Aktienindex und Volatilität

Transcription

Aktienindex und Volatilität
Martin Wallmeier
Universität Freiburg / Schweiz
Dahlem Lectures on FACTS
an der Freie Universität Berlin
am 10. Januar 2007
FACTS: Aktienindex und Volatilität
1
Gliederung
1. Volatilität als Risikomaß
2. Typische Verlaufsmuster der Volatilität
3. Volatilität als Handelsobjekt
4. Höhe der Risikoprämie für Volatilität
5. Volatilität als Anlageobjekt
6. Ergebnisse
2
Volatilität als Risikomaß: tägl. Log-Renditen 2005-2006
3
Tägliche Log-Rendite Allianz
-0.05
09.09.2006
09.05.2006
09.01.2006
09.09.2005
09.05.2005
09.01.2005
09.09.2004
09.05.2004
09.01.2004
09.09.2003
09.05.2003
09.01.2003
09.09.2002
09.05.2002
09.01.2002
09.09.2001
09.05.2001
09.01.2001
09.09.2000
09.05.2000
09.01.2000
09.09.1999
09.05.1999
09.01.1999
09.09.1998
09.05.1998
09.01.1998
09.09.1997
09.05.1997
09.01.1997
Clustering: Rendite Allianz-Aktie 1997-2006
0.15
0.1
0.05
0
-0.1
-0.15
-0.2
Datum
4
DAX und VDAX am 24. Juli 2006
DAX
VDAX
5
Kurs- und Volatilitätssprünge: 3. September 1999
DAX quotations on September 03, 1999
5340
5320
5300
5280
5260
5240
5220
5200
9
10
11
12
13
14
15
16
17
15
16
17
Implied ATM Volatility of DAX option with time-to-maturity of 14 days
0.235
0.23
0.225
0.22
0.215
0.21
0.205
0.2
0.195
0.19
9
10
11
12
13
14
6
Contemporaneous correlations 2004
for different sampling frequencies
7
5-minute cross-correlations 1995-2004
Corr(Log Return Implied ATM Volatility(t) , Log Return DAX(t-j))
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
-50
Implied ATM Volatility
-40
-30
-20
-10
0
10
20
30
40
50
lag j
Corr(Absolute Log Return DAX(t) , Log Return DAX(t-j))
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
-50
Absolute Log Return DAX
-40
-30
-20
-10
0
10
20
30
40
50
lag j
8
Gliederung
6. Ergebnisse
9
Volatility as an asset class
10
Varianzswap
Termingeschäft, bei dem die Vertragsparteien den
Austausch folgender Zahlungen vereinbaren:
• Käufer:
bezahlt den bei Vertragsabschluss vereinbarten Preis
• Verkäufer: bezahlt einen Betrag in Höhe der während der
Vertragslaufzeit realisierten annualisierten Varianz der
Log-Renditen des Basispapiers (hier: DAX)
Bei Vertragsabschluss wird der Preis so festgelegt, dass der faire Wert
des Geschäfts Null beträgt.
11
Variance swaps: replication
Variance Exposure as a function of stock price
Source:
Demeterfi/Derman/Kamal/Zhou
K VARS 
2
T
e 
rT
 e 
2
T
rT
F 0 T 1
0
K2
P BS K, T, 0 K, TdK

