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Transcript of 1 Intertemporal Futures Pricing with Expectation Heterogeneity and Adjustment Effect Simon H. Yen ...
1
Intertemporal Futures Pricing with Expectation Heterogeneity and
Adjustment Effect
Simon H. Yen and Jai Jen WangDepartment of FinanceNational Chengchi University
2
Abstract
Intertemporal futures pricing formulas accounting for expectation heterogeneity, adjustment effect and stochastic interest rate are derived.
Relationships among the 3 factors help to explain empirical results such as Contango or normal backwardation.
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Perfect Substitutes?
Owing to effective arbitrage linkage, a futures
contract and stock index can be viewed as
perfect substitutes.
Much literature does not conclude the
consistent empirical phenomenon for the cost of
carry model.
c y T tf t S t e
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Discrepancy Attributions
Market frictions
Tax timing options
Asymmetric transaction costs
Additional stochastic factors
Stochastic Convenience Yield
Stochastic Interest Rate
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Market Frictions
Tax timing options
futures traders lose tax timing options
Cornell & French (1983), Constantinides (1983), ……
Asymmetric transaction costs
No-arbitrage “band”
Modest & Sunderesan (1983), Klemkosky & Lee (1991), ……
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Stochastic Convenience Yield
Gibson and Schwartz (1990)
Important for pricing financial and real
assets contingent on the price of oil.
Bhatt and Cakici (1990)
Significant positive relationship between
S&P 500 index dividend and mispricing from
the cost of carry model.
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Stochastic Interest Rate
Differentiates futures and forward prices
CIR (1981), Jarrow and Oldfield (1981), Richard and Sundaresan (1981) ……
Cakici and Chatterjee (1991)
Perform better especially when far away from long-term mean
Not sensitive to the exact specification
11
Harrison & Kreps (1978)
Unless traders are all identical and
obliged to hold a stock forever,
speculation would not extinguish, and
heterogeneity in expectations yields
whereby.
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Harris & Raviv (1993)
Traders interpret common information differently
and each of them believes in him- or herself.
Empirical regularities
Absolute price changes and volume are positively
correlated.
Consecutive price changes exhibit negative serial
correlation.
Volume is positively auto-correlated.
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Frankel & Froot (1990)
Standard macroeconomic models can not explain
dollar path, especially from 1984/6 to 1985/2.
Unexpected deviations are so large to be
explained by rational revision such as taste or
technology change.
Wide-dispersed forecasts of participants surveyed
and tremendous trading volume reinforce the idea
of heterogeneous expectations.
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Ederington & Lee (1995)
Volatility remains higher after news releases
than normal times in T-Bond, Eurodollar, and
Deutschmark futures markets.
Such volatility is irrelevant with initial price
change.
It means that disagrees among participants
exist even in filtering common macroeconomic
news.
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Frechette & Weaver (2001)
Reject the representative agent hypothesis in
U.S. soybean futures market at the 95% level of
confidence.
Although the homogeneity assumption has been
maintained in the past to ensure model
tractability, it is incompatible with what we
know to be true about markets.
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Standard REE Models
Traders rationally respond to price changes by revisi
ng their estimates of other traders’ private signals
recursively.
Kyle (1985), Holden & Viswanathan (1992), Foster
& Viswanathan (1993), ……
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MacKinlay & Ramaswamy (1988)
Mispricing increases on average with maturity, because longer term means
Unanticipated variability of dividend payments;
Larger unexpected interest earnings or costs from marking-to-market flows;
More serious and more expensive replicating errors and adjustment costs.
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Yadav & Pope (1994)
Significant arbitrage opportunities after controlling for cash market settlement procedures.
Positive relationships between
Absolute mispricing and time to maturity
Mispricing and index option implied volatility.
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Ahn, Boudoukh, Richardson, and Whitelaw (2002)
Some subset of securities in an index may
partially adjust, or adjust more slowly, to
information because of different transmission
mechanisms or perturbation from noise trading.
Such “partial adjustment” effect imposes
restriction on trading and causes empirical
regularities.
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We take heterogeneity as different opinions on
future evolution of underlying asset price.
Traders are alike in the same group with the
same perspectives about spot price
dynamics, but with heterogeneous
viewpoints among different groups.
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Linear Combination
REE models: equilibrium price has a linear-combinati
on functional form of heterogeneities.
