Pricing of Stock Index Futures
under Trading Restrictions*
J.Q. Hu
Fudan University
* A joint work with Wenwei Hu, Jun Tong, and Tianxiang Wang
Outline





Introduction
Our Model
Main Results
Some Empirical Results
Discussions
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)
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.
The basis of HS300 stock index
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)
𝜔𝑗 is the weight of asset j in the stock index (j=1,…,J)
Model
Two periods (𝑡 = 1 and 2)

𝑝𝑗 : the price of asset 𝑗 at 𝑡 = 1

𝑞: the price of stock index futures 𝑡 = 1
𝑋𝑗 : 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

𝑛𝑗𝑘 : the initial endowment of asset 𝑗 for investor 𝑘


𝑥𝑗𝑘 : the position of asset 𝑗 held by investor 𝑘 after
rebalancing at the end of period 1 (decision variables)
Model
Assumptions:

All investors don’t hold any futures initially
No transaction costs and taxes will incur

The matrix Σ 𝑘 = 𝜎𝑖𝑗𝑘




𝐽×𝐽
, where 𝜎𝑖𝑗𝑘 = 𝑐𝑜𝑣 𝑘 (𝑋𝑖 , 𝑋𝑗 ), is
positively definite for all k
Dividends have been embedded in prices
𝑝0 = 1 and 𝑋0 = 1 + 𝑟
The index futures expires at 𝑡 = 2, therefore 𝜂 =
𝐽
𝑗=1 𝜔𝑗 𝑋𝑗 .
The model
𝑊 𝑘 (𝑡): the total wealth of investor 𝑘 at 𝑡, we have
𝐽
𝐽
𝑛𝑗𝑘 𝑝𝑗 = 𝑥0𝑘 +
𝑊 𝑘 1 = 𝑛0𝑘 +
𝑗=1
𝑊 𝑘 2 = 1 + 𝑟 𝑥0𝑘 +
𝑥𝑗𝑘 𝑝𝑗 + 𝑞𝑓 𝑘
𝑗=1
𝐽
𝑘
𝑥
𝑗=1 𝑗
𝑋𝑗 + 𝜂𝑓 𝑘
= 𝑊 𝑘 1 + 𝑋 − 𝑃 𝑇 𝑥 𝑘 + 𝜔𝑇 𝑋 − 𝑞 𝑓 𝑘 + 𝑟𝑥0𝑘
𝑇
𝑤ℎ𝑒𝑟𝑒 𝑋 = 𝑋1 , … , 𝑋𝐽 , 𝑃 = 𝑝1 , … , 𝑝𝐽
𝜔 = 𝜔1 , … , 𝜔𝐽 , 𝑥 𝑘 = (𝑥1𝑘 , … , 𝑥𝐽𝑘 )
𝑇
,
Model
For investor 𝑘, his utility is given by
𝑈𝑘 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.
Model
𝐸𝑘
𝑊𝑘
𝑉𝑎𝑟
𝑘
2
=
1 +
𝜇𝑘
−𝑃
𝑘 𝑇 𝑘 𝑘
𝑇 𝑘
𝑥
𝑘
= 𝑥
𝜇𝑘
𝑇
𝑘
𝑘
𝐸 [𝑋1 ], … , 𝐸 [𝑋𝐽 ]
𝑊 2
𝑤ℎ𝑒𝑟𝑒
𝑊𝑘
=
𝑘
+ 𝜔𝑇 𝜇𝑘 − 𝑞 𝑓 𝑘 + 𝑟𝑥0𝑘
𝑇 𝑘 𝑘
𝑇 𝑘
Σ 𝑥 + 2𝑓 𝜔 Σ 𝑥 + 𝜔 Σ 𝜔 𝑓
𝑘 2
The basic problem
In a perfect market (with no trading restriction), the
optimal portfolio selection problem for investor 𝑘 is:
(𝑃𝑀𝑘 )
𝑚𝑎𝑥𝑥 𝑘 ,…,𝑥 𝑘,𝑓𝑘 𝑈 𝑘 𝑊 𝑘 2
0
s.t.
𝐽
𝑃𝑇 𝑥 𝑘 + 𝑞𝑓 𝑘 + 𝑥0𝑘 = 𝑃𝑇 𝑁 𝑘 + 𝑁0𝑘
Note: additional constraint can be added on 𝑥𝑗𝑘 later.
Equilibrium
Definition: A vector (𝑃∗ , 𝑞) ∈ 𝑅 𝐽+1 is called an
equilibrium price of the market if there exist
𝑘 ∗ 𝑘∗ 𝑘 ∗
(𝑥0 , 𝑥 , 𝑓 ) ∈ 𝑅 𝐽+2 (𝑘 = 1, ⋯ , 𝐾) such that
∗
𝑘∗
𝑘
 (𝑥
,𝑓 )
∗ ∗
solves the optimization problem 𝑃𝑀𝑘 at
(𝑃 , 𝑞 ) for 𝑘 = 1, ⋯ , 𝐾

