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CHAPTER 10
Securities Futures Products
Refinements
In this chapter, we extend the discussion of stock index
futures. This chapter is organized into the following
sections:
1. Stock Index Futures Prices
2. Program Trading
3. Hedging with Stock Index Futures
4. Asset Allocation
5. Portfolio Insurance
6. Index Futures and Stock Volatility
7. Index Futures and Stock Market Crashes
Chapter 10
1
Stick Index Futures Prices
In this section, the following issues are explored:
1. The empirical evidence on stock index futures
efficiency.
– do stock index futures prices conform to the Cost-of-Carry
Model?
2. The effect of taxes on stock index futures prices.
3. The timing relationship between stock index futures
prices and the cash market index.
– Does the futures price lead the cash market index, or
does the cash market index lead the futures?
4. The seasonal impacts on stock index futures pricing.
Chapter 10
2
Stock Index Futures Efficiency
Recall that the success of an arbitrage opportunity can be
affected by:
– The use of short sale proceeds
– Transaction costs
– Dividend variability
Every real market has a range of permissible no-arbitrage
prices. This no-arbitrage band increases because of
transaction costs and restrictions on short selling.
Evidence suggests that the futures market was inefficient
in the early days of trading but now it conforms well to the
Cost-of-Carry Model.
Figure 10.1 shows the result of a study by Modest and
Sundaresan.
Chapter 10
3
Stock Index Futures Efficiency
Insert figure 10.1 here
Notice how the observed price is almost always within the
no arbitrage bounds and never deviates far from them.
Chapter 10
4
Effect of Taxes on Stock Index Futures
Prices
Because futures prices are marked-to-market at year end
for tax purposes, index futures contracts possess no taxtiming options.
In the futures markets, tax rules require all paper gains or
losses to be recognized as cash gains or losses each year.
In the cash market, an individual can time his tax gains or
losses.
In an empirical study of the effect of the tax-timing option,
Cornell concludes that the tax-timing option does not
appear to affect prices.
Chapter 10
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Timing Effect on Stock Index Futures
Prices
The Day of the Week Effect in Stock Index Futures
A great deal of evidence shows that returns on stocks differ
depending on the day of the week. In particular, Friday
returns are generally high.
Leads and Lags in Stock Index Prices
Leads and lags in stocks index prices refer to which market
drives the other.
– Does the futures price lead the cash market index, or
does the cash market index lead the futures market?
The question of leads and lags has been explored in
several studies, most of which find that futures prices lead
cash market prices.
Chapter 10
6
Program Trading
In Chapter 9, we examine index arbitrage through program
trading and how to engage in cash-and-carry and reverse
cash-and-carry strategies to exploit pricing differences
between the index and the index futures.
Recall further from Chapter 9 that the futures price that
conforms with the Cost-of-Carry Model is called the fairvalue futures price.
In this section, we determine the fair value of the
December 2001 S&P 500 stock index futures contract
traded on November 30, 2001.
Chapter 10
7
Program Trading
Assume that the December 2001 futures contract closed at
1140 index points on November 30. The cash index price
on this date was 1139.45. The value of the compounded
dividend stream expected to be paid out between the 30th
of November and December 21 totaled .9 index points.
The financing cost for large, credit-worthy borrowers was
approximately 1.90% annualized over a 365-day year
(0.1093% over the 21 days from Nov 30 to Dec 21).
Suppose that the December 2001 futures price on
November 30, 2001 had been 1143.00 instead of the
actual 1140. Using this information, we can apply the Costof-Carry Model to determine the fair-value futures price:
F0,t = 1139.45 (1 + .001093) -.9 = 1139.80 index points
Tables 10.1 and Table 10.2 show the transaction involved
in a cash-and-carry and reverse cash-and-carry arbitrages.
Chapter 10
8
Program Trading
A Real World Example
Table 10.1 shows how an arbitrage profit can be earned if
the futures price is 1143.
Table 10.1
CashBandBCarry Index Arbitrage
Date
Cash Market
Futures Market
November
30
Borrow $284,862.5
(1139.45 x $250) 21 days
at 1.9%. Buy stocks in the
S&P 500 for $284,862.5.
Sell one DEC S&P 500
index futures contract for
1143.00.
December
21
Receive accumulated
proceeds from invested
dividends of $225 (.9
index points x $250). Sell
stock for $285,000 (1140
index points x $250). Total
proceeds are $285,225.
Repay debt of $285,173.9.
