1. Pricing of Stock Index Futures under Trading Restrictions*
J.Q. Hu
Fudan University
* A joint work with Wenwei Hu, Jun Tong, and Tianxiang Wang

2. Outline Introduction Our Model Main Results Some Empirical Results
Discussions

3. Introduction
Stock Index Futures was first introduced in US in 1982 (SP500 futures)
Since then, stock index futures have been introduced in many markets in different countries, and HS300 index futures was introduced in China in 2010.
In general, stock index futures prices are determined by stock index prices, dividends, and interest rates (e.g., see Hall 2009)

4. Introduction
It has been observed stock index futures prices may deviate from their theoretical values from time to time (and it has also been studied quite extensively).
Our study offers a different perspective.

5. The basis of HS300 stock index

6. Model (based on the work by Robert Jarrow (1980), “Heterogeneous Expectations, Restrictions on Short Sales, and Equilibrium Asset Prices.” The Journal of Finance.)
Consider a market with 𝐾 investors, one risk-free asset (bond), J risky assets (stocks), and stock index futures
Investor 𝑘∈ 1,…𝐾
Asset 𝑗∈ 0,1,…𝐽 (0 is the risk-free asset)
𝜔 𝑗 is the weight of asset j in the stock index (j=1,…,J)

7. Model Two periods (𝑡=1 and 2) 𝑝 𝑗 : the price of asset 𝑗 at 𝑡=1
𝑞: the price of stock index futures 𝑡=1
𝑋 𝑗 : the price of asset 𝑗 at 𝑡=2 (a random variable)
𝜂: the price of stock index futures 𝑡=2
𝑟: the risk free interest rate
𝑛 𝑗 𝑘 : the initial endowment of asset 𝑗 for investor 𝑘
𝑥 𝑗 𝑘 : the position of asset 𝑗 held by investor 𝑘 after rebalancing at the end of period 1 (decision variables)

8. Model Assumptions: All investors don’t hold any futures initially
No transaction costs and taxes will incur
The matrix Σ 𝑘 = 𝜎 𝑖𝑗 𝑘 𝐽×𝐽 , where 𝜎 𝑖𝑗 𝑘 = 𝑐𝑜𝑣 𝑘 ( 𝑋 𝑖 , 𝑋 𝑗 ), is positively definite for all k
Dividends have been embedded in prices
𝑝 0 =1 and 𝑋 0 =1+𝑟
The index futures expires at 𝑡=2, therefore 𝜂= 𝑗=1 𝐽 𝜔 𝑗 𝑋 𝑗 .

9. The model 𝑊 𝑘 (𝑡): the total wealth of investor 𝑘 at 𝑡, we have
𝑊 𝑘 1 = 𝑛 0 𝑘 + 𝑗=1 𝐽 𝑛 𝑗 𝑘 𝑝 𝑗 = 𝑥 0 𝑘 + 𝑗=1 𝐽 𝑥 𝑗 𝑘 𝑝 𝑗 +𝑞 𝑓 𝑘
𝑊 𝑘 2 = 1+𝑟 𝑥 0 𝑘 + 𝑗=1 𝐽 𝑥 𝑗 𝑘 𝑋 𝑗 +𝜂 𝑓 𝑘
= 𝑊 𝑘 𝑋−𝑃 𝑇 𝑥 𝑘 + 𝜔 𝑇 𝑋−𝑞 𝑓 𝑘 +𝑟 𝑥 0 𝑘
𝑤ℎ𝑒𝑟𝑒 𝑋= 𝑋 1 ,…, 𝑋 𝐽 𝑇 , 𝑃= 𝑝 1 ,…, 𝑝 𝐽 𝑇 ,
𝜔= 𝜔 1 ,…, 𝜔 𝐽 , 𝑥 𝑘 =( 𝑥 1 𝑘 ,…, 𝑥 𝐽 𝑘 )

10. Model For investor 𝑘, his utility is given by
𝑈 𝑘 𝑊 𝑘 2 = 𝐸 𝑘 𝑊 𝑘 2 − 𝛼 𝑘 𝑉𝑎𝑟 𝑘 [ 𝑊 𝑘 2 ]
where 𝛼 𝑘 >0 is a constant measuring risk aversion, and 𝐸 𝑘 [∙] and 𝑉𝑎𝑟 𝑘 [∙] are the expectation and variance taken w.r.t the distribution of investor 𝑘’s belief regarding asset payoffs.

