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Anthony H. Tu (杜化宇 )

Education:PhD in Finance 1993.7University of Maryland-College Park

Positions:National Chengchi University (政治大學 )Associate professor 1996.8 ~ 2002.7Professor 2002.8 ~ 2012.1

New Huadu Business School (新華都商學院 )Professor 2012.2 ~ now

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Beyond Implementation Costs: Does Fear Expectation Behavior Explain the Index Futures Mispricing?

Wei-Shao Wu and Anthony H. Tu

Newhuadu Business School

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Purpose of this study

This study proposes a way of incorporating arbitragers' fear expectation behavior in arbitrage process in order to explain the substantial and persistent mispricing of S&P 500 index futures and spot.

• Figure 1

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Cost-of-carry model

(1)

where St is the spot index at time t; r is the risk-free interest rate; d is the dividend yield on the stock index portfolio; and T is the expiration date of the futures contract. The rate r-d is often referred to as carrying chargebecause it represents the opportunity cost of carrying the spot asset to maturity of the futures contract.

))((*,

tTdrtTt eSF

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Index futures mispricing , MPt , is defined as deviation of the observed futures price Ft from its theoretical value:

(2)

t

TtTtt S

FFMP

*,,

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No-arbitrage condition can be described as

(3)

where Ct is the time t present value of implementation costs incurred by traders to conduct arbitrage. The implementation costs include transaction costs and any costs due to market frictions. The transaction costs relevant to index arbitrage include round-trip commissions and market-impact costs for trading futures and spot assets.

tt CMP ||

In addition to transaction costs, other market frictions such as asymmetric information, the staleness and liquidity (volume)issues of the underlying spot asset, index-tracking error, taxes, up-tick rule, and short sales restrictions could widen the no-arbitrage band.

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Previous Tests of Index Futures Arbitrages

Mackinlay and Ramaswamy, 1998;

Buhler and Kempf, 1995;

Yadov and Pope, 1990, 1994;

Lafuente and Novales, 2003;

among others.

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Some Explanations on Previous Tests

1. De Long et al. (DSSW) (1990)

2. Abreu and Brunnermeier (AB)(2002)

3. Cao et al. (2011)

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AB(2002) points that there exists risk associated with time taken for arbitrageurs (transaction-lag risk), the no-arbitrage condition can be re-expressed as

(4)

where L>0 is the time lag inherent in the arbitrage process.

Ltt CMP ||

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New Insights in Our Paper

To correctly illustrate the transaction-lag risk, we claim that the no-arbitrage condition (4) has to be revised as

(5)

where E[MPt+L|t] denotes the investors’ expected price movements (spot or futures) at time t+L , conditional on the information available at time t. Its magnitude is determined by investors’ price expectation and risk aversion

]|[|| tLtLtt MPECMP

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The validity of the equation (5) relies heavily on the assumption of homogeneous arbitrageur behavior. However, this assumption is unrealistic with respect to at least three reasons.

First, it is highly likely that each arbitrageur will face different implementation costs and different levels of market frictions.

Second, the assumption of homogeneous arbitrageur behavior implies that there should be a simultaneous reaction to a given mispricing. It is an unrealistic assumption given the likely difference in the trading objectives of arbitrageurs (Kawaller, 1991).

Third, conditional on the same information available at time t, arbitrageurs will have different levels of price expectation at time t+L and different degrees of risk aversion.

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Heterogeneous Arbitrageur Behavior

The no-arbitrage condition has to re-expressed as follows:

(6)

where Ci,t+L is the implementation costs faced by the ith arbitrageur. E[MPi,i+L|t] is the expected price movements faced by the ith arbitrageur at time t+L, conditional on information available at time t.

]|[|| ,, tLtiLtit MPECMP

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Data

1. Our VIX sample period is from January 1990 to April 2013.

2. To match the frequency of VIX data, the daily closed (5883 observations) SP 500 futures prices are obtained from Datastream.

3. To calculate the SP 500 futures mispricing, only data for nearby contracts are used.

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VIX Index

1. Reported by the Chicago Board Options Exchange (CBOE), which is known as the “investor fear gauge” (Whaley, 2000).

2. The advantage of the VIX is that it is forward-looking.

3. It captures the market expectation of future volatility, since it estimates the expected market volatility of the SP 500 over the next 30 calendar days based on the implied volatility in the prices of options on the SP 500.

4. The rise or decline of VIX index (VIX>0 or VIX<0) can be regarded as investors’ fear of exuberance expectation, respectively.

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Literature Related to VIX

Fleming et al. (1995)

Low (2004)

Badshah (2012)∙∙∙∙

(7) The AMP is a function of implementation costs, arbitragers’ fear expectation and other control variables.

•Figure 2•Figure 3(a)•Figure 3(b)

)var,,( iablescontrolVIXCfAMP

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Econometric methods:Quantile Regression Model (QRM)

First, the QRM provides a natural generalization of the OLS model, which is particularly useful in that some of the statistical problems, such as errors in variables, sensitivity to outliers, and non-Gaussian error distribution , can be alleviables, (Barnes and Hughes, 2002). In those problems, the OLS model may not be adequate, while the QRM provides more robust and more efficient estimates.

Second, the QRM is a heterogeneity-consistent method, in which it effectively estimates the changes in all parts of the distribution of a response variable. Taylor (2007) found a time-varying heterogeneous arbitrager behavior underlying the futures-spot mispricing. He argued that it is highly likely that each arbitrager will face different implementation costs and different levels of capital constraint risk. Most importantly, there does not exist a simultaneous reaction to a given mispricing, since arbitragers should have different degree of risk aversion and trading objectives. In the heterogeneous environment, the QRM, which accounts for the whole distribution, can effectively describe the arbitrage behaviors across different quantiles.

Third, the OLS model assumes that the impact of VIX shocks is constant across different quantile levels of mispricing. As a consequence, it would miss important information across quantiles of mispricing and under- or overestimate the impact of VIX innovations, particularly in the context of lower and upper quantiles.

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Hypotheses

Hypothesis 1: contemporaneous (or lagged) VIX innovationis, beyond implementation cost, an important factor that determines the futures-sport mispricing. The stronger VIX shock, the larger AMP is.

• Appendix

• Table 4

• Table5

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Hypothesis 2: The impact of VIX shocks on AMP is more pronounced in the upper quantiles in comparison to lower quantiles.

• Figure 4

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Asymmetric Effect

Prior studies indicated that the price effect of VIX innovations differs as VIX increases (fear) and as VIX declines (exuberance) (Fleming et al. , 1995). We therefore propose that

]0|[]0|[ tLttLt VIXMPEVIXMPE

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Hypothesis 3: The effect of VIX innovations on futures-spot mispricing is asymmetric, the fear expectation (when VIX>0) has a much stronger impact than that of exuberance expectation (when VIX<0)

• Table 6

• Table 7

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Conclusion

1. The contemporaneous VIX innovation is, beyond implementation cost, an important factor that determines the index futures mispricing. The analysis concludes that behavioral incentive in the arbitrage process explain the external and persistence of index futures mispricing.

2. We employ the quantile regression to explore the full distributional impact of VIX innovations, and find that both VIX levels and movements strongly explain (at 1% significance level) the upper quantiles of index futures mispricing.

3. The effect also shows that the stronger VIX shock, the large AMP is.

4. This study proposes a way of incorporating arbitragers' fear expectation behavior in arbitrage process in order to explain the substantial and persistent mispricing of S&P 500 index futures and spot.

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To be continued

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Thank you!