Intertemporal Futures Pricing with Expectation Heterogeneity and Adjustment

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Intertemporal Futures Pricing with Expectation Heterogeneity and Adjustment Effect Simon H. Yen � and

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

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

I Introduction 3

Perfect Substitutes? ¡ Owing to effective arbitrage linkage, a futures contract and stock index

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. 4

Discrepancy Attributions Market frictions ¡ l Tax timing options l Asymmetric transaction costs Additional

Discrepancy Attributions Market frictions ¡ l Tax timing options l Asymmetric transaction costs Additional stochastic factors ¡ l Stochastic Convenience Yield l Stochastic Interest Rate 5

Market Frictions Tax timing options ¡ l futures traders lose tax timing options l

Market Frictions Tax timing options ¡ l futures traders lose tax timing options l Cornell & French (1983), Constantinides (1983), …… Asymmetric transaction costs ¡ l No-arbitrage “band” l Modest & Sunderesan (1983), Klemkosky & Lee (1991), …… 6

Stochastic Convenience Yield Gibson and Schwartz (1990) ¡ l Important for pricing financial and

Stochastic Convenience Yield Gibson and Schwartz (1990) ¡ l Important for pricing financial and real assets contingent on the price of oil. Bhatt and Cakici (1990) ¡ l 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 ¡ l CIR (1981), Jarrow and

Stochastic Interest Rate Differentiates futures and forward prices ¡ l CIR (1981), Jarrow and Oldfield (1981), Richard and Sundaresan (1981) …… Cakici and Chatterjee (1991) ¡ l Perform better especially when far away from long-term mean l Not sensitive to the exact specification 8

This Study 9

This Study 9

Heterogeneity 10

Heterogeneity 10

Harrison & Kreps (1978) ¡ Unless traders are all identical and obliged to hold

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

Harris & Raviv (1993) ¡ Traders interpret common information differently and each of them believes in him- or herself. ¡ Empirical regularities l Absolute price changes and volume are positively correlated. l Consecutive price changes exhibit negative serial correlation. l Volume is positively auto-correlated. 12

Frankel & Froot (1990) ¡ Standard macroeconomic models can not explain dollar path, especially

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

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

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

Adjustment 16

Standard REE Models Traders rationally respond to price changes by ¡ revising their estimates

Standard REE Models Traders rationally respond to price changes by ¡ revising their estimates of other traders’ private signals recursively. l Kyle (1985), Holden & Viswanathan (1992), Foster & Viswanathan (1993), …… 17

Mac. Kinlay & Ramaswamy (1988) Mispricing increases on average with maturity, because longer term

Mac. Kinlay & Ramaswamy (1988) Mispricing increases on average with maturity, because longer term means ¡ l Unanticipated variability of dividend payments; l Larger unexpected interest earnings or costs from marking-to-market flows; l More serious and more expensive replicating errors and adjustment costs. 18

Yadav & Pope (1994) ¡ Significant arbitrage opportunities after controlling for cash market settlement

Yadav & Pope (1994) ¡ Significant arbitrage opportunities after controlling for cash market settlement procedures. ¡ Positive relationships between l Absolute mispricing and time to maturity l Mispricing and index option implied volatility. 19

Ahn, Boudoukh, Richardson, and Whitelaw (2002) ¡ Some subset of securities in an index

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

II Model Specifications 21

Heterogeneity 22

Heterogeneity 22

¡ We take heterogeneity as different opinions on future evolution of underlying asset price.

¡ We take heterogeneity as different opinions on future evolution of underlying asset price. l 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.

Linear Combination ¡ REE models: equilibrium price has a linearcombination functional form of heterogeneities. l Kyle (1985), Holden & Subrahmanyam (1992), Foster & Viswanathan (1996), … ¡ Others: the similar result or setting l Figlewski (1978), Harris & Raviv (1993), Kogan, Ross, Wang, & Westerfield (2004), … 24

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Adjustment 26

Adjustment 26

Related Variables Number of investment analysts following ¡ l Brennan, Jegadeesh, & Swaminathan (1993)

Related Variables Number of investment analysts following ¡ l Brennan, Jegadeesh, & Swaminathan (1993) Realized mispricing ¡ l Figlewski (1978), Ahn, etc. al. (2002) Firm size ¡ l Merton (1987) and Lo & Mackinlay (1990) Time to maturity ¡ l Mac. Kinlay & Ramaswamy (1988), Yadav & Pope (1994), Hemler & Longstaff (1991) 27

Time Varying ξ(t) 28

Time Varying ξ(t) 28

Interest rate Dynamics ¡ Vasicek’s (1977) Ornstein-Uhlenbeck stochastic process: 29

Interest rate Dynamics ¡ Vasicek’s (1977) Ornstein-Uhlenbeck stochastic process: 29

PDE 30

PDE 30

III Closed-form Solutions and Comparative Statics 31

III Closed-form Solutions and Comparative Statics 31

Expectation heterogeneity with constant interest rate and without adjustment effect 32

Expectation heterogeneity with constant interest rate and without adjustment effect 32

PDE & Close-formed Solution 33

PDE & Close-formed Solution 33

¡ The cost of carry model is our special case when ξ= 0 or

¡ 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. 35

Comparative Statics ¡ A larger degree of heterogeneity reduces the futures prices. 35

Expectation heterogeneity with stochastic interest rate and constant adjustment effect 36

Expectation heterogeneity with stochastic interest rate and constant adjustment effect 36

PDE & Closed form Solution 37

PDE & Closed form Solution 37

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Comparative Statics 39

Comparative Statics 39

Expectation heterogeneity with stochastic interest rate and time-varying adjustment effect 40

Expectation heterogeneity with stochastic interest rate and time-varying adjustment effect 40

PDE & Closed form Solution 41

PDE & Closed form Solution 41

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Numerical Examples ¡ The signs of various results of comparative statics are dependent on

Numerical Examples ¡ The signs of various results of comparative statics are dependent on different combinations of parameters. 43

The Homogeneous Scenario 44

The Homogeneous Scenario 44

The Heterogeneous Scenario 45

The Heterogeneous Scenario 45

¡ Heterogeneity reduces the futures price relative to the cost-of-carry model. l Heterogeneity ~

¡ Heterogeneity reduces the futures price relative to the cost-of-carry model. l Heterogeneity ~ volatility (Frankel and Froot (1990) and Ederington and Lee (1995)) l Increased volatility lowers basis (f - S). (Chen, Cuny, and Haugen (1995)) 46

IV Conclusion 47

IV Conclusion 47

¡ Additional components are needed to advance futures pricing models ¡ l Not everybody

¡ Additional components are needed to advance futures pricing models ¡ l Not everybody holds the same perspective l Adjusting behavior happens as time goes by Heterogeneous expectations lowers futures price. And empirical phenomenon depend on the complicated relationships among these factors. 48