11. Model 𝐸 𝑘 𝑊 𝑘 2 = 𝑊 𝑘 1 + 𝜇 𝑘 −𝑃 𝑇 𝑥 𝑘 + 𝜔 𝑇 𝜇 𝑘 −𝑞 𝑓 𝑘 +𝑟 𝑥 0 𝑘
𝐸 𝑘 𝑊 𝑘 2 = 𝑊 𝑘 𝜇 𝑘 −𝑃 𝑇 𝑥 𝑘 + 𝜔 𝑇 𝜇 𝑘 −𝑞 𝑓 𝑘 +𝑟 𝑥 0 𝑘
𝑉𝑎𝑟 𝑘 𝑊 𝑘 2 = 𝑥 𝑘 𝑇 Σ 𝑘 𝑥 𝑘 +2 𝑓 𝑘 𝜔 𝑇 Σ 𝑘 𝑥 𝑘 + 𝜔 𝑇 Σ 𝑘 𝜔 𝑓 𝑘 2
𝑤ℎ𝑒𝑟𝑒 𝜇 𝑘 = 𝐸 𝑘 [ 𝑋 1 ],…, 𝐸 𝑘 [𝑋 𝐽 ] 𝑇

12. The basic problem
In a perfect market (with no trading restriction), the optimal portfolio selection problem for investor 𝑘 is:
(𝑃 𝑀 𝑘 )
𝑚𝑎𝑥 𝑥 0 𝑘 ,…, 𝑥 𝐽 𝑘 , 𝑓 𝑘 𝑈 𝑘 𝑊 𝑘 2
s.t 𝑃 𝑇 𝑥 𝑘 +𝑞 𝑓 𝑘 + 𝑥 0 𝑘 = 𝑃 𝑇 𝑁 𝑘 + 𝑁 0 𝑘
Note: additional constraint can be added on 𝑥 𝑗 𝑘 later.

13. Equilibrium
Definition: A vector (𝑃 ∗ ,𝑞)∈ 𝑅 𝐽+1 is called an equilibrium price of the market if there exist ( 𝑥 0 𝑘 ∗ ,𝑥 𝑘∗ , 𝑓 𝑘 ∗ )∈ 𝑅 𝐽+2 (𝑘=1,⋯,𝐾) such that
(𝑥 𝑘∗ , 𝑓 𝑘 ∗ ) solves the optimization problem 𝑃 𝑀 𝑘 at ( 𝑃 ∗ , 𝑞 ∗ ) for 𝑘=1,⋯,𝐾
𝑘=1 𝐾 𝑥 0 𝑘 ∗ = 𝑁 0 , 𝑘=1 𝐾 𝑥 𝑘 ∗ =𝑁 , and 𝑘=1 𝐾 𝑓 𝑘 ∗ =0,
where 𝑁 𝑗 = 𝑘=1 𝐾 𝑛 𝑗 𝑘 , 𝑁= 𝑁 1 ,⋯, 𝑁 𝐾 𝑇

14. Some related works
If investors are assumed to have homogeneous beliefs (the same expectation and covariance), then it is classical capital asset pricing model (CAPM) (Sharpe 1964, Lintner 1965, Mossin 1966)
It is also assumed that the market is efficient and trading is frictionless
There are some extensions
Mostly focusing on the impact of heterogeneous beliefs and/or short sale constraints on the market equilibrium.
Recently, we have proposed algorithms to calculate equilibrium prices (Tong, Hu and Hu 2017)
Our setting can be very general

15. Main Results 𝑚𝑎𝑥 𝑥 0 𝑘 ,…, 𝑥 𝐽 𝑘 , 𝑓 𝑘 𝐸 𝑘 𝑊 𝑘 2 − 𝛼 𝑘 𝑉𝑎𝑟 𝑘 [ 𝑊 𝑘 2 ]
(𝑃 𝑀 𝑘 )
𝑚𝑎𝑥 𝑥 0 𝑘 ,…, 𝑥 𝐽 𝑘 , 𝑓 𝑘 𝐸 𝑘 𝑊 𝑘 2 − 𝛼 𝑘 𝑉𝑎𝑟 𝑘 [ 𝑊 𝑘 2 ]
s.t 𝑃 𝑇 𝑥 𝑘 +𝑞 𝑓 𝑘 + 𝑥 0 𝑘 = 𝑃 𝑇 𝑁 𝑘 + 𝑁 0 𝑘
𝐸 𝑘 𝑊 𝑘 2 = 𝑊 𝑘 𝜇 𝑘 −𝑃 𝑇 𝑥 𝑘 + 𝜔 𝑇 𝜇 𝑘 −𝑞 𝑓 𝑘 +𝑟 𝑥 0 𝑘
𝑉𝑎𝑟 𝑘 𝑊 𝑘 2 = 𝑥 𝑘 𝑇 Σ 𝑘 𝑥 𝑘 +2 𝑓 𝑘 𝜔 𝑇 Σ 𝑘 𝑥 𝑘 + 𝜔 𝑇 Σ 𝑘 𝜔 𝑓 𝑘 2

