Replicating Options and Portfolio Insurance

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FINA 6220Professor Andrew Chen

• Replicating Options and Portfolio Insurance
• Lecture Note 5

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Outline

• Profit Loss for a Portfolio or Stock Without
  Insurance
• Profit Loss of Buy Put Option
• Profit Loss for a Portfolio With Insurance
• Some Useful Properties of Call and Put Options
• Replicating Options

AC 52
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Profit Loss of Buy Put Options
AC 53
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Profit and Loss of a Portfolio With Insurance

• (i.e., Buy Protective Put)

AC 54
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Some Useful Properties of Call and Put Options
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Some Useful Properties of Call and Put Options

• Call prices and stock prices change in the same
  direction
• One dollar change in the stock price will cause a
  change of less than one dollar in the call price
• For out of money call one-dollar change in the
  stock price will have little effect on the call
  price.
• For at the money call one-dollar change in the
  stock price will cause about .5 change in the
  call price.
• For in the money call one-dollar change in the
  stock price will cause about 1 change in the
  call price.

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Some Useful Properties of Call and Put Options
AC 57
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Some Useful Properties of Call and Put Options

• Implications
• From the put-call parity (i.e., S P C KB),
  we know the following implications in replicating
  call and put options with stock and cash
• Long Stock and Borrowing ? Long Call
• Short Stock and Lending (i.e., investing in
  risk-free assets) ? Long Put
• Replicating options involves a dynamic asset
  allocation

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Replicating Options

• The Binomial-Tree Approach
• The Black-Scholes OPM Approach

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Binomial-Tree Approach

• Assumptions
• Two Period Call
• K 40
• Rf 10 per period

t 0
t 1
t 2
Call (K 40)
70
30
60
50
50
10
40
30
0
AC 510
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Binomial-Tree Approach

• Take note of the following notations
• ? the number of shares of the stock to be
  purchased
• B the dollar amount to be borrowed and
• L the dollar amount to be lent or invested.
  
• To replicate the two-period European call option
  (K40), we simply search for a portfolio consists
  of stock and cash that will produce the same
  payoffs as that of the call option at its
  expiration date (t 2).
• This can be accomplished by solving three
  simple problems recursively as follows

AC 511
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Binomial-Tree Approach

• Step 1
• Finding a portfolio consists of stock and cash
  (at t1 and S160).
• It would produce a net value of 30 if stock price
  goes up to 70
• A net value of 10 if stock price goes down to 50.
  
• Formally, we have the following two simple
  equations and the solution.

70 x ?1 1.1B1 30
?1 1
50 x ?1 1.1B1 10
B1 40/1.1 36.36
20 x ?1 0B1 20
70 x 1 1.1B1 30
20?1 20
1.1B1 40
AC 512
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Binomial-Tree Approach

• Step 2
• Finding a portfolio consists of stock and cash
  (at t1 and S140)
• Will produce a net value of 10 if stock price
  goes up to 50
• Will produce a net value of 0 if stock price goes
  down to 30.
• Formally, we can set up the following simple
  problem and obtain its solution as follows

50 x ?1 1.1B1 10
?1 0.5
30 x ?1 1.1B1 0
B1 15/1.1 13.64
20 x ?1 0B1 10
50 x 0.5 1.1B1 10
20?1 10
1.1B1 15
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Binomial-Tree Approach

• Find a portfolio (at t 0 and S0 50) consists
  of stock and cash that will produce a net value
  of 23.64
• The amount of capital that is required to
  establish the portfolio of longing one share of
  stock and borrowing 36.36 found in Step 1) if
  stock price goes up to 60
• And a net value of 6.36
• If the stock price goes down to 40 (the dollar
  amount required to create a portfolio of longing
  .5 share of stock and borrowing 13.64 found in
  step 2).
• The problem in the last step and its solution are
  given in the next slide

AC 514
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Binomial-Tree Approach

• Step 3
• Therefore, we know that buying a two-period
  European call with K 40 is the same as the
  following portfolio

60 x ?0 1.1B0 23.64 (60 x 1 36.36)
40 x ?0 1.1B0 6.36 (40 x 0.5 13.64)

?0 17.28/20 0.864
B0 28.20/1.1 25.64

Buy 0.864 shares
Borrow 25.64
AC 515
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Binomial-Tree Approach

• Fair Value of the European Call(K40)
• 50 x 0.864 25.64
• 17.56
• Replicating a covered-call write (S-C) position
• S C S (0.864 x S 25.64)
• 0.136 x S 25.64
• We can replicate a covered-call-write position by
  creating a portfolio consists of buying .136
  shares of stock and lending 25.64.
• And the total cost of replicating the
  covered-call-write position is 32.44.

