Stock Index Futures: Their Effects on Stock Markets AWS
The Divergence of Stock Index Futures from Their Underlying Cash Values* J.Q. Hu Fudan University August 24, 2017
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 CSI 300 index futures was introduced in China in 2010. In general, stock index futures prices are based on the cost-to-carry model (Cornell and France 1983)
The cost-of-carry model:
It views a futures contract as a forward contract, assuming the risk-free interest rates are constant the markets are perfect in that any arbitrage strategy can be carried out it does not take into consideration the impact of the volatility of the stock market on the futures price In this model, the futures prices depend on the underlying index cash price, the risk-free interest rate, the dividend yield, and the time to maturity
Introduction
No-arbitrage model with stochastic interest rates (Ranaswamy and Sundaresan 1985) Models in which both interest rates and market volatility are stochastic (Bailey and Stulz 1989, Hemler and Longstaff 1991) However, very few study on the effects of trading and regulatory constraints on index futures prices.
The basis of CSI 300 stock index basis = futures price – cash price
The basic 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, J risky assets, and one risk-free asset ▪ Investor 𝑘 ∈ {1, … , 𝐾}
▪ Asset 𝑗 ∈ {0,1, … , J}, (0 is risk-free asset)
The basic model Two periods (𝑡 = 0 and 1)
▪ 𝑝𝑗 : the price of asset 𝑗 at 𝑡 = 1 ▪ 𝑋𝑗 : the price of asset 𝑗 at 𝑡 = 2 (a random variable) ▪ 𝑟: the risk free interest rate ▪ 𝑛𝑗𝑘 : the initial units of asset 𝑗 that investor 𝑘 has ▪ 𝑥𝑗𝑘 : the position of asset 𝑗 held by investor 𝑘 after rebalancing (decision variables)
The basic model 𝑊 𝑘 (𝑡): the total wealth of investor 𝑘 at 𝑡 𝐽
𝑊 𝑘 1 = 𝑛𝑗𝑘 𝑝𝑗 𝑗=0 𝐽
𝑊 𝑘 2 = 𝑥𝑗𝑘 𝑋𝑗 𝑗=0
We set 𝑋0 = 1 and 𝑝 = 1/(1 + 𝑟)
The basic 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.
The basic model The optimal portfolio selection problem for investor 𝑘 is: 𝐽
(𝑂1 )
𝐽
𝐽
𝑚𝑎𝑥𝜙𝑘,…,𝜙𝑘 𝑥𝑗𝑘 𝐸 𝑘 [𝑋𝑗 ] + 𝑥0𝑘 − 𝛼 𝑘 𝑥𝑗𝑘 𝑥𝑖𝑘 𝜎𝑗𝑖 0
𝐽
𝑗=1
𝑖=1 𝑗=1
s.t. 𝐽
𝐽
𝑥𝑗𝑘 𝑝𝑗 + 𝑥0𝑘 𝑝0 = 𝑛𝑗𝑘 𝑝𝑗 + 𝑛0𝑘 𝑝0 𝑗=1
𝑗=1
where 𝜎𝑗𝑖 is the investor 𝑘’s belief of covariance between assets 𝑖 and 𝑗.
Note: additional constraint can be added on 𝑥𝑗𝑘 .
Equilibrium Definition: A vector 𝑃∗ ∈ 𝑅 𝐽 is called an equilibrium price of the market if there exist 𝑥 𝑘∗ ∈ 𝑅 𝐽 for 𝑘 = 1, ⋯ , 𝐾 such that
𝑥 𝑘∗ solves the optimization problem 𝑂1 at 𝑃 = 𝑃∗ for 𝑘 = 1, ⋯ , 𝐾 𝑘∗ 𝑘 𝐾 σ𝐾 σ 𝑥 = 𝑁 for 𝑗 = 1, ⋯ , 𝐽. 𝑘=1 𝑘=1
where
𝑁𝑘
=
𝑘 𝐾 𝑇 𝑛1 , ⋯ , 𝑛𝐽
The basic model
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
Model (with stock index futures) 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 = σ𝐽𝑗=1 𝜔𝑗 𝑋𝑗
The model 𝑊 𝑘 (𝑡): the total wealth of investor 𝑘 at 𝑡, we have 𝐽
𝐽
𝑊 𝑘 1 = 𝑛0𝑘 + 𝑛𝑗𝑘 𝑝𝑗 = 𝑥0𝑘 + 𝑥𝑗𝑘 𝑝𝑗 + 𝑞𝑓 𝑘 𝑗=1
𝑗=1 𝐽
𝑊 𝑘 2 = 1 + 𝑟 𝑥0𝑘 + σ𝑗=1 𝑥𝑗𝑘 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 σ𝑘=1 𝑥0
=
where 𝑁𝑗 =
∗ ∗ 𝐾 𝐾 𝑘 𝑘 𝑁0 , σ𝑘=1 𝑥 = 𝑁 , and σ𝑘=1 𝑓 𝑘 𝑇 σ𝐾 𝑛 , 𝑁 = 𝑁 , ⋯ , 𝑁 1 𝐾 𝑘=1 𝑗
= 0,
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 basis𝑡 = 𝛼 basis𝑡−1 + 𝛽volatility𝑡 + 𝛾 Para\index 𝛼 𝛽 𝛶
SZ50 0.6135*** -5.6534*** -0.0002
CSI300 0.7055*** -6.9324*** 0.0012***
ZZ500 0.6315*** -7.0699*** -0.0023*
Significant codes: ***p
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 CSI 300 index futures was introduced in China in 2010. In general, stock index futures prices are based on the cost-to-carry model (Cornell and France 1983)
The cost-of-carry model:
It views a futures contract as a forward contract, assuming the risk-free interest rates are constant the markets are perfect in that any arbitrage strategy can be carried out it does not take into consideration the impact of the volatility of the stock market on the futures price In this model, the futures prices depend on the underlying index cash price, the risk-free interest rate, the dividend yield, and the time to maturity
Introduction
No-arbitrage model with stochastic interest rates (Ranaswamy and Sundaresan 1985) Models in which both interest rates and market volatility are stochastic (Bailey and Stulz 1989, Hemler and Longstaff 1991) However, very few study on the effects of trading and regulatory constraints on index futures prices.