At expiration, the futures
price is set equal to the
spot index value of 1140.00. This gives a profit of
3.00 index units. In dollar
terms, this is 3.00 index
points times $250 per
index point.
Gain: $311.40
Gain: $750
Total Profit: $311.40 + $750 = $1,061.40
By completing the arbitrage, the trader was able to
earn a 4.99% annualized return.
 1143 
Trader Profit  

 1139 .80 
365
21
 1  4.99 %
Since the financing cost was 1.9%, an arbitrage profit
was earned.
Chapter 10
9
Program Trading
A Real World Example
Now suppose that the futures price is 1138. Table 10.2
shows how an arbitrage profit can be earned.
Table 10.2
Reverse CashBandBCarry Index Arbitrage
Date
Cash Market
Futures Market
November
30
Sell stock in S&P 500 for
$284,862.5 (1139.45 x
$250). Lend $284,862.5 for
21 days at 1.9%.
Buy one DEC index futures
contract for 1138.00.
December
21
Receive proceeds from
investment of $285,173.9.
Buy stocks in S&P 500
index for $285,000
(1140.00 x $250). Return stocks to repay short sale.
At expiration, the futures
price is set equal to the
spot index value of
1140.00. This gives a profit
of 2.00 index points. In
dollar terms, this is 2.00
index points times $250
per index.
Gain: $173.9
Profit: $500
Total Profit: $173.9 + $500 = $673.9
The investor is earning a 2.78% annualized return.
 1139 .80 
Trader Profit  

 1138 
365
21
 1  2.78 %
Since the financing cost is 1.90%, an arbitrage profit
was earned.
Chapter 10
10
Real-World Impediments to Stock Index
Arbitrage
The Cost-of-Carry Model needs to be refined to account for
real-world impediments to arbitrage strategies.
An empirical study conducted by Sofianos reports that:
1. Existence of arbitrage opportunities depends on the level
of transaction costs. Lower transaction costs are
associated with more frequent arbitrage opportunities.
2. Arbitrageurs often use surrogate stock baskets containing
a subset of the index stocks instead of trading all the
stocks in the index.
3. Arbitrageurs frequently establish (or liquidate) their
futures and cash positions at different times.
Chapter 10
11
Hedging with Stock Index Futures
Recall from chapter 9 that a manager can determine the
number of contract to trade by using the following
equation:
VP
 P
 Number of Contracts
VF
Where:
VP = value of the portfolio
VF = value of the futures contract
βP = beta of the portfolio that is being hedged
Chapter 10
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Hedging with Stock Index Futures
The risk of a combined cash and futures position is
equal to:
 P2   S2  HR 2 F2  2 HR SF  S  F
Where:
 P2  Variance on the portfolio Pt
 S2  Variance of St
 F2  Variance of Ft
 SF  Correlatio n coefficien t between St and Ft
Chapter 10
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Hedging with Stock Index Futures
The risk-minimizing hedge ratio (HR) is:
HR RM = -
 SF  S  F
COV SF
=
 F2
 F2
Where:
COVSF = the covariance between S and F
The easiest way to find the risk-minimizing hedge ratio is
to estimate the following regression:
S t     RM Ft   t
St
=
Ft
=
Α
βRM
=
=
ε
=
the returns on the cash market position
in period t
the returns on the futures contract in
period t
the constant regression parameter
the slope regression parameter for the
risk-minimizing hedge
an error term with zero mean and
standard deviation of 1.0
Chapter 10
14
Hedging with Stock Index Futures
From the above equation, the negative of the estimated
Beta is the risk-minimizing hedge ratio.
Having found the risk-minimizing hedge ratio ( -βRM,),
Compute the number of contracts to trade, using:
-
RM
 VP 
  = number of contracts
 VF 
Chapter 10
15
Minimum Risk Hedging
Assume that today, November 28, a portfolio manager
has $10 million dollar invested in the 30 stocks of the
DJIA. The portfolio manager will hedge using S&P 500
JUN futures contract.
On Nov 27, the S&P futures closed at 354.75. The future
contract value is the index level times $250.
Compute the hedge ratio and determine the number of
contract to purchase.
Step 1: collect historical data
In order to perform the analysis the portfolio manager
collects historical data. The portfolio manager has
collected 100 paired observations of daily returns data
on her portfolio and the S&P 500 JUN futures contract.
The data covers from July 7 to November 27.
