Option Hedging Strategies

1
Option Hedging Strategies

• Lecture Notes for FIN 653
• Yea-Mow Chen
• Department of Finance
• San Francisco State University

2
I. Determinants of Option Pricing

• Option pricing using the Black-Scholes Model
•  
• c SN(d1) - (Ee-rt)N(d2)
•   where
•   d1 ln(S/E) (r ?2/2)t/ ?? t
•   d2 ln(S/E) (r -(?2 /2)t/ ?? t.

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I. Determinants of Option Pricing

• Key Option Pricing Determinants and their Impacts
  on Option Prices
• --------------------------------------------------
  --------------------------------------------------
  ------------------
• European American European American
• Calls Calls Puts Puts
• --------------------------------------------------
  --------------------------------------------------
  ------------------
• 1. Exercise Price - -
  
• 2. Time to Maturity NA NA
• 3. Underlying security price
  - -
• 4. Underlying security price
  
• Volatility
• 5. Dividend policy - -
• 6. The risk-free interest rate
  - -
• --------------------------------------------------
  --------------------------------------------------
  -------------------
•  

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II. OPTION DERIVATIVES

• 1. Delta The delta is defined as the rate of
  change of an option price with respect to the
  price of the underlying asset. It is the slope
  of the curve that relates the option price to the
  underlying asset price.
• Deltacall ?c/?S N(d1)
• where ?S a small change in the stock
  price
• ?c the corresponding change in the call
  price.
• Notice Delta for a put contract is
• Deltaput ?p/?S N(d1) -1.

5
Delta (See Figure 15.2, page 303)

• Delta (D) is the rate of change of the option
  price with respect to the underlying

6
II. OPTION DERIVATIVES

• For example If Eurodollar futures advanced 10
  ticks, a call option on the futures whose delta
  is .30 would increase only 3 ticks. Similarly,
  a call option whose delta is .11 would increase
  in value approximately 1 tick.
• The delta for a European call on a
  non-dividend-paying stock is N(d1), and for a
  European put is N(d1) -1. The delta for a call
  is positive, ranging in value from approximately
  0 for deep out-of-the-money calls to
  approximately 1 for deep in-the-money ones. In
  contrast, the delta for a put is negative,
  ranging from approximately 0 to -1.

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II. OPTION DERIVATIVES

• Deltas change in response not only to stock price
  changes, but also to the time to expiration. As
  the time to expiration decreases, the delta of an
  in-the-money call or put increases, while an
  out-of-the-money call or put tends to decrease.
•  
• Delta also can be used to measure the probability
  that the option will be in the money at
  expiration. Thus, the call with a delta N(d1)
  .40 has an approximately 40 chance that its
  stock price will exceed the options exercise
  price at expiration.

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II. OPTION DERIVATIVES

• Delta Neutral
• If the delta of a call is 0.4, then the short
  position in the call will lose .40 if the stock
  price increases by 1. Equivalently, if the
  short seller purchased 0.4 shares, then the
  position would be immunized against instantaneous
  local changes in the price. It is therefore
  possible to construct a strategy where the
  total delta position on the long side and total
  delta position on the short side are equal.

10
II. OPTION DERIVATIVES

• EX If an investor sold 20 call options with a
  delta of 0.6. The current premium on the option
  is 10 and the spot price of the underlying asset
  is 100. How can he hedge by creating a delta
  neutral hedge?
• Answer The investors position should be hedged
  by purchasing 0.62,000 1,200 shares. The gain
  (loss) on the option position would then be
  offset by the loss(gain) on the stock position.
  The delta of a stock is 1.00. The sum of deltas
•   Short 20 call options Long 1,200 shares
• -(20 0.6) (12 1.0)
• 0

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II. OPTION DERIVATIVES

• The investors position only remains delta hedged
  for relatively short period of time. This is
  because delta changes, in responding to changes
  in the spot price and the time to expiration. In
  practice when delta hedging is implementing, the
  hedge has to be adjusted periodically. This is
  known as rebalancing.
• For example, after 3 days, the stock price
  increased to 110, which increased the delta to
  0.65. This means that an extra 0.05 2,000
  100 shares would have to be purchased to maintain
  the hedge. Hedge schemes such as this that
  involves frequent adjustments are known as
  dynamic hedging schemes.