𝐾
𝑘∗
𝑘=1 𝑥0
=
where 𝑁𝑗 =
∗
∗
𝐾
𝐾
𝑘
𝑘
𝑁0 , 𝑘=1 𝑥 = 𝑁 , and 𝑘=1 𝑓
𝑘
𝐾
𝑇
𝑛
,
𝑁
=
𝑁
,
⋯
,
𝑁
1
𝐾
𝑘=1 𝑗
= 0,
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)


There are some extensions


It is also assumed that the market is efficient and trading is
frictionless
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
Main Results
𝑚𝑎𝑥𝑥 𝑘 ,…,𝑥 𝑘,𝑓𝑘 𝐸 𝑘 𝑊 𝑘 2 − 𝛼 𝑘 𝑉𝑎𝑟 𝑘 [𝑊 𝑘 2 ]
(𝑃𝑀𝑘 )
0
𝑃𝑇 𝑥 𝑘 + 𝑞𝑓 𝑘 + 𝑥0𝑘 = 𝑃𝑇 𝑁 𝑘 + 𝑁0𝑘
s.t.
𝐸𝑘
𝑊𝑘
𝑉𝑎𝑟
𝑘
2
𝑘
𝐽
=
𝑊 2
𝑊𝑘
1 +
= 𝑥
𝜇𝑘
−𝑃
𝑘 𝑇 𝑘 𝑘
𝑇 𝑘
𝑥
𝑘
+ 𝜔𝑇 𝜇𝑘 − 𝑞 𝑓 𝑘 + 𝑟𝑥0𝑘
𝑇 𝑘 𝑘
𝑇 𝑘
Σ 𝑥 + 2𝑓 𝜔 Σ 𝑥 + 𝜔 Σ 𝜔 𝑓
𝑘 2
Main Results
For (𝑃𝑀𝑘 ), we have the following necessary conditions for
its optimal solution:
𝑥𝑘
𝑓𝑘
1
= 𝑘 Σ𝑘
𝛼
=
−1
𝜇𝑘 − 1 + 𝑟 𝑃 − 𝑓 𝑘 𝜔
1 𝜔𝑇 𝜇𝑘 − 1+𝑟 𝑞
𝛼𝑘
𝜔𝑇 Σ𝑘 𝜔
𝜔𝑇 Σ𝑘 𝑥 𝑘
− 𝑇 𝑘
𝜔 Σ 𝜔
based on which we can obtain: 𝑞 = 𝑃𝑇 𝜔
Main Results (with trading restrictions)
(𝑇𝑅𝑘 )
𝑚𝑎𝑥𝑥 𝑘 ,…,𝑥 𝑘,𝑓𝑘 𝐸 𝑘 𝑊 𝑘 2 − 𝛼 𝑘 𝑉𝑎𝑟 𝑘 [𝑊 𝑘 2 ]
0
s.t.
𝐽
𝑃𝑇 𝑥 𝑘 + 𝑞𝑓 𝑘 + 𝑥0𝑘 = 𝑃𝑇 𝑁 𝑘 + 𝑁0𝑘
𝐿𝑘 ≤ 𝑥 𝑘 ≤ 𝑈 𝑘
when 𝐿𝑘 = 0, there is no short selling is allowed.
Main Results (with trading restrictions)
For (T𝑅𝑘 ), we have the following Lagrangian functions:
𝑘
𝛼
𝑇
2
+ 𝑤 𝑇 𝜇𝑘 𝑓 𝑘 −
𝑥 𝑘 Σ𝑘 𝑥 𝑘 + 2𝑓 𝑘 𝜔𝑇 Σ𝑘 𝑥 𝑘 + 𝑓 𝑘 𝜔𝑇 Σ𝑘 𝜔
2
𝑇
+ 1 + 𝑟 𝑥0𝑘 + 𝜆𝑘 𝑃𝑇 𝑥 𝑘 + 𝑞𝑓 𝑘 + 𝑥0𝑘 − 𝑃𝑇 𝑁 𝑘 − 𝑁0𝑘 + 𝜃 𝑘 𝑥 𝑘 − 𝐿𝑘
𝑇
𝜇𝑘 𝑥 𝑘
+ 𝜉𝑘
𝑇
𝑈𝑘 − 𝑥 𝑘
where 𝜆𝑘 ∈ 𝑅, 𝜉 𝑘 ≥ 0, 𝜃 𝑘 ≥ 0 are Lagrangian multipliers