Chapter 10
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Minimum Risk Hedging
Step 2: estimate the hedging beta using:
S t     RM Ft   t
The regression results are:
βRM = 0.8801
R2 = 0.9263
Step 3: compute the futures position using:
-
RM
 VP 
  = number of contracts
 VF 
 $10,000,000
- 0.8801 
 (354.75)($250)


 =  99.2361


The estimated risk-minimizing futures position is -99.24
contracts, so the portfolio manager decides to sell 100
contracts.
Chapter 10
17
Minimum Risk Hedging
Step 4: evaluate the hedging results.
Figure 10.2 illustrates the results.
Insert Figure 10.2 here
The hedged portfolio maintained its value while the unhedged portfolio declined in value substantially. Clearly,
the hedge worked well.
Chapter 10
18
Minimum Risk Hedging
Using historical data or ex-ante (before the fact), the best
ratio that the portfolio manager had was βRM = 0.8801.
Using data after the fact or ex-post (data from Nov 28 to
Feb 22), the best beta ratio that the portfolio manager had
was βRM =0.9154. This beta was calculated after the
investment was made using data from Nov 28 to Feb 22.
Figure 10.3 illustrates the differences in performance using
ex-ante and ex-post data.
Insert figure 10.3 here
While the ex-post hedge ratio is superior, the ex-ante
hedge is the best estimate that is available at the time
the decision must be made.
Chapter 10
19
CAPM and Portfolio Beta
Portfolio managers often adjust the CAPM betas of their
portfolios in anticipation of bull and bear markets.
– Bull market: increase the beta of the portfolio to take
advantage of the expected rise in stock prices.
– Bear market: reduce the beta of a stock portfolio as a
defensive maneuver.
From the CAPM, all risk is defined as either systematic or
unsystematic.
– Systematic risk is associated with general movements in
the market and affects all investments.
– Unsystematic risk is particular to a investment or range of
investments.
Diversification can almost eliminate unsystematic risk from
a portfolio. The remaining systematic risk is unavoidable.
A portfolio with zero systematic risk should earn the riskfree rate of interest.
Chapter 10
20
CAPM and Portfolio Beta
Portfolio managers can use hedging to eliminate only a
portion of the systematic risk or they can use stock index
futures to increase the systematic risk of a portfolio.
Risk-Minimizing Hedge
A risk-minimizing hedge matches a long position in stock
with a short position in stock index futures in an attempt to
create a portfolio whose value does not change with
fluctuations in the stock market.
To reduce, but not eliminate the systematic risk, a portfolio
manager could sell some futures, but fewer than the riskminimizing amount.
To increase the systematic risk of the portfolio, a manager
could buy some futures contracts.
Figure 10.4 shows the price paths of two portfolios.
Chapter 10
21
CAPM and Portfolio Beta
The first portfolio is an unhedged portfolio. Its value starts
with $10,000,000 and finished at $9,656,090 in a period of
declining markets. The second portfolio includes the same
$10,000,000 of stocks from the first portfolio plus a long
position of 52 futures contracts. This combination doubles
the systematic risk of the portfolio. In this case, the value of
the portfolio declined to $9,052,340 in the same period of
declining markets.
Insert figure 10.4 here
Chapter 10
22
Asset Allocation
In asset allocation, an investor decides how to allocate and
shift funds among broad asset classes.
Recall that for financial futures the cost of carry essentially
equals the financing cost.
In a full carry market, a cash-and-carry strategy should
earn the financing rate, which equals the risk-free rate of
interest. This can be expressed as:
Short-Term Riskless Debt = Stock - Stock Index Futures
A trader might create a synthetic T-bill by holding stock and
selling futures:
Synthetic T-bill = Stock - Stock Index Futures
This is a synthetic T-bill rather than an actual T-bill. While
the portfolio will mimic the price movements of a T-bill, no
T-bills were purchased. This technique is useful for a trader
that wishes to temporarily reduce the risk of a portfolio
without selling stocks.
A futures portfolio with no systematic risk has an expected
return that equals the risk-free rate.
Rearranging the second equation, a synthetic stock
portfolio can be created.
Synthetic Stock Portfolio = T-bills + Stock Index Futures
Chapter 10
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Portfolio Insurance
For a given well-diversified portfolio, selling stock index
futures can create a combined stock/futures portfolio with
reduced risk.
Portfolio insurance refers to a collection of techniques for
managing the risk of an underlying portfolio.
The goal of portfolio insurance is to manage the risk of a
portfolio to ensure that the value of the portfolio does not
drop below a specified level.