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II. OPTION DERIVATIVES

• Ex Dynamic Delta Hedge
• A stock is priced at 50. Its volatility is 38
  percent per year. Interest rates are 5 percent
  per year. A five-week at-the money European call
  option is priced at 2.47. The delta value of
  the option is 0.5625.
• To construct a delta hedge requires purchasing ?
  shares of stock. Consider an investor who has
  sold 10,000 call option. To immunize this
  position against a small instantaneous change in
  the stock price, the investor needs to purchase
  5,625 shares of the stock. Assume all these
  shares are financed by borrowing at the risk free
  rate.

13
II. OPTION DERIVATIVES

• With four weeks for maturity, the stock price
  increased by 50 cents, and the delta value
  changed by 0.0103. This implied that 103
  additional shares had to be purchased to maintain
  the delta-neutral position. All purchases are
  financed by borrowing.
• In this example, the option expired in the money,
  and the total number of shares held by the trader
  increased from 5,625 to 10,000. The trader
  receives 50 per share for these stocks. This
  leaves a net obligation of 13,985. Offsetting
  this loss is the premium taken in from the sale
  of the 10,000 call options (assuming one share
  per option). This revenue is 24,700, which, if
  invested at the riskless rate over the five
  weeks, would grow to 24,819. Hence, the delta
  hedging scheme leads to a profit of 10,834.

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II. OPTION DERIVATIVES

•  
• Time to Stock Delta Change in
  Shares Cost of Cumulative
• Expiration Price Delta Purchased
  Shares Cost
• (weeks) () or
  Sold () ()
• __________________________________________________
  _____________
• 5 50.00 0.5625 -
  5,625 281,250.00 281,250
• 4 50.50 0.5728 0.0103 103
  5,201.50 286,722
• 3 51.25 0.6361 0.0633 633
  32,441.25 319,439
• 2 51.00 0.6289 -0.0072
  -72 -3,672.00 316,074
• 1 52.25 0.8108 0.1819 1,819
  95,042.75 411,421
• 0 54.00 1 0.1892 1,892
  102,168.00 513,985
•  _________________________________________________
  _____________

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II. OPTION DERIVATIVES

• Calculation Cumulative Cost 513,985
• - Call Excise Price 500,000
  (5010,000)
• ________________
• Loss - 13,985
• Premium Income 24,819
• ________________
• Net profit 10,834

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II. OPTION DERIVATIVES

• 2. Gamma
• Gamma is the second derivative of the option
  premium with respect to the stock price. It tells
  you how much the delta will change when the stock
  price increases or decreases.
• The gamma value is also refereed to as the
  curvature, since it measures the curvature of the
  option price with respect to the stock price.
•   ?2 C N?(d1)
• ? ------------ ----------------
• ? S2 S0 ?? T

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II. OPTION DERIVATIVES

• The gamma of a call or put varies with respect to
  the stock price and time to maturity. It can
  increase dramatically as the time to expiration
  decreases. Gamma values are largest for
  at-the-money options and smallest for
  deep-in-the-money and deep-out-of-the-money
  options.

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II. OPTION DERIVATIVES

• Ex Suppose delta 50 and gamma 5 if the
  stock price increases by 1.00 then the delta
  will increase by 5 percentage points to 55 (50
  5). In other words, the option premium will
  increase or decrease in value at the rate of 50
  of the stock price before the 1.00 point move,
  and 55 after the 1.00 point move.

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II. OPTION DERIVATIVES

• Another Example
• A stock is priced at 50. The volatility of the
  stock is 30 percent, and interest rates are 5
  percent. A three-month at-the-money European call
  option trades at 3.27. The delta value is
  0.5625, and the gamma value is 0.0529. If the
  stock changes price by 1 cent, the change in the
  option price should be (0.5625)(0.01)
  0.005625. The new delta value will not be
  0.5625 but will be 0.5625 (0.0529)(0.01)
  0.563029.

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II. OPTION DERIVATIVES

• 3. Theta The theta is the first derivative of
  the option premium with respect to time. It
  measures time decay - the amount of premium lost
  as another day passes.
•  
• ?C S0N(d1)?
• ?c - -------- -------------- - r
  E e-rT N(d2)
• ?T 2? T
•  

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II. OPTION DERIVATIVES

• The theta value for call options on nondividend
  stocks is always negative. This is because as
  time to maturity decreases, the option becomes
  less valuable. Stock options with large negative
  theta values can lose their time premium rapidly.
  The value changes the most as maturity
  approaches.
• Put options usually have negative thetas as well.
  However, deep-in-the-money European puts could
  have positive thetas.