16. Main Results
For (𝑃 𝑀 𝑘 ), we have the following necessary conditions for its optimal solution:
𝑥 𝑘 = 1 𝛼 𝑘 Σ 𝑘 −1 𝜇 𝑘 − 1+𝑟 𝑃 − 𝑓 𝑘 𝜔
𝑓 𝑘 = 1 𝛼 𝑘 𝜔 𝑇 𝜇 𝑘 − 1+𝑟 𝑞 𝜔 𝑇 Σ 𝑘 𝜔 − 𝜔 𝑇 Σ 𝑘 𝑥 𝑘 𝜔 𝑇 Σ 𝑘 𝜔
based on which we can obtain: 𝑞= 𝑃 𝑇 𝜔

17. Main Results (with trading restrictions)
(𝑇 𝑅 𝑘 )
𝑚𝑎𝑥 𝑥 0 𝑘 ,…, 𝑥 𝐽 𝑘 , 𝑓 𝑘 𝐸 𝑘 𝑊 𝑘 2 − 𝛼 𝑘 𝑉𝑎𝑟 𝑘 [ 𝑊 𝑘 2 ]
s.t 𝑃 𝑇 𝑥 𝑘 +𝑞 𝑓 𝑘 + 𝑥 0 𝑘 = 𝑃 𝑇 𝑁 𝑘 + 𝑁 0 𝑘
𝐿 𝑘 ≤ 𝑥 𝑘 ≤ 𝑈 𝑘
when 𝐿 𝑘 =0, there is no short selling is allowed.

18. Main Results (with trading restrictions)
For (T 𝑅 𝑘 ), we have the following Lagrangian functions:
𝜇 𝑘 𝑇 𝑥 𝑘 + 𝑤 𝑇 𝜇 𝑘 𝑓 𝑘 − 𝛼 𝑘 𝑥 𝑘 𝑇 Σ 𝑘 𝑥 𝑘 +2 𝑓 𝑘 𝜔 𝑇 Σ 𝑘 𝑥 𝑘 + 𝑓 𝑘 2 𝜔 𝑇 Σ 𝑘 𝜔 + 1+𝑟 𝑥 0 𝑘 + 𝜆 𝑘 𝑃 𝑇 𝑥 𝑘 +𝑞 𝑓 𝑘 + 𝑥 0 𝑘 − 𝑃 𝑇 𝑁 𝑘 − 𝑁 0 𝑘 + 𝜃 𝑘 𝑇 𝑥 𝑘 − 𝐿 𝑘 + 𝜉 𝑘 𝑇 𝑈 𝑘 − 𝑥 𝑘
where 𝜆 𝑘 ∈𝑅, 𝜉 𝑘 ≥0, 𝜃 𝑘 ≥0 are Lagrangian multipliers

19. Main Results (with trading restrictions)
For (𝑇 𝑅 𝑘 ), we have the following necessary conditions for its optimal solution:
𝑥 𝑘 = 1 𝛼 𝑘 Σ 𝑘 −1 𝜇 𝑘 − 1+𝑟 𝑃+ 𝜃 𝑘 − 𝜉 𝑘 − 𝑓 𝑘 𝜔
𝑓 𝑘 = 1 𝛼 𝑘 𝜔 𝑇 𝜇 𝑘 − 1+𝑟 𝑞 𝜔 𝑇 Σ 𝑘 𝜔 − 𝜔 𝑇 Σ 𝑘 𝑥 𝑘 𝜔 𝑇 Σ 𝑘 𝜔
𝜃 𝑗 𝑘 𝑥 𝑗 𝑘 − 𝐿 𝑗 𝑘 = 𝜉 𝑗 𝑘 𝑈 𝑗 𝑘 − 𝑥 𝑗 𝑘 =0
based on which we can obtain: 𝑞= 𝑃 𝑇 𝜔− 1 1+𝑟 𝜃 𝑘 − 𝜉 𝑘 𝑇 𝜔

20. Main Results (with trading restrictions)
Hence, in general, we have 𝑞≠ 𝑃 𝑇 𝜔. In particular, if 𝑈 𝑘 =∞, then 𝜉 𝑘 =0, we have 𝑞≤ 𝑃 𝑇 𝜔

21. Main Results (with margin requirement)
(𝑀 𝑅 𝑘 )
𝑚𝑎𝑥 𝑥 0 𝑘 ,…, 𝑥 𝐽 𝑘 , 𝑓 𝑘 𝐸 𝑘 𝑊 𝑘 2 − 𝛼 𝑘 𝑉𝑎𝑟 𝑘 [ 𝑊 𝑘 2 ]
s.t 𝑃 𝑇 𝑥 𝑘 +𝑚𝑞 𝑓 𝑘 + 𝑥 0 𝑘 = 𝑃 𝑇 𝑁 𝑘 + 𝑁 0 𝑘
𝐿 𝑘 ≤ 𝑥 𝑘 ≤ 𝑈 𝑘
where 0<𝑚≤1 is the margin requirement for trading stock index futures, i.e., if an investor trades (either longs or shorts) one unit of stock index futures, then his margin requirement is m units of cash.

