Higher Moment Trading Using ARCH
Presentation by Ingo Jungwirth
40789 Nonlinear Time Series
University of Vienna
INTRODUCTIONFirst moment trading
• Common case: trading the mean
0
500
1000
1500
2000
2500
3000
3500
1951 1956 1961 1966 1971 1976 1981 1986 1991 1996 2001 2006 2011 2016
Source: Thomson Reuters, own calculations
INTRODUCTIONBlack and Scholes (1973)
T - t ... time to maturity
S ... spot price of the underlying asset
K ... strike price
r … risk free interest rate
σ² ... volatility in the log-returns of the underlying
INTRODUCTIONOptions
• Po= f [(S-K), (T-t), σ²]
0
2
4
6
8
10
12
14
78 80 82 84 86 88 90
Profit
Loss
Profit
-4
-2
0
2
4
6
8
10
12
74 78 82 86 90 94 98
Profit
Loss
Loss
Profit
Loss
Put Option Call Option
INTRODUCTIONOption-strategy: Straddle
• Ps= f [(S-K), σ²]
The combination of
a straddle and a
future contract with
the same
expiration date and
future price = K
allows trading a
derivative which
only depends on
the underlying‘s σ²
DATAStraddle
3000
3500
4000
4500
5000
5500
6000
6500
7000
7500
Feb-08 May-08 Aug-08 Nov-08 Feb-09 May-09 Aug-09 Nov-09
0
4
8
12
16
20
24
28
32
36
DAX Index Straddle 6600 (r.h.s.)
Source: Thomson Reuters, own calculations
• Trading the variance/kurtosis (higher moment trading)
0
500
1000
1500
2000
2500
3000
3500
1951 1956 1961 1966 1971 1976 1981 1986 1991 1996 2001 2006 2011 2016
INTRODUCTIONHigher moment trading
Source: Thomson Reuters, own calculations
The combination of
option strategies
and
future/foreward
contract with the
same expiration
date allows trading
a derivative which
only depends on
the underlying‘s σ²
DATAVolatility Index
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1992 1994 1996 1998 2000 2002 2004 2006 2008
0
10
20
30
40
50
60
70
80
90
DAX Index VDAX New (r.h.s.)
Source: Thomson Reuters
DATASubstantial leptokurtosis
Source: own calculations
DATASkewed volatility
The news impact
curve* for equity
markets is typically
skewed. Negative
newsflow increases
the volatility
stronger than
positive one.
* Engle and Ng (1993)
Source: Engle and Ng (1993)
ModelFirst learning, then doing
0
10
20
30
40
50
60
70
80
90
1992 1994 1996 1998 2000 2002 2004 2006 2008
Estimation period Evaluation period VDAX New
Source: Thomson Reuters
ModelARCH
• ARCH (5)
DAX=C(1)+C(2)*DAX(-1)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-2)^2 + C(6)*RESID(-3)^2
+ C(7)*RESID(-4)^2 + C(8)*RESID(-5)^2
ModelResults
DAX=C(1)+C(2)*DAX(-1)
GARCH = C(3) + C(4)*RESID(-1)^2 +
C(5)*RESID(-2)^2 + C(6)*RESID(-3)^2 +
C(7)*RESID(-4)^2 + C(8)*RESID(-5)^2
Source: own calculations
ModelEstimated standard deviation
Source: own calculations
ModelBetter in 2008
0
20000
40000
60000
80000
100000
120000
Jan/08 Apr/08 Jul/08 Oct/08 Jan/09 Apr/09 Jul/09 Oct/09
15
30
45
60
75
90
105
ARCH (DAX) VDAX New (r.h.s.)
Source: Thomson Reuters, own calculations
EvaluationThe trading rule
• Evaluation period: 02/01/2008 – 21/12/2009 (515 trading days; daily market close of VDAX New)
• XOR strategy: always invested; either long or short, never neutral
• All in trading: always fully invested
• Long variance if the model forecasts the next day’s variance to increase, short variance if the model forecasts a decrease
• Stay long or short as long as the model tells you to
EvaluationTrading signals
0
10
20
30
40
50
60
70
80
90
Dec/07 Mar/08 Jun/08 Sep/08 Dec/08 Mar/09 Jun/09 Sep/09
Signals VDAX New
Long: 316 days (61% of time)
Short: 200 days (39% of time)
152 trades
1 trade every 3.4
days (≈ one trade a
week)
Source: Thomson Reuters, own calculations
EvaluationPerformance: better in 2008
0
100
200
300
400
500
600
Dec/07 Mar/08 Jun/08 Sep/08 Dec/08 Mar/09 Jun/09 Sep/09
Signals XOR strategy (performance: 01/01/08 = 100)
0
100
200
300
400
500
600
Jan/08 Apr/08 Jul/08 Oct/08 Jan/09 Apr/09 Jul/09 Oct/09
15
30
45
60
75
90
105
XOR strategy (performance: 01/01/08 = 100) VDAX New (r.h.s.)
Source: Thomson Reuters, own calculations
EvaluationToo good to be true?
Sharpe (1966) ratio
2008 2009 Total
Performance (in % p.a.) 202 -14 61
Maximum (in % p.a.) 447 31 447
Minimum (in % p.a.) 0 -30 0
Sharpe ratio 30 -3 10.4
Source: own calculations
ConclusionToo good to neglect
• The ARCH model is able to produce fairly realistic results for implied volatilities of options.
• The applied trading strategy (XOR) based on ARCH(5) generates an impressive performance.
• However, most of this performance is due to the strong increases in volatility during 2008. A simple buy-and-hold would have resulted in a performance of 141% p.a. (XOR: 202% p.a.)
• 2009 XOR fails (-14% p.a.). However, a buy-and-hold would have resulted in a performance of -50% p.a.
• The strategy is biased towards a long-position because the implied volatility of options is heavily skewed.
• There seems to be a structural break in the data.
• More dynamic estimation methods (rolling estimation window) may overcome these shortfalls.
• TARCH and other asymmetric ARCH models promise to be useful. These models adjust for the observed asymmetric shocks to volatility, which could increase the performance during an environment of decreasing volatility.
REFERENCES
Black, Fischer and Myron Scholes (1973). The Pricing of Options and Corporate
Liabilities. Journal of Political Economy, 81, 637-654.
Engle, Robert and Viktor Ng (1993). Measuring and Testing the Impact of News on
Valatility. Journal of Finance, 84, 1022-1082.
Sharpe, William (1966). Mutual Fund Performance. Journal of Business, 119-138.
APPENDIXOption strategies