Insurance, Collars, and Other Strategies

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Insurance, Collars, and Other Strategies
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Part One Basic Insurance Strategies

• 1st Options can be used to insure long or short
  asset positions
• 2nd Options can be written against an asset
  position?option writer is selling insurance.

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Insuring a Long Position Floors

• Floor-The purchase of a put option.
• We are guaranteeing a minimum sale price for the
  value of the index or underlying stock.

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Insuring a Short Position Caps

• Buying a call option is known as a Cap
• Insure a short position by purchasing a call
  option to protect against a high price of
  repurchasing the index or underlying stock.

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Selling Insurance

• Covered Writing-writing an option when there is a
  corresponding long position in the underlying
  asset
• Also known as option overwriting, or selling a
  covered call.
• Naked Writing-the writer of an option does not
  have a position in the asset.

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Covered Call Writing

• Covered Call-own the stock and simultaneously
  sell a call option
• The covered call will have limited profitability
  if the stock increases because the option writer
  is obligated to sell the index for the strike
  price.
• If the stock decreases the loss on the stock is
  offset by the premium earned from selling the
  call.

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Covered call writing

• Stock1100 at expiry K1000, interest rate2
• Profit
• 1100 (10001.02) (93.8091.02) 100 75.68
• Profit on stock Profit on written call

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Covered Call Writing

• Reason to write a covered call
• You think that the stock will neither move up or
  down and you wrote the call, then you get to keep
  the premium.
• If the stock appreciates, you miss out on gains
  you would have had if you did not write the call.

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Payoff and profit diagram

• Buy stock and sell 1000 strike call option with
  premium of 93.809. Interest rate is 2
• 1st column is positive numbers who one hundred
  below strike and end two hundred above strike.
  This number is positive since we bought the
  stock.
• 2nd column is Max((S-K,0) since we have a short
  call (we sold a call).
• 3rd column is 1st column plus 2nd column
• 4th column is -(K-premium)1.02 since we
  bought stock and sold 1000 strike call for a
  premium of 93.809.
• 5th column is 3rd column plus 4th column.

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Payoff at Expiration

• Stock Short Call Payoff -(costinterest) Profit
  
• 900 0 900 -924.32 -24.32
• 950 0 950 -924.32 25.68
• 1000 0 1000 -924.32 75.68
• 1050 -50 1000 -924.32 75.68
• 1100 -100 1000 -924.32 75.68
• 1150 -150 1000 -924.32 75.68
• 1200 -200 1000 -924.32
  75.68

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Covered Puts

• Achieved by writing a put against a short
  position on the stock.
• The written put obligates you to buy the stock
  (for a loss) if it goes down
• For stock price below strike price, the loss on
  the written put offsets the short stock.
• For stock price above strike price, you lose the
  short stock.

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Part Two Synthetic Forwards

• Synthetic long forward-purchase of a call and
  sale of a put

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Synthetic forward vs. forward

• Forward contract has a zero premium, while the
  synthetic forward requires that we pay the net
  option premium.
• With forward contract we pay forward price, while
  with the synthetic forward we pay the strike
  price.

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Synthetic forward vs. forward

• If K is low, we buy stock at a discount relative
  to the forward price.
• To obtain this benefit we have to pay a positive
  net option premium which stems from the call
  being more expensive than the put

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Synthetic forward vs. forward

• If K is high, we buy the stock at a high price
  relative to the forward price.
• To offset the extra cost of acquiring the stock
  using a high strike option, we should receive
  payment initially. This happens if the put that
  we sell is more expensive than the call we buy.

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Synthetic forward vs. forward

• If K is equal to the forward price, then in order
  to mimic the forward, the initial premium must
  equal zero.
• In this situation, the put and call premiums must
  be equal

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Put-Call Parity

• The net cost of buying the stock using options
  must equal the net cost of buying the index using
  a forward contract.
• At time 0 we enter into a long forward position
  expiring at time T, we are to buy the stock at
  the forward price, F0,T .
• The present value of buying the stock in the
  future is just the present value of the forward
  price. PV(F0,T)

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Put-Call Parity

• If we instead buy a call and sell a put to
  guarantee the purchase price for the stock in the
  future, the present value of the cost is the net
  option premium for buying the call and selling
  the put.
• Call(K,T)-Put(K,T), plus the present value of the
  strike price, PV(K)
• Call(K,T) and Put(K,T) denote the premiums of the
  options with strike price K and with T periods
  until expiration.