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II. OPTION DERIVATIVES

• EX Assume a premium of 1.00 and a theta of
  0.04. You would expect the premium to lose 4
  points by tomorrow, to 0.96, assuming that no
  other variables have changed.

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II. OPTION DERIVATIVES
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II. OPTION DERIVATIVES

• Example
• Reconsider our stock priced at 50. The
  volatility of the stock is 30 percent and
  interest rates are 5 percent. A three-month
  at-the-money European call option trades at
  3.27. The theta value of this option is -7.196
  per year. If this number is divided by 52, the
  resulting number (0.1384) represents the
  expected drop in the price of the option if the
  stock price remains unchanged for one week.

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II. OPTION DERIVATIVES

• 4. Vega The vega is the first derivative of the
  option premium with respect to volatility. It
  measures the dollar change in the value of option
  when the underlying implied volatility increases
  by one percentage point.
•   ?C
• ? ----------- S ?T N(d1)
• ??
• European puts with the same terms have the same
  vega values. A change in volatility will give
  the greatest total dollar effect on at-the-money
  options and the greatest percentage effect on
  out-of-the-money options.

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II. OPTION DERIVATIVES

• EX If the implied volatility is 20, the call
  premium is 2.00, and the vega is 0.12, then you
  would expect the premium to increase to 2.12
  (2.00 0.12) when implied volatility moves up
  to 21. Vega gives you an idea of how sensitive
  the option premium is to perceived changes in
  market value.
•  

28
II. OPTION DERIVATIVES

• Reconsider our stock priced at 50, with
  volatility of 30 percent and interest rates at 5
  percent. A three-month at-the-money call option
  is priced at 3.27 and has a vega value of
  9.7833. This implies that if the volatility
  increases from 0.30 to 0.31, the price will
  change by (9.7833)(0.01) 0.097833.

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II. OPTION DERIVATIVES

• The vega value can be viewed as a volatility
  hedge ratio. A trader with an opinion on
  volatility can choose a position that increases
  in value if the opinion is correct.
• For example, if the trader believes the implied
  volatility is low and is about to increase, then
  a position with a positive vega value can be
  established.
• Like delta, the vega approximation is valid only
  for short ranges of volatility estimates.
• Vega changes with the stock price and with time
  to expiration and is maximized for options that
  are near the money.

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II. OPTION DERIVATIVES

• Volatility Trading
• Some traders believe that the market is efficient
  with respect to prices but inefficient with
  respect to volatility. In such a market,
  information about future volatility could be used
  in designing successful trading rules. Trading
  rules that exploit opinions on volatility are
  referred to as volatility trading rules.
•  

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II. OPTION DERIVATIVES

• One strategy for implementing a volatility
  trading rule is based on the vega rule.
• A trader who thinks that volatility will increase
  above the current levels implied by the market
  should invest in a positive-vega position. Since
  all options have positive vegas, the investor
  should purchase calls and puts.
• If the trader also believes the stock is
  currently underpriced (overpriced), then
  clearly, the best strategy is to purchase call
  (put) options.
• However, if the investor has no information on
  the direction if future price movement, then a
  risk-neutral position, with a zero delta value,
  may be desirable.

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II. OPTION DERIVATIVES

• Example A Vega Delta Trading Strategy
• The current information on three-month at the
  money European call and put option is shown in
  Exhibit below.
• Call Put
• ________________________________________
• Price 3. 27 2. 65
• Delta 0.5625 -0.4375
• Gamma 0.0529 0.0529
• Vega 9.7833 9.7833
• ________________________________________
•  

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II. OPTION DERIVATIVES

• Suppose a trader has established ? to be the
  target position delta and v to be the target
  position vega. To initiate a strategy that meets
  the target, the trader must purchase Nc calls and
  Np puts, where Nc and Np are chosen such that
•   ? c N c ? p N p ?
• v c N c v p N p v
•  For European options, ? c ? p - 1 and v c
  v p. hence
• ? c N c (?c -1 ) N p ?
• N c N p v/v c

