1

Chapter 22

Benching the Equity Players

2

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

3

Outline Introduction Using options Using futures contracts Dynamic hedging

4

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

5

Using Options Introduction Equity options with a single security Index options

6

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

7

Equity Options with A Single Security

Importance of delta Protective puts Protective put profit and loss diagram Writing covered calls

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

where option premium

stock price

P

S

P

S

9

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

10

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

11

Protective Put Profit and Loss Diagram

Assume the following information for ZZX:

12

Protective Put Profit & Loss Diagram (cont’d) Long position for ZZX stock:

0

-50$50

Stock Price at Option Expiration

Profit or Loss

13

Protective Put Profit & Loss Diagram (cont’d) Long position for SEP 45 put ($1

premium):

0

44

$45

Stock Price at Option Expiration

Profit or Loss Maximum Gain = $44

Maximum Loss = $1

-1

14

Protective Put Profit & Loss Diagram (cont’d) Protective put diagram:

0$45

Stock Price at Option Expiration

Profit or Loss Maximum Gain is unlimited

Maximum Loss = $6

-6

15

Protective Put Profit & Loss Diagram (cont’d) Observations:

• The maximum possible loss is $6

• The potential gain is unlimited

16

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

17

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

18

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

19

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

20

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(d1)

21

Index Options (cont’d) The hedge ratio is a calculated value

indicating the number of puts necessary:

Portfolio value 1Portfolio beta

Contract "value" Delta

where contract value = strike price 100

HR

22

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?

23

Index Options (cont’d)Example (cont’d)

Solution: You should buy 25 puts to hedge the portfolio:

Portfolio value 1Portfolio beta

Contract "value" Delta

$150,000 11.20

$31,500 0.235

24.32

HR

24

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

25

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

26

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

27

S&P 500 Stock Index Futures Contract

28

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

29

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

30

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

31

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

32

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

33

Computation (cont’d) The futures hedge ratio is:

Dollar value of portfolio Beta

Dollar value of S&P contractHR

34

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?

35

Computation (cont’d)Example (cont’d)

Solution: Calculate the hedge ratio:

Dollar value of portfolio Beta

Dollar value of S&P contract$90,000,000

1.50$250 $353

1,529.75

HR

36

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.

37

The Market Falls If the market falls:

• There is a loss in the stock portfolio

• There is a gain in the futures market

38

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.

39

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

40

The Market Rises If the market rises:

• There is a gain in the stock portfolio

• There is a loss in the futures market

41

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.

42

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

43

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)

44

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

45

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

46

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

47

Dynamic Hedging Example Assume the following information for ZZX:

48

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

49

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

50

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

51

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

52

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)

53

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

54

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

55

Dynamic Hedging with Futures Contract

Appropriate for large portfolios

Stock index futures have a delta of +1.0

56

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

57

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 Beta

Dollar value of S&P contract$52,500,000

1.0$250 700

300 contracts

HR

58

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