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.

3

I Introduction

4

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

5

Discrepancy Attributions

Market frictions

Tax timing options

Asymmetric transaction costs

Additional stochastic factors

Stochastic Convenience Yield

Stochastic Interest Rate

6

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), ……

7

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.

8

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

9

This Study

10

Heterogeneity

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.

12

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.

13

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.

15

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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Adjustment

17

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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II Model Specifications

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Heterogeneity

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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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Adjustment

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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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Interest rate Dynamics

Vasicek’s (1977) Ornstein-Uhlenbeck stochastic

process:

r rdr m r dt 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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III Closed-form Solutions and Comparative Statics

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Expectation heterogeneity withconstant interest rateand without adjustment effect

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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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Expectation heterogeneitywith stochastic interest rateand constant adjustment effect

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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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Expectation heterogeneity with stochastic interest rateand time-varying adjustment effect

T tt

T

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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

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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))

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IV Conclusion

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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.