Kyle (1985), Holden & Subrahmanyam (1992), Fost
er & Viswanathan (1996), …
Others: the similar result or setting
Figlewski (1978), Harris & Raviv (1993), Kogan, Ros
s, Wang, & Westerfield (2004), …
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11 1 1 1
12 22 2 2 12 12
2
1
dSdt d z
Sd zdS
dtd zS
1 1 2 2
21 2 12 1 2 12 2
1
1 1 1
dSt t dt
S
t t d z t d z
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Related Variables
Number of investment analysts following Brennan, Jegadeesh, & Swaminathan (1993)
Realized mispricing Figlewski (1978), Ahn, etc. al. (2002)
Firm size Merton (1987) and Lo & Mackinlay (1990)
Time to maturity MacKinlay & Ramaswamy (1988), Yadav & Pope (1994),
Hemler & Longstaff (1991)
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Time Varying ξ(t)
t
0 1 2 1 2 2 2 0
22 21 2 12 2 12
t
1 2 12 1 0
t 22 12 2 0
exp
11 1 1
2
1
1 1
tS S t
t t t dt
t t d z
t d z
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PDE
21 2 2
221 2 12
22
2 12
1 1 2 1 12
22 2 12
1 2 2
1
21
12
1 1
1
1 1
S r r r r r r
S S
S r r r r
r
f S r t f m r f
f S t t
t
f S t t
t
f r t f
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1 21,
r T t
tf S t S e
22
1 2 2 1 2 2
22 2
2 1 2 2
1
2
11 1
2
S S S
S S
f S r f S
f S f r f
PDE & Close-formed Solution
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The cost of carry model is our special case when ξ= 0 or some constant.
Heterogeneity in expectations affects futures pricing through heterogeneous perspectives of dividend yield but not the drift and diffusion terms.
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Comparative Statics
A larger degree of heterogeneity reduces the
futures prices.
1 2
1 2
10
11 0
fT t for
F
fT t for
F
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1 2 2
221 2 12
22 2 2
2 12 1 1
22 1 12 2 2 12
1 2 2
11
21 1
1 12 2
1 1 1
S
r r r S S
S S r r r S r r r
S r r r S r r r
f S r
f m r f S
f S f f S
f S f S
f r f
PDE & Closed form Solution
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1 2 1 2 31m T t L t L t L tf t S t e
21 1 1 2 1 12 2 2 12
23
2
23 1 1 2 1 12 2 2 12
1 1 1
3 424
1 1 1
1
rr r r
r
rr r r
T t
r r
L t T t
L t r t m H t H t H t
L t H t
eH t
m m
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Comparative Statics
1 r S r
S
fH
F
21 12 2 2 1
2 r S r Sr
fH H H
F
1 2
121 1
2 212
2
1 1, 1
, 1
dSfH Cov dr
F S
dSCov dr
S
21 1 2 1 12 2 2 121 1 1S r S r r r
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1 2 1
2 22 2
1 2 12 1 2 12
12 2 2
21 2 12 1 2 12
21 2 1
11
2
1
1
2
t S r r r
S S
S r S r r
r r r
tf f S r f m r
T
t tf S
T T
t tf S
T T
tf r f
T
PDE & Closed form Solution
42
1 2 2 1 2 3
1
2et
m T t T t K t K t K tT Tf t S t
2
2 21 3
21
2 12
12 2
212
3 2
2
3
2 24
2 2
2 2 2
2 1
2 1
r
r r
r r
K t r t m H H H a T t
K t H T t T tT
H H T T t T t T t
H T tK t
T H T t H T t
T
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Numerical Examples
The signs of various results of
comparative statics are dependent on
different combinations of parameters.
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The Homogeneous Scenario
/f S 1 r 2 r 1 2 1 2
0.92 -100% -100% 5% 5% 5% 5%
0.96 -50% -50% 4% 4% 4% 4%
1.00 0% 0% 3% 3% 3% 3%
1.03 50% 50% 2% 2% 2% 2%
1.06 100% 100% 1% 1% 1% 1%
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The Heterogeneous Scenario
/f S 1 r 2 r 1 2 1 2
0.975 0% 0% 6% 5% 6% 5%
0.962 25% -25% 7% 4% 7% 4%
0.951 50% -50% 8% 3% 8% 3%
0.943 75% -75% 9% 2% 9% 2%
0.937 100% -100% 10% 1% 10% 1%
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Heterogeneity reduces the futures price relative to the cost-of-carry model.
Heterogeneity ~ volatility (Frankel and Froot (1990) and Ederington and Lee (1995))
Increased volatility lowers basis (f - S).(Chen, Cuny, and Haugen (1995))