Chapter 22
Benching the Equity Players
1
Don’t be discouraged by a failure. It can be a
positive experience. Failure is, in a sense, the
highway to success, inasmuch as every discover of
what is false leads us to seek earnestly after what
is true, and every fresh experience points out some
form of error which we shall afterwards carefully
avoid.
- John Keats
2
Outline
 Introduction
 Using
options
 Using futures contracts
 Dynamic hedging
3
Introduction
 Portfolio
protection involves adding
components to a portfolio in order to
establish a floor value for the portfolio
using:
• Equity or stock index put options
• Futures contracts
• Dynamic hedging
4
Using Options
 Introduction
 Equity
options with a single security
 Index options
5
Introduction
 Options
enable the portfolio manager to
adjust the characteristics of a portfolio
without disrupting it
 Knowledge
of options improves the
portfolio manager’s professional
competence
6
Equity Options with
A Single Security
 Importance
of delta
 Protective puts
 Protective put profit and loss diagram
 Writing covered calls
7
Importance of Delta
 Delta
is a measure of the sensitivity of the
price of an option to changes in the price of
the underlying asset:
P
Delta   
S
where P  option premium
S  stock price
8
Importance of Delta (cont’d)
 Delta:
• Equals N(d1) in the Black-Scholes OPM
• Allows us to determine how many options are
needed to mimic the returns of the underlying
security
• Is positive for calls and negative for puts
• Has an absolute value between 0 and 1
9
Protective Puts
 A protective
put is a long stock position
combined with a long put position
 Protective
puts are useful if someone:
• Owns stock and does not want to sell it
• Expects a decline in the value of the stock
10
Protective Put
Profit and Loss Diagram
 Assume
the following information for ZZX:
11
Protective Put
Profit & Loss Diagram (cont’d)
 Long
position for ZZX stock:
Profit or Loss
0
Stock Price at
Option Expiration
-50
$50
12
Protective Put
Profit & Loss Diagram (cont’d)
 Long
Profit or Loss
44
0
position for SEP 45 put ($1 premium):
Maximum
Gain = $44
$45
-1
Maximum
Loss = $1
Stock Price at
Option Expiration
13
Protective Put
Profit & Loss Diagram (cont’d)
 Protective
put diagram:
Profit or Loss
0
-6
Maximum
Gain is unlimited
$45
Maximum
Loss = $6
Stock Price at
Option Expiration
14
Protective Put
Profit & Loss Diagram (cont’d)
 Observations:
• The maximum possible loss is $6
• The potential gain is unlimited
15
Protective Put
Profit & Loss Diagram (cont’d)
 Selecting
the striking price for the
protective put is like selecting the
deductible for your stock insurance
• The more protection you want, the higher the
premium
16
Writing Covered Calls
 Writing
covered calls is an alternative to
protective puts
• Appropriate when an investor owns the stock,
does not want to sell it, and expects a decline in
the stock price
• An imperfect form of portfolio protection
17
Writing Covered Calls (cont’d)
 The
premium received means no cash loss
occurs until the stock price falls below the
current price minus the premium received
 The
stock price could advance and the
option could be called
18
Index Options
 Investors
buying index put options:
• Want to protect themselves against an overall
decline in the market or
• Want to protect a long position in the stock
19
Index Options (cont’d)
 If
an investor has a long position in stock:
• The number of puts needed to hedge is
determined via delta (as part of the hedge ratio)
• He needs to know all the inputs to the BlackScholes OPM and solve for N(d1)
20
Index Options (cont’d)
 The
hedge ratio is a calculated value
indicating the number of puts necessary:
Portfolio value
1
HR 
 Portfolio beta 
Contract "value"
Delta
where contract value = strike price 100
21
Index Options (cont’d)
Example
OEX 315 OCT puts are available for premium of $3.25.
The delta for these puts is –0.235.
How many puts are needed to hedge a portfolio with a
market value of $150,000 and a beta of 1.20?
22
Index Options (cont’d)
Example (cont’d)
Solution: You should buy 25 puts to hedge the portfolio:
Portfolio value
1
HR 
 Portfolio beta 
Contract "value"
Delta
$150, 000
1