It involves adjusting the number of futures contracts in the
portfolio over time as the value of the portfolio changes.
Dynamic hedging refers to implementing portfolio
insurance strategies using futures. It requires continually
monitoring the portfolio.
While portfolio insurance can be desirable, it is not free.
Chapter 10
24
Portfolio Insurance
Assume that a stock index futures contract has an
underlying value of $100 million. A trader wishes to insure
a minimum value for the portfolio of $90 million. Initially the
trader sells futures contracts to cover $50 million of the
value of the portfolio. Thus, in the initial position, the trader
is long $100 million in stock and short $50 million in
futures, so 50% of the portfolio is hedged. Table 10.4
shows the basic strategy of portfolio insurance with
dynamic hedging.
Table 10.4
Portfolio Insurance Transactions and Results
Time
0
1
2
3
4
5
6
Gain/Loss $ millions
StocksFutures
0.00
B2.00
B2.00
B2.00
B4.00
B35.86
B10.00
0.00
1.00
1.11
1.22
2.88
30.65
10.00
Total
Value
Futures
Position
Portion
Hedged
100.00
99.00
98.11
97.33
96.21
90.00
90.00
B50
B55
B60
B70
B80
B90
B90
.50
.56
.61
.72
.83
1.00
1.00
Notice that the value of the portfolio does not drop below
the $90 million floor, so the insurance worked.
Chapter 10
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Implementing Portfolio Insurance
Choosing the initial futures position depends on:
A. The floor that is chosen relative to the initial value.
The lower the floor, the lower the portion the portfolio to be
initially hedged.
B. The volatility of the stock portfolio.
The higher the volatility of the stock portfolio, the higher the
proportion of the portfolio to be initially hedged.
Adjustments to the futures position depends upon:
A. The floor that is chosen relative to the portfolio value.
B. New information about the volatility of the stock price.
Higher the volatility leads to larger futures positions.
Chapter 10
26
Index Futures and Stock Market
Volatility
Has stock market volatility increased since the introduction
of stock index futures trading?
1. Stock index futures have been alleged to cause market
volatility due to index arbitrage and portfolio insurance
practices.
– Evidence suggest that worldwide financial volatility has
generally decreased.
Even if proven that stock index futures trading did increase
stock market volatility, is that bad?
– In an efficient market, the price quickly adjusts to reflect
new information.
– Price volatility results from the arrival of new information in
the market.
– Economists often interpret volatile prices as evidence of
functioning efficient market.
Chapter 10
27
Index Arbitrage and Stock Market
Volatility
2. Critics argue that index arbitrage may lead to dramatic
volatility in the market and disrupted trading.
Recall that in index arbitrage, traders search for
discrepancies between stock prices and futures prices.
When the discrepancies are large enough to cover the
transaction costs, index arbitrageurs enter the market to
sell the overpriced side and buy the underpriced side.
This action may put large orders on the market at critical
times.
Chapter 10
28
Portfolio Insurance and Stock Market
Volatility
3. Portfolio insurance can also contribute to potential order
imbalances that might affect stock prices.
Assume a large drop in stock prices. This will cause the
following chain reaction:
A. Future prices will fall.
B. Portfolio insurers will place large numbers of orders
to sell index futures.
Critics argue that the large sell orders from portfolio
insurers might temporarily depress futures prices below the
price justified by the Cost-of-Carry Model, creating
disruptive chain reactions.
Chapter 10
29
Index Futures and Stock Crashes
October 19, 1987 Stock Market Crash
Dow Jones value drops by 22.61%
Heavy trading volume brought trade processing to a virtual
halt.
The inability of cash markets to handle the incredible order
flow contributed to the market turmoil.
The Cascade Theory was introduced from the Brady
Report.
The Cascade Theory was described as a vicious cycle
cause by index arbitrage and portfolio insurance.
Chapter 10
30
Index Futures and Stock Crashes
Cascade Theory
Low equity price 1
Portfolio Insurers liquidate equity
exposure by selling index futures
Future prices drop below equilibrium price.
Created reverse cash-and-carry arbitrage opportunity
Depress equity prices
New below equity price 2
Chapter 10
31
Index Futures and Stock Crashes
Figure 10.6 shows the spread between the cash and
futures using Chicago time.
Insert figure 10.6 here
Figure 10.6 indicates that on October 19, 1987 the stock
and futures basis did respond to the information that was
available.
Chapter 10
32