22. Main Results (with margin requirement)
For (M 𝑅 𝑘 ), we have the following Lagrangian functions:
𝜇 𝑘 −𝑃 𝑇 𝑥 𝑘 + 𝑤 𝑇 𝜇 𝑘 −𝑞 𝑓 𝑘 − 𝛼 𝑘 𝑥 𝑘 𝑇 Σ 𝑘 𝑥 𝑘 +2 𝑓 𝑘 𝜔 𝑇 Σ 𝑘 𝑥 𝑘 + 𝑓 𝑘 2 𝜔 𝑇 Σ 𝑘 𝜔 +𝑟 𝑥 0 𝑘 + 𝜆 𝑘 𝑃 𝑇 𝑥 𝑘 +𝑚𝑞 |𝑓 𝑘 |+ 𝑥 0 𝑘 − 𝑃 𝑇 𝑁 𝑘 − 𝑁 0 𝑘 + 𝜃 𝑘 𝑇 𝑥 𝑘 − 𝐿 𝑘 + 𝜉 𝑘 𝑇 𝑈 𝑘 − 𝑥 𝑘
where 𝜆 𝑘 ∈𝑅, 𝜉 𝑘 ≥0, 𝜃 𝑘 ≥0 are Lagrangian multipliers

23. Main Results (with margin requirement)
For (M 𝑅 𝑘 ), we have the following necessary conditions for its optimal solution:
𝑥 𝑘 = 1 𝛼 𝑘 Σ 𝑘 −1 𝜇 𝑘 − 1+𝑟 𝑃+ 𝜃 𝑘 − 𝜉 𝑘 − 𝑓 𝑘 𝜔
𝑓 𝑘 = 1 𝛼 𝑘 𝜔 𝑇 𝜇 𝑘 − 1+𝑟𝑚 𝑣 𝑘 𝑞 𝜔 𝑇 Σ 𝑘 𝜔 − 𝜔 𝑇 Σ 𝑘 𝑥 𝑘 𝜔 𝑇 Σ 𝑘 𝜔
𝜃 𝑗 𝑘 𝑥 𝑗 𝑘 − 𝐿 𝑗 𝑘 = 𝜉 𝑗 𝑘 𝑈 𝑗 𝑘 − 𝑥 𝑗 𝑘 =0
We then have: 𝑞= 𝑃 𝑇 𝜔+ 𝑟 1−𝑚 𝑣 𝑘 𝑃− 𝜃 𝑘 − 𝜉 𝑘 𝑇 𝜔 1+𝑟𝑚 𝑣 𝑘

24. Some Empirical Results
HS300 Index

25. Some Empirical Results
SZ50 Index

26. Some Empirical Results
ZZ500 Index

27. Some Empirical Results
basi s 𝑡 =𝛼 basi s 𝑡−1 +𝛽volatilit y 𝑡 +𝛾
Para\index
SZ50
HS300
ZZ500 𝛼
0.6135***
0.7055***
0.6315*** 𝛽
***
***
*** 𝛶
0.0012*** *
Significant codes: ***p<0.001, **p<0.01, *p<0.05, p<0.1

28. Some Empirical Results
Margin requirements for SZ50 and HS300
Times
m（%）
2015/4/ /8/25 10
2015/8/ /8/31 20
2015/9/1-2015/9/6 30
2015/9/7-2017/2/16 40
2017/2/ /4/13

29. Some Empirical Results
basi s 𝑡 =𝛼∗basi s 𝑡−1 +𝛽∗volatilit y 𝑡 +𝛾+𝑘∗ Dummy t
as m increases from 10% to 40% …
Para\index
SZ50
HS300 𝛼
0.5683***
0.6031*** 𝛽
***
*** 𝛶
0.0034*
0.0016 k ** *

30. Some Empirical Results
basi s 𝑡 =𝛼∗basi s 𝑡−1 +𝛽∗volatilit y 𝑡 +𝛾+𝑘∗ Dummy t
as m decreases from 40% to 20% …
Para\index
SZ50
HS300 𝛼
0.3716***
0.5098*** 𝛽
***
*** 𝛶
***
*** k
0.0015*
0.0018*

31. Some Empirical Results
KOSPI 200

32. Some Empirical Results
basi s 𝑡 =𝛼 basi s 𝑡−1 +𝛽volatilit y 𝑡 +𝛾
Para\index
KOSPI 200 𝛼
0.6044*** 𝛽 * 𝛶
0.0010***

33. Discussions Chinese markets are highly regulated and inefficient
Very difficult to short stocks
T+1
Government interventions …
Stock index futures prices …

34. Thank you!