C BS K, T, 0 K, TdK
1
F 0 T K 2
12
Previous papers
On variance swaps:
On the variance risk premium:
13
Set-up of the empirical study
1. Estimate the strike price structure of option prices (smile).
2. Calculate the fair value of variance swaps.
3. Measure realized (ex post) variance during the time to maturity.
4. Calculate the profit or loss for the buyer of a variance swap as the
difference between realized variance and KVARS.
5. Initiate such a variance swap trade on each trading day over the
10 year period from January 1995 to December 2004.
14
Strike price structure of option prices
0.26
0.24
where M is time to maturity-adjusted moneyness
and D a dummy variable for M>0.
0.18
0.20
0.22
   0  1 M  2M 2  3 D  M3  
0.14
0.16
Implied volatility
DAX option on Dec. 10, 2004 with time to maturity of 42 days
0.8
0.9
1.0
1.1
1.2
Strike price
/ Index level
Moneyness
15
Effect of the volatility skew on variance swap rates
 imp  b 0  b 1 M  b 2 M 2  
M
X/F−1
T
K VARS ≈  2  3b 21  2  2b 2  3
2
: Variance swap price when smile is flat at level
b0
16
Variance swap rates over time
0.4
Russian crisis
Financial crisis
DAX
ESX
Iraq war
0.2
Asian crisis
0.0
0.1
Variance swap rate
0.3
WTC attack
1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
17
20
40
60
80
DAX
ESX
0
Spread Swap rate over ATM implied variance (in %)
100
Spread swap rate over ATM implied variance
1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
18
1.0
Histograms of log returns variance swap DAX and ESX
1.0
1.0
1.0
0.8
0.8
0.8
0.8
0.6
0.6
0.6
Density
Density
0.6
0.4
0.4
0.4
0.2
0.2
0.0
0.0
0.0
0.0
0.2
0.2
0.4
-1.5
-1.0
-0.5
0.0
0.5
1.0
Log return variance swap D AX
Log return
variance swap DAX
1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
Log return variance swap ESX
Log return
variance swap ESX
19
0.5
0.0
-0.5
-1.0
-1.5
Log return variance swap ESX
1.0
Log returns var swap DAX versus ESX 2000 to 2004
-1.5
-1.0
-0.5
0.0
0.5
1.0
Log return variance swap DAX
20
0.0
-0.5
-1.5
-1.0
Log return
variance
DAX
Log return
varianceswap
swap DAX
1.0
0.5
0.0
-0.5
-1.0
Log return
variance
DAX
Log return
varianceswap
swap DAX
0.5
1.5
Relationship with index returns
-0.4
-0.3
-0.2
Logreturn
return ODAX
Log
DAX
-0.1
0.0
0.00
0.05
0.10
0.15
0.20
0.25
Log return
ODAX
Log
return
DAX
21
Equilibrium analysis of variance swap returns
Leland modification of CAPM:
 L,i  ER ei  − ER eM L,i
 L,i 
covRi ,−1RM  − 
covRM ,−1RM  − 
22
Gliederung
6. Ergebnisse
23
Mean variance analysis (DAX 1995-2004)
Return volatility
0.14
H3L
H1L
H2L
0.12
0.1
Var-Swap
DAX
0.08
H4L
0.06
0.04
H5L
0.02
Expected excess return
- 0.02
-0.01
0.01
0.02
24
Characteristics of mean-variance efficient portfolios
25
Optimal portfolio weights under power utility
26
Polynomial Goal Programming
Method to explicitly consider higher moments in portfolio optimization.
Definition of sharpe ratio (SR), skewness (SK) and kurtosis (KT):
SR 
ER

; SK 
E R−ER 3
3
; KT 
E R−ER 4
4
27
Polynomial Goal Programming
Maximiere
Zx VARS , x S   1  d SRx VARS , x S  
 1  d SK x VARS , x S  
 1  d KT x VARS , x S   
unter den Nebenbedingungen
SR max − SRx VARS , x S 
,
d SR x VARS , x S  
max
SR
SK max − SKx VARS , x S 
d SK x VARS , x S  
,
max
SK
KTx VARS , x S  − KT min
.
d KT x VARS , x S  
min
KT
28
Ergebnis der Optimierung
KT min
xS
a)
b)
SR max
1 − d SR
10
SK max
1
1 − d SK
0.5
5
SK max
-10
-5
5
SR max
10
x VARS
-10
-5
5
10
x VARS
-5
-0.5
-10
-1
KT min
c)
Zx VARS , x S 
d)
10
Setting: , , 
 1, , 0
5.7
x VARS
5.6
5
5.5
0.5
-5
1
1.5
2

5.4
5.3
xS
5.2
5.1
-10
-10
e)
Setting: , , 
 1, 1. 33, 0
1 − d SR
SR max
f)
1
T
5
10
10
Setting: , , 
 1, 1. 33, 
5
0.5
x VARS
-5

1 − d SK
0.2
-1
-0.5
0.5
0.4
0.6
0.8
1
1
-5
-0.5
-10
-1
SK max
x VARS
xS
29
Conclusion
¾ The profile of log variance swap returns against log index returns on average
resembles the payoff of a long put position. Æ Crash protection.
¾ Due to the option-like profile of returns it is crucial to account for the
non-normality of returns in measuring the performance of variance swap
investments.
¾ The volatility risk premium at the German as well as the European stock
market is strongly negative.
¾ Its magnitude is not compatible with standard equilibrium pricing models.
¾ Our backtests result in significant short volatility positions in optimal portfolios
during the sample period. Typically, the stock index weight is also negative,
since the diversification gain exceeds the loss in expected return.
30
Conclusion
¾ The objectives of a high Sharpe ratio, high skewness and low kurtosis are
highly conflicting. In particular, the short positions required to profit from
negative variance swap returns inevitably lead to an undesired negative
skewness.
¾ A balanced tradeoff between Sharpe ratio and skewness often does not exist.
Investors tend to the extreme portfolios (Sharpe ratio driven, skewness driven
or kurtosis driven) and avoid to be stuck in the middle.
¾ This ‘all-or-nothing' characteristic is reflected in jumps of asset weights when
certain thresholds of preference parameters are crossed.
31

Aktienindex und Volatilität

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