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II. OPTION DERIVATIVES

• Solving for N c and N p yields
•  
• N c ? - (?c - 1) v/v c
•   N p v/v c - N c
•  
• For the case where the investor has no
  information on the direction of the stock price,
  ? 0. In this case the solution simplifies to
• N c (1-? c)v/v c
• N p (? c / ? p ) N c

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II. OPTION DERIVATIVES

• If the trader set the target vega value at 1.2
  times the current call vega value (v/v c 1.2),
  then the actual number of calls to buy is
• N C (1-0.5625)1.2 0.525, and
• the number of puts to buy is
• N p (0.5625)/(0.4372)0.525 0.675.
• A trader who purchases 525 calls and 675 put has
  created a position that has a delta value of zero
  but will profit if volatility expands.

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II. OPTION DERIVATIVES

• Derivative Exercise
•  
•  
• Option Value Delta
  Gamma Theta Vega
• __________________________________________________
  _______
• Deutsche mark 58 call 2.29 60 14
  -.04 .05
• Eurodollar 92 put .24 -50
  2 001 .03
• Japanese Yen 75 call 1.15 20
  3 -.012 .22
• SP 500 250 put 70 -30
  9 -.007 .13
• Swiss Franc 65 call 6.20 90
  2 -.002 .08
• __________________________________________________
  _______

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II. OPTION DERIVATIVES

• 1. If Deutsche Mark futures rally one full
  point, the 58 call will advance from 2.29 to
  2.89
• 2. If volatility in the Yen futures contract
  increases from 12 to 13, the Yen 75 call will
  advance from 1.15 to 1.37
• 3. If the SP 500 futures decline 1.00 point,
  the delta on the 250 put will move from -30
  to -39
• 4. If six days pass with the Japanese Yen
  futures contract remaining unchanged (and all
  other parameters remain unchanged), how much
  value will the Yen 75 call lose? 1.32

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II. OPTION DERIVATIVES

• 5. If implied volatility in Eurodollar futures
  drops from 9 to 7, the 92 put will decline from
  .24 to
• .18
•  
• 6. If a trader sells 10 Deutsche Mark 58 calls,
  how much futures contracts will he have to
  buy/sell in order to establish a delta neutral
  position? buy 6 futures contracts

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III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS

•  A. Bond Portfolio Protection
• Suppose that on Feb 15, 2005, a bond portfolio
  manager holds 50 Treasury Bonds (100,000 par
  value each) with a coupon rate of 10.75 and
  maturity of Feb. 15, 2023. The manager seeks a
  strategy to protect the portfolio against rising
  interest rates and falling bond prices over the
  next three months. Further, although protecting
  the value of the portfolio is important, the
  manager would like to retain the opportunity to
  profit from an increase in bond prices. The
  current market yield is 11.63 and each is worth
  93,422.

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III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS

• Spot Market Futures
  Options Market
• __________________________________________________
  _______
• Today Holds 5m T-bonds with Today Buy 50
  June 2005
• 10.75 coupon and 18 years futures put options
  at a
• maturity. The current market price strike price
  of 72.
• is 93,422 to yield 11.63 The current T-bond
  futures, trading
• at 70.64, are priced at 2,594 each
• (total premium 129,700).
• May If interest rate rises to 12.14 May
  T-bond futures option
• bonds are traded at 92,611 each. settled at
  68.04.
• __________________________________________________
  _______
• Loss (93,422-92,611) Gain (72
  68.04)50
• (5,000,000/100,000)
  100,000/100 198,000
• 198,000
•  
• Net Loss 198,000 129,700 - 198,000
  129,700

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III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS

• May If interest rates fall to 11.12 May
  T-bond futures
• and bonds are priced at 99,835 settled at
  73.88.
• each Exercise the puts?
• __________________________________________________
  ______
• Gain (99,835 - 93,422) Loss 2,594
  50
• (5,000,000 100,000)
  129,700
• 320,650
•  
• Net Gain 320,650 - 129,700
• 190,950
•  _________________________________________________
  _______

42
III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS
92,611
93,422
72,000
68.04
43
III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS

• B. Asset/Liability Management
• Support that on March 2, 2005, a bank funded 75
  million in loans that reprice every six months
  with three-month Eurodollar CDs at an annual
  rate of 9.30. For each 100 basis points increase
  in interest rates, the bank would have to pay
  additional 187,500. To hedge, the bank writes
  30 June 2005 Eurodollar futures call options at
  a strike price of 89.50. Since the Eurodollar
  futures settled at 89.78, the calls are
  in-the-money and priced at 14.50 each.
• If by June 1, 2005, Eurodollar CD rate dropped to
  7.6 and Eurodollar futures price settled at
  92.44. What is the net result of the this hedging
  strategy?
• If Eurodollar CD rate increased to 10.30 instead
  and futures price settled at 88.00, what is the
  net result?

