1. Risk Management using Index Options and Futures

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. 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.

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. Black-Scholes Formula (European Options)

9. 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:

10. 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.

11. Importance of Delta (cont’d)
Equals N(d1) in the Black-Scholes Call price.
Equals -N(-d1) 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

12. 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

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

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

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

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

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

18. 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

19. 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

20. 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

21. 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.

22. 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

23. 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)

24. Index Options (cont’d)
The hedge ratio is a calculated value indicating the number of puts necessary:

25. 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.

26. 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
How many puts are needed to hedge a portfolio with a market value of $150,000 and a beta of 1.20?

27. Index Options (cont’d)
Example (cont’d)
Solution: You should buy 23 puts to hedge the portfolio:

28. 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.

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)
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?

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

32. Example #2 (Cont’d) Thus: Rindex = (Rportfolio-Rf)/beta + Rf
So Rindex = (-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.

33. 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.

34. 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 ( )(10)(100) = $80,000
Adding $80,000 to $370,000 brings the net terminal value to $450,000: our required level.

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

36. 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

37. 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

38. 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).

39. 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

40. 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

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

42. 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

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

44. Computation (cont’d) Example
You are managing a $90 million portfolio with a beta of 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
How many S&P 500 stock index futures should you short to hedge the portfolio?

45. Computation (cont’d) Example (cont’d)
Solution: Calculate the hedge ratio:

46. 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.

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

48. The Market Falls (cont’d)
Example
Consider the previous example. Assume that the S&P 500 index is currently at a level of Over the next few months, the S&P 500 index falls to
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

49. 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

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

51. The Market Rises (cont’d)
Example
Consider the previous example. Assume that the S&P 500 index is currently at a level of Over the next few months, the S&P 500 index rises to to
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

52. 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

53. 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)

54. 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

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

56. 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

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

58. 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

59. 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

60. 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

61. 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

62. 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 ( = 509 shares)

63. 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

64. 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

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

66. 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

67. 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:

68. 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