Main Results (with trading restrictions)
For (𝑇𝑅𝑘 ), we have the following necessary conditions for
its optimal solution:
𝑥𝑘
𝑓𝑘
1 𝑘
= 𝑘 Σ
𝛼
=
−1
𝜇𝑘 − 1 + 𝑟 𝑃 + 𝜃 𝑘 − 𝜉 𝑘 − 𝑓 𝑘 𝜔
1 𝜔𝑇 𝜇𝑘 − 1+𝑟 𝑞
𝛼𝑘
𝜔𝑇 Σ𝑘 𝜔
𝜔𝑇 Σ𝑘 𝑥 𝑘
− 𝑇 𝑘
𝜔 Σ 𝜔
𝜃𝑗𝑘 𝑥𝑗𝑘 − 𝐿𝑗𝑘 = 𝜉𝑗𝑘 𝑈𝑗𝑘 − 𝑥𝑗𝑘 = 0
based on which we can obtain: 𝑞 =
𝑃𝑇 𝜔
−
1
1+𝑟
𝜃𝑘
−
𝑇
𝑘
𝜉
𝜔
Main Results (with trading restrictions)
Hence, in general, we have 𝑞 ≠ 𝑃𝑇 𝜔.
In particular, if 𝑈 𝑘 = ∞, then 𝜉 𝑘 = 0, we have 𝑞 ≤ 𝑃𝑇 𝜔
Main Results (with margin requirement)
(𝑀𝑅𝑘 )
𝑚𝑎𝑥𝑥 𝑘 ,…,𝑥 𝑘,𝑓𝑘 𝐸 𝑘 𝑊 𝑘 2 − 𝛼 𝑘 𝑉𝑎𝑟 𝑘 [𝑊 𝑘 2 ]
0
s.t.
𝐽
𝑃𝑇 𝑥 𝑘 + 𝑚𝑞 𝑓 𝑘 + 𝑥0𝑘 = 𝑃𝑇 𝑁 𝑘 + 𝑁0𝑘
𝐿𝑘 ≤ 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.
Main Results (with margin requirement)
For (M𝑅𝑘 ), we have the following Lagrangian functions:
𝜇𝑘
−𝑃
𝑇 𝑘
𝑥
+ 𝑤 𝑇 𝜇𝑘 − 𝑞 𝑓 𝑘
𝛼𝑘
𝑇
2
−
𝑥 𝑘 Σ𝑘 𝑥 𝑘 + 2𝑓 𝑘 𝜔𝑇 Σ𝑘 𝑥 𝑘 + 𝑓 𝑘 𝜔𝑇 Σ𝑘 𝜔 + 𝑟𝑥0𝑘
2
𝑇
𝑘
𝑘
𝑘
𝑇
𝑘
𝑘
𝑇
𝑘
𝑘
+ 𝜆 𝑃 𝑥 + 𝑚𝑞|𝑓 | + 𝑥0 − 𝑃 𝑁 − 𝑁0 + 𝜃
𝑥 𝑘 − 𝐿𝑘
+ 𝜉
𝑘 𝑇
𝑈𝑘 − 𝑥 𝑘
where 𝜆𝑘 ∈ 𝑅, 𝜉 𝑘 ≥ 0, 𝜃 𝑘 ≥ 0 are Lagrangian multipliers
Main Results (with margin requirement)
For (M𝑅𝑘 ), we have the following necessary conditions for
its optimal solution:
𝑥𝑘
𝑓𝑘
1 𝑘
= 𝑘 Σ
𝛼
=
−1
𝜇𝑘 − 1 + 𝑟 𝑃 + 𝜃 𝑘 − 𝜉 𝑘 − 𝑓 𝑘 𝜔
1 𝜔𝑇 𝜇𝑘 − 1+𝑟𝑚𝑣 𝑘 𝑞
𝛼𝑘
𝜔𝑇 Σ𝑘 𝜔
−
𝜔𝑇 Σ𝑘 𝑥 𝑘
𝜔𝑇 Σ𝑘 𝜔
𝜃𝑗𝑘 𝑥𝑗𝑘 − 𝐿𝑗𝑘 = 𝜉𝑗𝑘 𝑈𝑗𝑘 − 𝑥𝑗𝑘 = 0
We then have: 𝑞 =
𝑃𝑇 𝜔
+
𝑟 1−𝑚𝑣 𝑘 𝑃− 𝜃𝑘 −𝜉 𝑘