1.20 
$31,500
0.235
 24.32
23
Using Futures Contracts
 Importance
of financial futures
 Stock index futures contracts
 S&P 500 stock index futures contract
 Hedging with stock index futures
24
Importance of
Financial Futures
 Financial
futures are the fastest-growing
segment of the futures market
 The
number of underlying assets on which
futures contracts are available seems grows
every year
25
Stock Index Futures Contracts

A stock index futures contract is a promise to buy
or sell the standardized units of a specific index at
a fixed price by a predetermined future date

Stock index futures contracts are similar to the
traditional agricultural contracts except for the
matter of delivery
• All settlements are in cash
26
S&P 500 Stock
Index Futures Contract
27
Hedging with
Stock Index Futures
 With
the S&P 500 futures contract, a
portfolio manager can attenuate the impact
of a decline in the value of the portfolio
components
 S&P 500 futures can be used to hedge:
• Endowment funds
• Mutual funds
• Other broad-based portfolios
28
Hedging with
Stock Index Futures (cont’d)
 To
hedge using S&P stock index futures:
• Take a position opposite to the stock position
– E.g., if you are long in stock, short futures
• Determine the number of contracts necessary to
counteract likely changes in the portfolio value
using:
– The value of the appropriate futures contract
– The dollar value of the portfolio to be hedged
– The beta of your portfolio
29
Hedging with
Stock Index Futures (cont’d)
 Determine
the value of the futures contract
• The CME sets the size of an S&P 500 futures
contract at $250 times the value of the S&P 500
index
• The difference between a particular futures
price and the current index is the basis
30
Calculating A Hedge Ratio
 Computation
 The
market falls
 The market rises
 The market is unchanged
31
Computation
 A futures
hedge ratio indicates the number
of contracts needed to mimic the behavior
of a portfolio
 The hedge ratio has two components:
• The scale factor
– Deals with the dollar value of the portfolio relative
to the dollar value of the futures contract
• The level of systematic risk
– I.e., the beta of the portfolio
32
Computation (cont’d)
 The
futures hedge ratio is:
Dollar value of portfolio
HR 
 Beta
Dollar value of S&P contract
33
Computation (cont’d)
Example
You are managing a $90 million portfolio with a beta of
1.50. The portfolio is well-diversified and you want to
short S&P 500 futures to hedge the portfolio. S&P 500
futures are currently trading for 353.00.
How many S&P 500 stock index futures should you short
to hedge the portfolio?
34
Computation (cont’d)
Example (cont’d)
Solution: Calculate the hedge ratio:
Dollar value of portfolio
HR 
 Beta
Dollar value of S&P contract
$90, 000, 000