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III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS

• Cash Market Futures
  Options Market
• __________________________________________________
  _______
• Today 6-month 75m loans March Write 30
  June 2005 matched with 3-month
  Eurodollar 3m-Eurodollar futures call options
• CDs at 9.3. at a strike price of 89.50.
• (If rates rise by 1, the bank will have Since
  the Eurodollar futures
• to pay an additional 187,500 settled at
  89.78, calls earn a
• 75M 1 3/12). premium of 14.50 each.
• June  If 3-month Eurodollar CD rate June
  Eurodollar futures
• rises to 10.3 price settled at 88.00. The
  calls are out-of-the-money and will not
• be exercised by holders.
• __________________________________________________
  _______
• Loss Additional funding costs Gain premium
  14.50 100
• 187,500
  30 43,500
• Net Loss 187,500 - 43,500 144,000

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III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS

• Cash Market Futures
  Options Market
• __________________________________________________
  _______
• Today 6-month 75m loans March Write 30
  June 2005 matched with 3-month
  Eurodollar 3m-Eurodollar futures call options
• CDs at 9.3. at a strike price of 89.50.
• (If rates rise by 1, the bank will have Since
  the Eurodollar futures
• to pay an additional 187,500 settled at
  89.78, calls earn a
• 75M 1 3/12). premium of 14.50 each.
• June  If 3-month Eurodollar CD rate June
  Eurodollar futures dropped to 7.6 price
  settled at 92.44. The calls are
  in-the-money and will be exercised by
  holders.
• __________________________________________________
  _______
• Gain 318,750 Loss (92.44 - 89.50)
  (75m (9.3 - 7.6) 3/12) 2500 30 7,350
  30
• savings in financing. 220,500
• Net Gain 318,750 43,500 - 220,500
  141,750

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III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS

• C. Mortgage Prepayment Protection
•  
• The prepayment option of fixed-rate mortgage
  contracts essentially gives borrowers a call
  option written by banks over the life of the
  mortgages.
• It will be exercised when it is in-the-money,
  i.e., when mortgage rates fall below the
  contractual rate minus any prepayment penalties
  or new loan origination costs.
• To manage the risk of mortgage prepayment if
  rates should fall, SLs should buy interest rate
  call options.

47
III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS

• A SL has five mortgage loans on its books, each
  earning a fixed rate of 14.25 with 20 years to
  maturity on an outstanding principal of 100,000.
  These loans are funded with three-month CDs. On
  Nov. 5, 2004, the three-month CD rate was 9.2.
  The SL imposes a 2.5 fees on new loan
  origination.
• To hedge the risk of a fall in mortgage rates
  and mortgage prepayment, management decides to
  buy five March 2005 T-bond futures call options
  at a strike price of 70. On November 5, 2004,
  each T-bond futures call option has a premium
  of 851 (March 2005 T-bond futures are priced at
  69.78).

48
III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS

• On Feb. 15, 2005, mortgage rates have fallen to
  11.7 and 3-month CDs earn 8.7 interest, while
  the T-bond futures price rose to 72.11, what is
  the net result of the hedging strategy?
• Cash Market Future
  Options Market
• __________________________________________________
  _____
• Today 500,000 mortgage loans at Today Buy
  five March
• 14.25 fixed, with 20 year maturity 2005
  T-bond futures call financed with 3-month CDs at
  9.2. options at a strike price
• Want to hedge against falling interest of 70.
  On this day, T-bond
• rates futures were at 69.78 (at-the-
• money) and T-bond futures
• call option has a premium
• 851 per contract
• Profit 500,000 (14.25 -9.2) Premium Cost
  851 5
• 3/12 6,313/Quarter
  4,255

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III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS

• (Case I Falling Interest Rates Mortgages are
  refinanced)
•  
• Feb. 15 Mortgage rates have fallen Feb.
  15 T-bond future price 2005 to 11.7 and
  3-month CDs 2005 rises to 72.11
• earn 8.7 interest The five
  futures call options
• can be offset to return 2,110
• per option
• __________________________________________________
  _______
• Profit 500,000 (11.7 - 8.7) Profit
  2,110 5
• 3/12 3,750/Quarter
  10,550
• Loss of profit 6,313 - 3,750
• 2,563/Quarter
• Net Result 10,550 - 4,255 - 2,563
  3,732.