The basis of CSI 300 stock index basis = futures price – cash price
The basic 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, J risky assets, and one risk-free asset ▪ Investor 𝑘 ∈ {1, … , 𝐾}
▪ Asset 𝑗 ∈ {0,1, … , J}, (0 is risk-free asset)
The basic model Two periods (𝑡 = 0 and 1)
▪ 𝑝𝑗 : the price of asset 𝑗 at 𝑡 = 1 ▪ 𝑋𝑗 : the price of asset 𝑗 at 𝑡 = 2 (a random variable) ▪ 𝑟: the risk free interest rate ▪ 𝑛𝑗𝑘 : the initial units of asset 𝑗 that investor 𝑘 has ▪ 𝑥𝑗𝑘 : the position of asset 𝑗 held by investor 𝑘 after rebalancing (decision variables)
The basic model 𝑊 𝑘 (𝑡): the total wealth of investor 𝑘 at 𝑡 𝐽
𝑊 𝑘 1 = 𝑛𝑗𝑘 𝑝𝑗 𝑗=0 𝐽
𝑊 𝑘 2 = 𝑥𝑗𝑘 𝑋𝑗 𝑗=0
We set 𝑋0 = 1 and 𝑝 = 1/(1 + 𝑟)
The basic 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.
The basic model The optimal portfolio selection problem for investor 𝑘 is: 𝐽
(𝑂1 )
𝐽
𝐽
𝑚𝑎𝑥𝜙𝑘,…,𝜙𝑘 𝑥𝑗𝑘 𝐸 𝑘 [𝑋𝑗 ] + 𝑥0𝑘 − 𝛼 𝑘 𝑥𝑗𝑘 𝑥𝑖𝑘 𝜎𝑗𝑖 0
𝐽
𝑗=1
𝑖=1 𝑗=1
s.t. 𝐽
𝐽
𝑥𝑗𝑘 𝑝𝑗 + 𝑥0𝑘 𝑝0 = 𝑛𝑗𝑘 𝑝𝑗 + 𝑛0𝑘 𝑝0 𝑗=1
𝑗=1
where 𝜎𝑗𝑖 is the investor 𝑘’s belief of covariance between assets 𝑖 and 𝑗.
Note: additional constraint can be added on 𝑥𝑗𝑘 .
Equilibrium Definition: A vector 𝑃∗ ∈ 𝑅 𝐽 is called an equilibrium price of the market if there exist 𝑥 𝑘∗ ∈ 𝑅 𝐽 for 𝑘 = 1, ⋯ , 𝐾 such that
𝑥 𝑘∗ solves the optimization problem 𝑂1 at 𝑃 = 𝑃∗ for 𝑘 = 1, ⋯ , 𝐾 𝑘∗ 𝑘 𝐾 σ𝐾 σ 𝑥 = 𝑁 for 𝑗 = 1, ⋯ , 𝐽. 𝑘=1 𝑘=1
where
𝑁𝑘
=
𝑘 𝐾 𝑇 𝑛1 , ⋯ , 𝑛𝐽
The basic model
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
Model (with stock index futures) 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 = σ𝐽𝑗=1 𝜔𝑗 𝑋𝑗
The model 𝑊 𝑘 (𝑡): the total wealth of investor 𝑘 at 𝑡, we have 𝐽
𝐽
𝑊 𝑘 1 = 𝑛0𝑘 + 𝑛𝑗𝑘 𝑝𝑗 = 𝑥0𝑘 + 𝑥𝑗𝑘 𝑝𝑗 + 𝑞𝑓 𝑘 𝑗=1
𝑗=1 𝐽
𝑊 𝑘 2 = 1 + 𝑟 𝑥0𝑘 + σ𝑗=1 𝑥𝑗𝑘 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 σ𝑘=1 𝑥0
=
where 𝑁𝑗 =
∗ ∗ 𝐾 𝐾 𝑘 𝑘 𝑁0 , σ𝑘=1 𝑥 = 𝑁 , and σ𝑘=1 𝑓 𝑘 𝑇 σ𝐾 𝑛 , 𝑁 = 𝑁 , ⋯ , 𝑁 1 𝐾 𝑘=1 𝑗
= 0,
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 basis𝑡 = 𝛼 basis𝑡−1 + 𝛽volatility𝑡 + 𝛾 Para\index 𝛼 𝛽 𝛶
SZ50 0.6135*** -5.6534*** -0.0002
CSI300 0.7055*** -6.9324*** 0.0012***
ZZ500 0.6315*** -7.0699*** -0.0023*
Significant codes: ***p