AC 516
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Replicating Put Option Portfolio Insurance

• Similar to replicating a call option, we can use
  stock and cash to replicate a two-period European
  put option with K 40 and other related hedge
  positions such as buying a protective put
  position.
• As we have noted earlier, a portfolio consists of
  a short position in stock and an investment in
  risk-free assets can be used to replicate a put
  option.
• In the following, we will show how to replicate a
  two-period European put option with K 40.

AC 517
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Replicating Put Option Portfolio Insurance

• Step 1
• Finding a portfolio consists of stock and lending
  in cash (at t1 and S160)
• It will produce a net value of zero if stock
  price goes up to 70.
• And a net value of zero if stock price goes down
  to 50.
• This problem is set up in the following two
  equations and the solution is given below.

70 x ?1 1.1L1 0
?1 0
50 x ?1 1.1L1 0
L1 0
AC 518
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Replicating Put Option Portfolio Insurance

• Step 2
• Finding a portfolio consists of stock and lending
  cash (at t1 and S140)
• Will produce a net value of zero if the stock
  price goes up to 50.
• And that will produce a net value of 10 if the
  stock price goes down to 30.
• We can set up this simple problem and show its
  solution as follows

50 x ?1 1.1L1 0
?1 -0.5
30 x ?1 1.1L1 10
L1 25/1.1 22.73
AC 519
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Replicating Put Option Portfolio Insurance

• Step 3
• Finding a portfolio consists of stock and lending
  cash (at t 0 and S0 50
• Producing a net value of 0 if stock price goes up
  to 60
• And produce a net value of 2.73 40(-.5)
  22.73.

60 x ?0 1.1L0 0
40 x ?0 1.1L0 2.73 (40 x (-0.5) 22.73)

?0 -2.73/20 -0.1365
L0 8.19/1.1 7.45
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Replicating Put Option Portfolio Insurance

• Therefore, we know that buying a two-period
  European put option with K 40 is the same as
  the following portfolio

Short 0.1365 shares
Lend 7.45
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Replicating Put Option Portfolio Insurance

• Fair Value of the European Put(K40)
• 50 x (-0.1365) 7.45
• 0.625
• Replicating a protective put position
• S P S (-0.1365 x S 7.45)
• 0.8635 x S 7.45
• We can replicate a hedge position with portfolio
  insurance (S P) by holding .8635 shares of
  stock and investing 7.45 in the risk-free
  assets.
• The total cost of replicating buying a protective
  put equals to 50.625.

AC 522
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Comparison of Different Option Strategies

• Writing a Covered Call (S - C)
• Writing a Ratio Covered Call (S 2C)

Buy 0.136 Shares of stock
Total Cost 32.44
Invest 25.64 in Rf assets

Short 0.728 shares of stock
Total Cost 14.88
Invest 51.28 in Rf assets
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Comparison of Different Option Strategies

• Buying a Protective Put (S P)
• Buying a Straddle (P C)

Buy 0.8635 Shares of stock
Total Cost 50.625
Invest 7.45 in Rf assets

Buy 0.7275 Shares of stock
Total Cost 18.185
Borrow 18.19 at Rf
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The Black-Scholes OPM Approach
C

• Short Recap from LN 4
• where

C
(5.1)
(5.2)
d1
(5.3)
d2 d1
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The Black-Scholes OPM Approach

• CoveredCall Write (S C)
• Hedge Portfolio S C

(5.4)
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The Black-Scholes OPM Approach

• 2/1 Ratio Covered Call Write (S 2C)
• Hedge Portfolio S 2C

S 2

(5.5)
AC 527
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The Black-Scholes OPM Approach

• Replicating Put Options

P -SN(-d1)
(5.6)
AC 528
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The Black-Scholes OPM Approach

• Purchase a Protective Put (S P)
• Hedged Portfolio S P

S
S

(5.7)
AC 529
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The Black-Scholes OPM Approach

• Replicating Spreading Positions
• Bullish Call Spread
• therefore

CL CH
(5.7)

AC 530