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III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS

• (Case II Falling Interest Rates Mortgages are
  not refinanced)
•  
• Feb. 15 Mortgage rates have fallen Feb.
  15 T-bond futures
• 2005 to 13.25 and 3-month 2005 price
  rises to 70.70.
• CDs earn 9.0 interest The five
  futures call
• options can be
• offset to return 700
• per option
• __________________________________________________
  _______
• Profit 500,000 (14.25 - 9.0) Profit
  700 5
• 3/12 6,562.50/Quarter
  3,500
• Loss of Profit 6,313 - 6,562.50
• -249.50/Quarter
• Net Result 249.50 3,500 - 4,255
  -505.50.

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III. MICRO-HEDGING WITH FINANCIAL FUTURES OPTIONS

• (Rising Interest Rates Options are not
  exercised)
•  
• Feb. 15 Mortgage rates have been rising Feb.
  15 T-bond futur
• 2005 to 15 and 3-month CDs 2005
  price falls to 69
• earn 9.8 interest The five
  futures call
• options are not exercised
• __________________________________________________
  ______
• Profit 500,000 (14.25 - 9.8) 3/1 Loss
  of premiums 4,255
• 5,562.5/Quarter
• Loss 6,313 - 5,562.5 750.5
• Net Result -750.5 - 4,255
  -5,005.5

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IV. MACRO-HEDGING WITH OPTIONS

• An FI's net worth exposure to an interest rate
  shock could be represented as
•  
• ?R
• ?E -(DA - kDL) A ---------
• (1R)
• Now we want to adopt a put option position to
  generate profits that just offset the loss in net
  worth due to a rate shock , given a positive
  duration gap for the FI.

53
IV. MACRO-HEDGING WITH OPTIONS

• Let ?P be the total change in the value of the
  put position in T-bonds. This can be decomposed
  into
•   ?P (Np ? p) (1)
•  Where Np is the number of 100,000 put option
  on T-bond contracts to be purchased (the number
  for which we are solving) and ? p is the change
  in the dollar value for each 100,000 face value
  T-bond put option contract.
•  

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IV. MACRO-HEDGING WITH OPTIONS

• The change in dollar value for each contract (?
  p) can be further decomposed into
• ? p (dp/dB) (dB/dR) (? R/1R) (2)
• The first term (dp/dB) shows how the value of a
  put option change for each 1 dollar change in
  the underlying bond. This is called the delta of
  an option (? ) and lies between 0 and 1. For put
  option, the delta is negative.
• The second term (dB/dR) shows how the market
  value of a bond changes if interest rates rise
  by one basis point. The value of a basis point
  can be linked to duration.

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IV. MACRO-HEDGING WITH OPTIONS

• The value of a basis point can be linked to
  duration.
•   dB/B - MD dR (3)
•  Equation (3) can be arranged by cross
  multiplying as
•   dB/B - MD B (4)
• As a result, we can rewrite Equation (2) as
• ? p (-?) MD B (? R/1R) (5)

56
IV. MACRO-HEDGING WITH OPTIONS

• Thus the change in the total value of a put
  option ? P is
•  ? P Np (-?) MD B (? R/1R) (6)
• The term in squared brackets is the change in the
  value of one 100,000 face value T-bond put
  option as rates change and Np is the number of
  put option contracts.

57
IV. MACRO-HEDGING WITH OPTIONS

• To hedge net worth exposure, we require the
  profit on the off-balance sheet put option to
  just offset the loss of on balance sheet net
  worth when rates rise (or bond prices fall). That
  is
•   ? P ? E
•   Np (-?) MD B (? R/1R)
• (DA-kDL) A (? R/1R)
•  Solving for Np the number of put option to buy-
  we have
•   Np (DA-kDL) A / (-?) MD B