 1.50
$250  $353
 1,529.75
35
Computation (cont’d)
Example (cont’d)
Solution: The hedge ratio indicates that you need 1,530
S&P 500 stock index futures contracts to hedge the
portfolio.
36
The Market Falls
 If
the market falls:
• There is a loss in the stock portfolio
• There is a gain in the futures market
37
The Market Falls (cont’d)
Example
Consider the previous example. Assume that the S&P 500
index is currently at a level of 348.76. Over the next few
months, the S&P 500 index falls to 325.00.
Show the gains and losses for the stock portfolio and the
S&P 500 futures, assuming you close out your futures
position when the S&P 500 index is at 325.00.
38
The Market Falls (cont’d)
Example (cont’d)
Solution: For the $90 million stock portfolio:
-6.81% x 1.50 x $90,000,000 = $9,193,500 loss
For the futures:
(353 – 325) x 1,530 x $250 = $10,710,000 gain
39
The Market Rises
 If
the market rises:
• There is a gain in the stock portfolio
• There is a loss in the futures market
40
The Market Rises (cont’d)
Example
Consider the previous example. Assume that the S&P 500
index is currently at a level of 348.76. Over the next few
months, the S&P 500 index rises to to 365.00.
Show the gains and losses for the stock portfolio and the
S&P 500 futures, assuming you close out your futures
position when the S&P 500 index is at 365.00.
41
The Market Rises (cont’d)
Example (cont’d)
Solution: For the $90 million stock portfolio:
4.66% x 1.50 x $90,000,000 = $6,291,000 gain
For the futures:
(365 – 353) x 1,530 x $250 = $4,590,000 loss
42
The Market Is Unchanged
 If
the market remains unchanged:
• There is no gain or loss on the stock portfolio
• There is a gain in the futures market
– The basis will deteriorate to 0 at expiration (basis
convergence)
43
Hedging in Retrospect
 Futures
hedging is never perfect in practice:
• It is usually not possible to hedge exactly
– Index futures are available in integer quantities only
• Stock portfolio seldom behave exactly as their
betas say they should
 Short
hedging reduces profits in a rising
market
44
Dynamic Hedging
 Definition
 Dynamic
hedging example
 The dynamic part of the hedge
 Dynamic hedging with futures contracts
45
Definition
 Dynamic
hedging strategies:
• Attempt to replicate a put option
• By combining a short position with a long
position
• To achieve a position delta equal to that which
would be obtained via protective puts
46
Dynamic Hedging Example
 Assume
the following information for ZZX:
47
Dynamic Hedging
Example (cont’d)
You own 1,000 shares of ZZX stock
 You are interested in buying a JUL 50 put for
downside protection
 The JUL 50 put expires in 60 days
 The JUL 50 put delta is –0.435
 T-bills yield 8 percent
 ZZX pays no dividends
 ZZX stock’s volatility is 30 percent

48
Dynamic Hedging
Example (cont’d)
 The
position delta is the sum of all the
deltas in a portfolio:
• (1,000 x 1.0) + (1,000 x –0.435) = 565
– Stock has a delta of 1.0 because it behaves like itself
– A position delta of 565 behaves like a stock-only
portfolio composed of 565 shares of the underlying
stock
49
Dynamic Hedging
Example (cont’d)
 With
the puts, the portfolio is 56.5 percent
as bullish as without the puts
 You
can sell short 435 shares to achieve the
position delta of 565:
• (1,000 x 1.0) + (435 x –1.0) = 565
50
The Dynamic Part of the Hedge
 Suppose
that one week passes and:
• ZZX stock decline to $49
• The delta of the JUL 50 put is now –0.509
• The position delta has changed to:
– (1,000 x 1.0) + (1,000 x –0.509) = 491
51
The Dynamic Part
of the Hedge (cont’d)
 To
continue dynamic hedging and to
replicate the put, it is necessary to sell short
74 shares (435 + 74 = 509 shares)
52
The Dynamic Part
of the Hedge (cont’d)
 Suppose
that one week passes and:
• ZZX stock rises to $51
• The delta of the JUL 50 put is now –0.371
• The position delta has changed to:
– (1,000 x 1.0) + (1,000 x –0.371) = 629
53
The Dynamic Part
of the Hedge (cont’d)
 To
continue dynamic hedging and to
replicate the put, it is necessary to cover 64
of the 435 shares you initially sold short
54
Dynamic Hedging with
Futures Contract
 Appropriate
 Stock
for large portfolios
index futures have a delta of +1.0
55
Dynamic Hedging with
Futures Contract (cont’d)
 Assume
that:
• We wish to replicate a particular put option
with a delta of –0.400
• We manage an equity portfolio with a beta of
1.0 and $52.5 million market value
• A futures contract sells for 700
– The dollar value is $250 x 700 = $175,000
56
Dynamic Hedging with
Futures Contract (cont’d)
 We
must sell enough futures contracts to
pull the position delta to 0.600
 The hedge ratio is:
Dollar value of portfolio
HR 
 Beta
Dollar value of S&P contract
$52,500, 000

1.0
$250  700
 300 contracts
57
Dynamic Hedging with
Futures Contract (cont’d)
 If
the hedge ratio is 300 contracts, we must
sell 40% x 300 = 120 contracts to achieve a
position delta of 0.600
58