Hedging risk with Derivatives

1
Hedging risk with Derivatives

• Review of equity options
• Review of financial futures
• Using options and futures to hedge portfolio risk
• Introduction to Hedge Funds

2
Options -- Contract

• Calls and Puts
• Underlying Security (Number of Units)
• Exercise or Strike Price
• Expiration date
• Option Premium
• American, European, Asian, etc.

3
Options -- Markets

• 1 Buyer 1 Seller (writer) 1 Contract
• Examples of Price Quotations
• Premium Intrinsic Value Time Prem
• Options available on
• Equities
• Indicies
• Foreign Currencies
• Futures

4
Options -- Basic Strategies

• Buy Call
• Sell (write) Call
• Buy Put
• Sell (write) Put

5
Options -- Advanced Strategies

• Straddle
• Strips and Straps
• Vertical Spreads
• Bullish
• Bearish

6
Options - Determinants of Value

• Value of Underlying Asset
• Exercise Price
• Time to Expiration
• VOLATILITY
• Interest Rates
• Dividends

7
Options -- Black Scholes Option Pricing Model

• C SN(d1) - Xe-rTN(d2) ln(S/X)
  (rs2/2)T d1 ---------------------------
  sT1/2 d2 d1 - sT1/2
• Put-Call Parity P C Xe-rT - S

8
Futures Contract

• Agreement to make (sell) or take (buy) delivery
  of a prespecified quantity of an asset at an
  agreed upon price at a specific future date.
• ex. SP 500 Index Futures
• Price 1126.10 Delilvery month June
• Buyer agrees to purchase a portfolio representing
  the SP 500 (or its cash equivalent) for 1126.10
  x 250 281,525 on Thursday prior to 3rd Friday
  in June. (Buyer is locking in the purchase price
  for the portfolio.)
• Seller agrees to deliver the portfolio described
  above.
• Note since this is a cash settled contract, if
  the price was 1116.10 on the delivery date, the
  buyer would pay the seller 2,500 ( 10 x 250).
  If the price was 1136.10, the seller would pay
  the buyer 2,500

9
Futures Contract Marking to Market

• Marking to market
• Price of Futures contract is reset every day
• Gains/Losses versus previous day are posted to
  buyer and seller margin accounts
• Futures a bundle of consecutive 1-day forward
  contracts
• If futures held to expiration, effective delivery
  price is same as when contract initiated

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Futures Contract Marking to Market example (C
contract)
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Index Futures Market

• Speculators often sell index futures when they
  expect the underlying index to depreciate, and
  vice versa.

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Index Futures Market

• Index futures may be sold by investors to hedge
  risk associated with securities held.

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Index Futures Market

• Most index futures contracts are closed out
  before their settlement dates (99).
• Brokers who fulfill orders to buy or sell futures
  contracts earn a transaction or brokerage fee in
  the form of the bid/ask spread.

14
Hedging with Derivatives

• Basic option strategies
• Covered call
• Protective put
• Synthetic short
• Basic futures strategies
• Using interest rate futures to reduce risk

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Covered Call

• Sell call on stock you own. (Long stock, short
  call)
• Good
• As value of stock falls, loss is partially offset
  by premium received on calls sold.
• Essentially costless since hedge generates a cash
  inflow
• Bad
• Maximum inflow from call premium Hedge is less
  effective for large drop in stock price
• If stock price rises, call will be exercised
  Investor transfers gains on stock to holder of
  call.

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Protective Put

• Buy put on stock you own. (Long stock, long put)
• Good
• As value of stock falls, loss is partially offset
  by gain in value of put. Gain from put continues
  to grow as stock price falls.
• If stock price rises, maximum loss on put
  premium Investor keeps all stock gains less
  fixed put premium.
• Bad
• More expensive to hedge with put

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Synthetic Short

• Sell call and buy put on stock you own. (Long
  stock, short call, long put)
• Good
• As value of stock falls, loss is offset by gain
  in value of put. Gain from put continues to grow
  as stock price falls.
• If stock price rises, gain is offset by loss on
  call. Loss from call continues to grow as stock
  price rises.
• Very effective hedging device
• Can be self-financing (premium received on put
  sold offsets premium paid on call purchased)
• Bad
• Often more expensive than simply shorting the
  stock itself.

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Delta Hedging with Options

• Call Delta DC dC/dS
• From Black-Scholes model,
• DC N(d1)
• Ex. If S74.49, X75, r1.67, s 38.4,
• t0.1589 yrs.
• Then, C 4.40 and N(d1) 0.5197
• If S increases by 1, C increases by 0.5197
• Hedge Ratio H 1/DC 1/0.5197 1.924
• Sell 1.924 calls per share of stock held to
  hedge!

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Example of Call Hedge Held to Expiration, 1000
share stock position
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Delta Hedging - Puts

• Put Delta DP dP/dS
• From Black-Scholes model and Put-Call Parity,
• DP DC 1 N(d1) - 1
• Ex. If S74.49, X75, r1.67, s 38.4,
• t0.1589 yrs.
• Then, C 4.40, P 4.71, N(d1) 0.5197,
• and N(d1) -1 -0.4803
• If S increases by 1, P decreases by 0.4803
• Hedge Ratio H 1/D 1/0.4803 2.082
• Buy 2.082 puts per share of stock held to hedge!

