Intertemporal Futures Pricing with
Expectation Heterogeneity and
Adjustment Effect
Simon H. Yen
and Jai Jen Wang
Department of Finance
National Chengchi University
1
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.
2
I Introduction
3
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.
f t   S t  e 
c  y  T  t 
4
Discrepancy Attributions
Market frictions


Tax timing options

Asymmetric transaction costs
Additional stochastic factors


Stochastic Convenience Yield

Stochastic Interest Rate
5
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), ……
6
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.
7
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
8
This Study
9
Heterogeneity
10
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.
11
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.
12
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.
13
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.
14
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.
15
Adjustment
16
Standard REE Models
Traders rationally respond to price changes by

revising their estimates of other traders’ private
signals recursively.

Kyle (1985), Holden & Viswanathan (1992),
Foster & Viswanathan (1993), ……
17
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.
18
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.
19
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.
20
II Model Specifications
21
Heterogeneity
22

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.
23
Linear Combination

REE models: equilibrium price has a linearcombination functional form of heterogeneities.

Kyle (1985), Holden & Subrahmanyam (1992),
Foster & Viswanathan (1996), …

Others: the similar result or setting

Figlewski (1978), Harris & Raviv (1993),
Kogan, Ross, Wang, & Westerfield (2004), …
24
d S1
S
d S2
   1   1  dt   1 d z 1
   2   2  dt   2   12

S
1 
2
12
d z1 


  d z 2 
dS
   t    1   1   1    t     2   2   dt
S
    t   1  1    t    2  12 d z 1  1    t    2



1   122 d z 2 

25
Adjustment
26
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)
27
Time Varying ξ(t)


S t  S 0 exp 



t
0



  t   
1
  2    1   2      2   2 
2
1
    t   1  1    t    2  12   1    t    2

2


t
0
t
0

 
1      dt
  
2
12
2
  t   1  1    t    2  12  d z 1
1    t   
2

1   122  d z 2






28
Interest rate Dynamics

Vasicek’s (1977) Ornstein-Uhlenbeck stochastic
process:
d r    m  r  dt   r d z r
29
PDE
1
f S S  r    t   1   2    2   f r   m  r    r  r   f r r  r2
2

1
fSS S 2
2
 f Sr S  r

  t   1  1    t    2  12 
   
t
 1    t    2

1
2
1  

2
12
2



 1r  1    t    2  1r  12
 1    t    2  2 r
1   122

 f    r    t   1   2    2  f
30
III
Closed-form Solutions
and Comparative Statics
31
Expectation heterogeneity with
constant interest rate
and without adjustment effect
32
PDE & Close-formed Solution
fS S


2
1
2
r    1   2    2   f S S S   1   2     2  
2
1
 f S S S 2 1     2

2

1     f   r    1   2    2 

f  S, t   S t e
2
2
f
r   1 1  2  T t 


33

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.
34
Comparative Statics

A larger degree of heterogeneity reduces the
futures prices.






1 f
  T  t   0
F  
for  1   2
1 f
  1    T  t   0
F  
for  1   2
35
Expectation heterogeneity
with stochastic interest rate
and constant adjustment effect
36
PDE & Closed form Solution
fS S
 r   
1
  2    2 

2
1
2
 f r   m  r    r  r   f S S S   1  1     2  12 
2
2
1
1
2 
2 
 f S S S 1     2 1   12  f r r  r2  f S r  S  1  r  1r


2
2
 f S r 1    S  2  r  1r  12  f S r 1    S  2  r  2 r

 f   r    1   2    2 
f
1   122
37
f t   S t  e
m

   1  1   2 
 T  t   L  t   L
1
2
 t  L 3 t 
r 
L1  t  
  1  1r  1     2  1r  12  1     2  2 r
 
L 2  t    r  t   m   H  t  
 r2
3  H  t   4   H  t  
24 
3
r 
L3  t   
  1  1r  1     2  1r  12  1     2  2 r
 
H t  
1 e
m  m 

1   122  T  t 

1   122  H  t 

 T t 

 r r

38
Comparative Statics
 r Sr
1 f

  H 
F  S

1 f
1
 2  2 H  H 2   r  2 1  H   S r  S 


F  r 2 


d S1  
2
1 f
1
      H      Cov  d r ,
   1   12

F 

S1  
1



dS2
Cov  d r ,

S2


 


1 
2
12







 S r  S    1  1r  1     2  1r  12  1     2  2 r
1   122
39
Expectation heterogeneity
with stochastic interest rate
and time-varying adjustment effect
T t
 t  
T
40
PDE & Closed form Solution

t


f t  f S S  r   1   2   1     f r   m  r    r  r 
T



2

1
t
 t
2  
 f S S S   1   2  12   1      2
2
T
 T
 
 f Sr Sr


S r 

2

1  

t

 t
 1  T  2  12   1     T  2
2
12
2




1  

2
12
2



1
2
1

t


f r r  r2   r   1   2   1    f
2
T



41
f t   S t  e
t 
   1
 1  2  T  t    2    T  t   K 1  t   K 2  t   K 3  t 
m  
T 
 2T



K 1  t    r  t   m  H 
 r2
2
2

2
H


H

 2 a T  t  
3 
4
 r  1r 
2

K 2 t   2
  1  2 H  2 T  t    T  t  
 T 

  12  2  2 H  2 H T  2 T  t    T  t T  t   

K 3 t    r  2r

 2



1   122   H  T  t  


 2  2 1   T  H   T  t   H  T  t   


3
 T


42
Numerical Examples

The signs of various results of
comparative statics are dependent on
different combinations of parameters.
43
The Homogeneous Scenario
f /S
1 r
2r
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%
44
The Heterogeneous Scenario
f /S
1 r
2r
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%
45

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))
46
IV Conclusion
47

Additional components are needed to advance futures
pricing models


Not everybody holds the same perspective

Adjusting behavior happens as time goes by
Heterogeneous expectations lowers futures price .
And empirical phenomenon depend on the
complicated relationships among these factors.
48