Risk Management using
Index Options and Futures
1

Outline
 Introduction
 Using options
 Using futures contracts
 Dynamic hedging
2

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
3


Hedging
Hedging removes risk. Hedging involves establishing a second position whose price behavior will likely offset the price behavior of the original portfolio.
 The objective of portfolio protection is the temporary removal of some or all the market risk associated with a portfolio. Portfolio protection techniques are generally more economic in terms of commissions and managerial time than the sale and eventual replacement of portfolio components.
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

Black-Scholes Formula
(European Options)
C

S e
0
 qT
( )
1

Ke
 rT
(
2
)
P

Ke
 rT
N (
 d
2
)

S e
0
 qT
N (
 d
1
) where d
1
 ln( S K r q
0
/ ) (

T

2
/ 2) T d
2
 d
1
 
T
8

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:
Delta
  

P

S where P

option premium
S

stock price
9

Importance of Delta (cont’d)
 Delta enables the portfolio manager to figure out the number of option contracts necessary to mimic the returns of the underlying security. This statistic is important in the calculation of many hedge ratios.
10

Importance of Delta (cont’d)
 Delta:
•
Equals N(d
1
) in the Black-Scholes Call price.
•
Equals N(-d
1
) in the Black-Scholes Put price.
•
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
11

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
12

Protective Put
Profit and Loss Diagram
 Assume the following information for ZZX:
13

Protective Put
Profit & Loss Diagram (cont’d)
 Long position for ZZX stock:
Profit or Loss
0
-50
$50
Stock Price at
Option Expiration
14

Protective Put
Profit & Loss Diagram (cont’d)
 Long position for SEP 45 put ($1 premium):
Profit or Loss
44
0
-1
Maximum
Gain = $44
$45
Maximum
Loss = $1
Stock Price at
Option Expiration
15

Protective Put
Profit & Loss Diagram (cont’d)
 Protective put diagram:
Profit or Loss
Maximum
Gain is unlimited
$45
0
-6
Maximum
Loss = $6
Stock Price at
Option Expiration
16

Protective Put
Profit & Loss Diagram (cont’d)
 Observations:
•
The maximum possible loss is $6
•
The potential gain is unlimited
17

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
18

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
19

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
20

Differences
 Protective puts provide protection against large price declines, whereas covered calls provide only limited downside protection.
Covered calls bring in the option premium, while the protective put requires a cash outlay.
21

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
22

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 Black-
Scholes OPM and solve for N(d
1
)
23

Index Options (cont’d)
 The hedge ratio is a calculated value indicating the number of puts necessary:
HR

Portfolio value
Contract "value"

Portfolio beta

1
Delta
24

Portfolio Insurance Example #1:
S&P 100 index Options (OEX)
 The OEX contract is the tool of choice for many professional portfolio risk managers.
 While the S&P 100 futures contract is similar to traditional agricultural futures, delivery does not occur, nor does it need to occur for this to be an effective hedging tool.
25

Index Options (cont’d)
Example
OEX 315 OCT puts are available for premium of $3.25.
The delta for these puts is –0.235, and the current index level is 327.19.
How many puts are needed to hedge a portfolio with a market value of $150,000 and a beta of 1.20?
26

Index Options (cont’d)
Example (cont’d)
Solution: You should buy 23 puts to hedge the portfolio:
HR

Portfolio value
Contract "value"

Portfolio beta

1
Delta

$150, 000
$32,719

1.20

1
0.235

23.41
27

Portfolio Insurance Example #2
 Suppose the manager must protect a
$500,000 portfolio with a beta of 2.
 Current riskless rate is 12%.
 Dividend yield (on both the portfolio and the market index) is 4%.
 S&P 500 index is currently 1,000.
 The manager must hedge the value that the portfolio will take three months from now.
28

Example #2 (Cont’d)
 Suppose that we want to insure $450,000 of the portfolio, i.e. the manager is willing to let the portfolio go down in value by
$50,000 but no further.
 What strike price for the protective put option should we choose?
29

Example #2 (Cont’d)
 Suppose that we want to insure $450,000 of the portfolio, i.e. the manager is willing to let the portfolio go down in value by
$50,000 but no further.
 What strike price for the protective put option should we choose?
30

Example #2 (Cont’d)

Terminal Portfolio Value “that we can live with”: $450,000.
 Change in value = (450-500)/500 = -10%
 Dividends Earned = (3/12)(4) = 1%
 Total Portfolio Return = -10+1 = -9%
 Note that 3-month R f
= (3/12)(12) = 3%
 CAPM: R portfolio
= R f
+beta(R index
-R f
)
31

Example #2 (Cont’d)
 Thus: R index
= (R portfolio
-R f
)/beta + R f
 So R index
= (-9-3)/2+3 = -3%
 Dividends from index = (3/12)(4) = 1%
 Change in index value = -3-1 = -4%
 Terminal index value = 1000(1-.04) = 960
 Therefore we need an index put option with a strike price of 960.
32

Example #2 (Cont’d)
 And the number of put option contracts that need to be purchased is:
 N = (2)500,000/[(1,000)(100)]=10

Note that we do not care about “delta” this time because we hold the option until maturity.
33

Example #2 (end)
 Also note that if the value of the index falls to
880 for example, we can compute the corresponding terminal portfolio value (using
CAPM) as about $370,000.
 The options pay (960-880)(10)(100) = $80,000
 Adding $80,000 to $370,000 brings the net terminal value to $450,000: our required level.
34

Using Futures Contracts
 Importance of financial futures
 Stock index futures contracts
 S&P 500 stock index futures contract
 Hedging with stock index futures
35

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 grows every year
36

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
37

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 most broad-based portfolios (ex: mutual funds).
38

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
39

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
40

Calculating A Hedge Ratio
 Computation
 The market falls
 The market rises
 The market is unchanged
41

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
42

Computation (cont’d)
 The futures hedge ratio is:
HR

Dollar value of portfolio
Dollar value of S&P contract

Beta
43

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?
44

Computation (cont’d)
Example (cont’d)
Solution: Calculate the hedge ratio:
HR

Dollar value of portfolio
Dollar value of S&P contract

Beta

$90, 000, 000

1.50

1, 529.75
45

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.
46

The Market Falls
 If the market falls:
•
There is a loss in the stock portfolio
•
There is a gain in the futures market
47

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.
48

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
49

The Market Rises
 If the market rises:
•
There is a gain in the stock portfolio
•
There is a loss in the futures market
50

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.
51

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
52

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 )
53

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
54

Dynamic Hedging
 Definition
 Dynamic hedging example
 The dynamic part of the hedge
 Dynamic hedging with futures contracts
55

Definition
Dynamic hedging involves monitoring a portfolio's position delta and readjusting this value as it deviates from a target number.
Example:
• 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
56

Dynamic Hedging Example
 Assume the following information for ZZX:
57

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
58

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
59

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
60

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
61

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)
62

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
63

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
64

Dynamic Hedging with
Futures Contract
 Appropriate for large portfolios
 Stock index futures have a delta of +1.0
65

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
66

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:
HR

Dollar value of portfolio
Dollar value of S&P contract

Beta

$52, 500, 000

1.0

300 contracts
67

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
68