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Example of Put Hedge Held to Expiration, 1000
share stock position
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Delta Hedging with Options

• Delta changes over time!
• S changes
• Time declines
• Other factors (r, s) may change

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True Delta Hedging Adjust hedge when S changes

• Scenarios 1 2
• IBM stock drops by 1 to 73.49 gt Loss of
  1000
• Call options also drop by 0.5197 gt Gain of
  1037.97 gtNet change 37.97
• IBM stock rises by 1 to 75.49 gt Gain of
  1000
• Call options also rise by 0.5193 gt Loss of
  1037.97
• gt Net change (37.97)

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True Delta Hedging Adjust hedge when t changes

• Scenario 3
• One week passes, IBM stock at 71.49 gt Loss
  of 3000
• Call options now worth 2.73 gt
  Gain of 3173 gtNet change 173
• New call delta 0.4029
• New hedge ratio 1/0.4029 2.482 gt Sell 5
  more contracts!
• Scenario 4
• One week passes, IBM stock at 77.49 gt Gain
  of 3000
• Call options now worth 5.82 gt
  Loss of 2698
• gt Net change (302)
• New call delta 0.6238
• New hedge ratio 1/0.6238 1.603 gt Buy 3
  contracts!

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True Delta Hedging Adjust hedge when S changes

• Scenarios 1 2
• IBM stock drops by 1 to 73.49 gt Loss of
  1000
• Put options also rise by 0.4803 gt Gain of
  1008.63 gtNet change 8.63
• IBM stock rises by 1 to 75.49 gt Gain of
  1000
• Put options also fall by 0.4803 gt Loss of
  1008.63
• gt Net change (8.63)

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True Delta Hedging Adjust hedge when t changes

• Scenario 3
• One week passes, IBM stock at 71.49 gt Loss
  of 3000
• Put options now worth 6.06 gt
  Gain of 2835 gtNet change (165)
• New put delta 0.4028 1 -0.5972
• New hedge ratio 1/0.5972 1.674 gt Sell 4
  contracts!
• Scenario 4
• One week passes, IBM stock at 77.49 gt Gain
  of 3000
• Put options now worth 3.15 gt
  Loss of 3276
• gt Net change (276)
• New put delta 0.6238 1 -0.3762
• New hedge ratio 1/0.3762 2.658 gt Buy 5
  more contracts!

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Delta Hedging with options

• Delta represents response of call (or put) price
  with change in the stock price
• Delta changes as stock price, time to expiration,
  interest rates, volatility change
• It is too expensive to hedge individual stock
  positions with matching options. It is more
  common to hedge a portfolio with index options
  (cross hedging)
• Most managers monitor delta itself to decide when
  to rebalance.

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A True Protective Put

• Puts can be used to build a floor under the value
  of a long position
• Buy 1 put per long share
• Ex. Long 1000 shares of IBM at 74.49
• Buy 1000 puts at 4.71
• Puts guarantee a value of 75 per share
• This is insurance, not a hedge!

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A True Protective Put
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Hedging with Futures (example from May 2001)

• There are futures on the SP500. Suppose I have
  a portfolio that is currently worth 1,117,672.
  The portfolio has a beta of 1.3.
• June SP500 futures are at 1430.70
• gt contract is worth 500 x 1430.70 715,350
• Hedge ratio
• (Value of portfolio / Value of Futures
  contract)(Portfolio Beta)
• (1,117,672/715,350)(1.3) 2.031 gt Sell 2
  Contracts !

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Hedging with Futures (example from May 2001)
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Adjusting Systematic Risk with Futures

• PM may choose to adjust systematic exposure up or
  down to reflect
• investor desires
• expectations of market movements
• About index futures
• Represents contract to make/take delivery of a
  portfolio represented by the index
• Since index itself may be non-investable, most
  index futures contracts are cash-settled
• example
• SP500 futures CME contract value 250 x index
• Initial margin 6K for spec, 2.5K for hedgers.

33
Adjusting Systematic Risk with Futures

• I have an 11 million stock portfolio with
  b1.05. I want to increase b to 1.2.
• Value of Futures 1314.50 x 250 328,625
• bf 1.0.
• Target b contribution from portfolio
  contribution from futures
• 1.2 (1.0)(1.05) (F x 328,625)/11,000,000(1.
  0)
• F (bT - Wsbs)(Vs/VF)
• F 5.02 gt buy 5 contracts
• What have we done?
• Used futures contracts to leverage holdings and
  increase exposure to market risk

34
Adjusting Systematic Risk with Futures

• Suppose target b .90
• 0.90 (1.0)(1.05) (F x 328,625)/11,000,000(1
  .0)
• F (.90 - 1.05)(33.4728)(1.0) -5.02 contracts
  (sell)
• We have shorted futures to reduce systematic
  exposure.

35
Hedging with Interest Rate Futures

• How do you reduce duration for a bond portfolio?
• Sell high D, buy low D
• Sell bonds, buy Tbills
• Sell interest rate futures
• Interest rate futures agreement to make/take
  delivery of a fixed income asset on a particular
  date for an agreed upon price
• ex Sept Tbond futures contract
• 100K FV US Treas bonds with 15-years to maturity
  and 8 coupon (what if they don't exist?)
• Price 99-27 99 27/32 of 100,000
  998,437.50
• (Tick 31.25) D 8.64 years

36
Hedging with Interest Rate Futures

• I own an 11,000,000 face value portfolio of high
  grade US corporate bonds with an aggregate value
  of 101-08 (or 11,137,500) and a duration of 7.7
  years.
• I expect rates to rise. How can I immunize my
  portfolio?
• Target D contribution of bond port
  contribution of fut.
• 0 (1.0)(7.7) (F x 998,437.50)/11,137,500(8.6
  4)
• F (0.0 - (1.0)(7.7))(11,137,500/998,437.50)/8.64
  
• F -9.94 contracts gt short 10 Tbond futures
  contracts
• This is the weighted average duration approach