) ( ) ( ) (1)
( , ,1) 0
( ) ( ) ( ) ( ) (1)
n n n n
n n n n
n
n n n n n n n n n n
n n n n n n
z X z X X z z
d z X d z X dX d z d z d
dz dz dz dz dz dz
f z X
d z X d z X d X d z d z d
dz dz dz dz dz dz
Sample data:
US Dollar – €uros
Time interval:
1999-01-04 to 2011-03-11
The data can be downloaded from here:
http://www.ecb.int/stats/exchange/eurofxref/html/index.en.html
Data Analysis
Data Analysis
volatility clusters volatility evolves over time in a continuous
manner volatility varies within some fixed range leverage effect
Data Analysis
Stylized-facts of financial return series
The {rt2}
is highly correlated
The changes in { rt } tend to be clustered.
{ rt } is heavy tailed
stylized-facts of financial
return series
Data Analysis: Correlation
{Xt}{Xt
2}
The log returns are independent
The square of log returns are highly correlated
.
H0:H1: for some
.
Example:US Dollar/Euros Exchange Rate
Estimate Std. Error t-value Pr( >|t| )
μ 7.365e-05 4.584e-05 1.607 0.1081
α0 3.136e-08 1.334e-08 2.350 0.0188
α1 3.147e-02 4.240e-03 7.421 1.16e-13
β1 9.651e-01 4.583e-03 210.588 <2e-16
.
Estimate of
Std. Errors are based on Hessian. Significance at 1, 5, 10 percent are indicated by (***), (**), (*).
Example:US Dollar/Euros Exchange Rate
Residuals Tests
The Ljung-Box Test are performed for standardized residuals and squared standardized residuals respectively
Trading strategy
Simulate data (ACF)
ACF of historical return ACF of simulated return
Green:| rt | Red: rt ^2 Blue: rt
Trading strategy
Take the 10-day historical volatility (HV) reading.
Take the 50-day historical volatility (HV) reading.
If (VAR(10) < 0.5*VAR(50)) Display(“a big move is likely near!”)
Trading strategy
Using historical data to test:
If(VAR(+n)>VAR(-10))
The strategy is efficient
VAR(-10):volatility between trading signal and10 days before the trading signal VAR(+n):volatility between trading signal and n days after the trading signal
Trading strategy
n VAR(+n) > VAR(-10)
1 0.4194
2 0.5161
3 0.6419
4 0.7161
5 0.7548
6 0.7871
7 0.8161
8 0.8419
9 0.8484
10 0.8581
Trading strategy
Make an assumption:
when we face the trading signal : exchange our US dollar to Euro dollar at current time t. exchange Euro dollar back to US dollar n days after.
By using the historical 3024 days’ exchange rate data,
the program gave us 310 trading signals.
Trading strategy n Probability of making profit
1 0.4645
2 0.4645
3 0.4484
4 0.4677
5 0.4806
6 0.4806
7 0.5065
8 0.4968
9 0.5161
10 0.4935
11 0.5129
12 0.5290
13 0.5484
14 0.5290
15 0.5355
Trading strategy
Property of GARCH:
Large shocks tend to be followed by another large shock;
Small shocks tend to be followed by another small shock.
Trading signal
Market will volatile in next days
Trading strategy
Problems we faced by now
Know some significance moves are about to take place.
Not sure what the direction will turn out to be.
using STRADDLE or STRANGLE
Trading strategy
Straddle: purchase the same number of call and put options at
the same strike price with the same expiration date.
Strangle: purchase the same number of call and put options at
different strike prices with the same expiration date.
Trading strategy
Steps of trading:Straddle : Buy an ATM (At-The-Money) Put Buy an ATM CallStrangle: Buy an OTM (Out-The-Money) Put Buy an OTM Call
Trading strategy
Risk and Reward : The maximum risk of a Straddle/ Strangle is equal to the
amount that you paid for the two option contracts. If the stock moves nowhere, and volatility drops to nothing, you lose.
The reward is that same as for calls and puts - unlimited.
Trading strategy
Tricks to buy the straddle/ strangle
Buy options while volatility is relatively slow Sell as volatility increase either just before a news report or
soon after.