Higher Moment Trading Using ARCH

Presentation by Ingo Jungwirth

40789 Nonlinear Time Series

University of Vienna

INTRODUCTIONFirst moment trading

• Common case: trading the mean

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500

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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]

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78 80 82 84 86 88 90

Profit

Loss

Profit

-4

-2

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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

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3500

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5500

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6500

7000

7500

Feb-08 May-08 Aug-08 Nov-08 Feb-09 May-09 Aug-09 Nov-09

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36

DAX Index Straddle 6600 (r.h.s.)

Source: Thomson Reuters, own calculations

• Trading the variance/kurtosis (higher moment trading)

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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

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9000

1992 1994 1996 1998 2000 2002 2004 2006 2008

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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

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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

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120000

Jan/08 Apr/08 Jul/08 Oct/08 Jan/09 Apr/09 Jul/09 Oct/09

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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

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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

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Dec/07 Mar/08 Jun/08 Sep/08 Dec/08 Mar/09 Jun/09 Sep/09

Signals XOR strategy (performance: 01/01/08 = 100)

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Jan/08 Apr/08 Jul/08 Oct/08 Jan/09 Apr/09 Jul/09 Oct/09

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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