1+𝑟𝑚𝑣 𝑘
𝑇
𝜔
Some Empirical Results
HS300 Index
Some Empirical Results
SZ50 Index
Some Empirical Results
ZZ500 Index
Some Empirical Results
basis𝑡 = 𝛼 basis𝑡−1 + 𝛽volatility𝑡 + 𝛾
Para\index
𝛼
𝛽
𝛶
SZ50
0.6135***
-5.6534***
-0.0002
HS300
0.7055***
-6.9324***
0.0012***
ZZ500
0.6315***
-7.0699***
-0.0023*
Significant codes: ***p<0.001, **p<0.01, *p<0.05, p<0.1
Some Empirical Results
Margin requirements for SZ50 and HS300
Times
2015/4/16-2015/8/25
2015/8/26-2015/8/31
2015/9/1-2015/9/6
2015/9/7-2017/2/16
2017/2/17-2017/4/13
m（%）
10
20
30
40
20
Some Empirical Results
basis𝑡 = 𝛼 ∗ basis𝑡−1 + 𝛽 ∗ volatility𝑡 + 𝛾 + 𝑘 ∗ Dummyt
as m increases from 10% to 40% …
Para\index
𝛼
𝛽
𝛶
k
SZ50
0.5683***
-8.4435***
0.0034*
-0.0035**
HS300
0.6031***
-9.8030***
0.0016
-0.0024*
Some Empirical Results
basis𝑡 = 𝛼 ∗ basis𝑡−1 + 𝛽 ∗ volatility𝑡 + 𝛾 + 𝑘 ∗ Dummyt
as m decreases from 40% to 20% …
Para\index
𝛼
𝛽
𝛶
k
SZ50
HS300
0.3716*** 0.5098***
-5.8229*** -6.6331***
-0.0026*** -0.0033***
0.0015*
0.0018*
Some Empirical Results
KOSPI 200
Some Empirical Results
basis𝑡 = 𝛼 basis𝑡−1 + 𝛽volatility𝑡 + 𝛾
Para\index
𝛼
𝛽
𝛶
KOSPI 200
0.6044***
-1.5683*
0.0010***
Discussions

Chinese markets are highly regulated and
inefficient





Very difficult to short stocks
T+1
Government interventions
…
Stock index futures prices …