Anthony H. Tu (杜化宇)
Education:
PhD in Finance 1993.7
University of Maryland-College Park
Positions:
National Chengchi University (政治大學)
Associate professor 1996.8 ~ 2002.7
Professor 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
2
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
Ft*,T  St e ( r d )(T t )
(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.
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Index futures mispricing , MPt , is defined as
deviation of the observed futures price Ft from
its theoretical value:
MPt 
Ft ,T  Ft*,T
(2)
St
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No-arbitrage condition can be described as
| MPt | Ct
(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.
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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.
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
| MPt | Ct  L
(4)
where L>0 is the time lag inherent in the arbitrage
process.
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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
| MPt | Ct  L  E[MPt  L | t ]
(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
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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.
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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.
Heterogeneous Arbitrageur Behavior
The no-arbitrage condition has to re-expressed as
follows:
| MPt | Ci ,t  L  E[MPi ,t  L | t ]
(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.
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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) SP 500 futures prices
are obtained from Datastream.
3. To calculate the SP 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
SP 500 over the next 30 calendar days based on the
implied volatility in the prices of options on the SP 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)
∙
∙
∙
∙
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AMP  f (C ,VIX , control var iables )
(7)
The AMP is a function of implementation costs,
arbitragers’ fear expectation and other control variables.
•Figure 2
•Figure 3(a)
•Figure 3(b)
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.
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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.
Hypotheses
Hypothesis 1: contemporaneous (or lagged)
VIX innovationis, beyond implementation cost,
an important factor that determines the futuressport 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
E[MPt  L | VIX t  0]  E[MPt  L | VIX t  0]
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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.
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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.
To be continued 
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Thank you!
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