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Put-Call Parity

• PV(F0,T)Call(K,T)-Put(K,T)PV(K)
• Rewritten
• Call(K,T)-Put(K,T)PV(F0,T-K)
• This equation is known as the put-call parity.

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Ex.1 Put-call parity

• Consider buying the 6 month, 1000-strike call for
  a premium of 93.809, with interest equal to 2,
  and selling the 6 month 1000 strike put for a
  premium of 74.201. This creates a synthetic
  forward allowing us to buy the stock in 6 months
  for 1000.
• Since the actual forward price is 1020, this
  synthetic forward permits us to buy the stock at
  a bargain of 20, the present value which is
  20/1.0219.61

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Ex.1 Put-Call Parity

• The difference in option premiums must therefore
  be 19.61. In fact, 93.809-74.20119.61
• Put into put-call parity equation
  93.809-74.201PV(1020-1000)

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Put-Call Parity

• A forward contract which the premium is not zero
  is sometimes called an off-market forward. This
  comes from the fact that a true forward by
  definition has a zero premium.
• Unless the strike price equals the forward price,
  buying a call and selling a put creates an
  off-market forward.

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Equivalence of Different Positions

• Selling a covered call (buying stock and selling
  a call) generates the same profit as selling a
  put.
• Recall in EX.1, we have the forward price of
  1020 and stock equal to 1000. Present value of
  the forward price equals the index price.
  Rewriting the equation give us
• PV(F0,T)Put(K,T)Call(K,T)PV(K)
• 1000 74.201 93.809 980.39

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Equivalence of different positions

• In the case of writing a covered call we have
• PV(F0,T)-Call(K,T)PV(K)-Put(K,T)
• Writing a covered call has the same profit as
  lending PV(K) and selling a put.

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No Arbitrage

• If the put-call parity equation did not hold
  there would be both low cost and high cost ways
  to acquire the stock at time T.
• Simultaneously buy the stock at low cost and sell
  the stock at high cost. This transaction has no
  risk and generates a positive cash flow.
• Taking advantage of such is called arbitrage.

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Part Three Spreads and Collars

• Spread-position consisting of only calls or only
  puts, in which some options are purchased and
  some are written

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Spreads and Collars

• You think stock will appreciate
• Enter into a long forward
• Buy a call option with strike price equal to
  forward price.
• Forward contract has zero premium and the call
  has a positive premium.
• The difference in payoffs explains the difference
  in premiums.

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Spreads and Collars

• If stock price at expiry is greater than the
  forward price, the forward contract and the call
  have the same payoff.
• If stock price at expiry is less than forward
  price, the forward contract has a loss and the
  call is worth zero

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Bull Spread

• Position in which you buy a call and sell an
  otherwise identical call with a higher strike
  price.
• Can be constructed using puts. Achieve the same
  results by either buying a low strike call and
  sell a high strike call, or buy a low strike put
  and sell a high strike put.

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Bull Spread

• Spreads constructed with either calls or puts are
  sometimes called Vertical Spreads.

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Premiums

• Stock price40, interest rate8.33 (8
  continuously compounded), and dividend0
• Strike Call Put
• 35 6.13 .44
• 40 2.78 1.99
• 45 .97 5.08

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Ex.2 Bull Spread

• Speculate stock price increasing.
• Buy call with k40 and 3 month to expiration.
  Premium2.78 We can reduce the cost of the
  position by selling a 45 strike call with premium
  of 0.97.

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EX.2

• Initial net cost of the two options is
• 2.78-.971.81
• With 3 months interest the total cost at
  expiration is 1.81(1.0833)3/12 1.85
• Table shows the cash flow at expiration for both
  options and computes profit on the position by
  subtracting the future value of the net premium

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Ex.2
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Ex.2 Bull Spread Graph
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Bear Spread

• The Bear spread is the exact opposite of a bull
  spread.
• In the above example we would create a bear
  spread by selling the 40 strike call and buying
  the 45 strike call.

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Bear Spread
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Box Spread

• Done by using options to create a synthetic long
  forward at one price and a synthetic short
  forward at a different price.
• Guarantees a cash flow in the future.
• A means of borrowing or lending money.
• Costly, but has no stock price risk.

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Ex.3 Box Spread

• Buy 40 strike call and sell a 40 strike put with
  premiums of 2.78 and 1.99.
• Sell a 45 strike call and buy a 45 strike put
  with premiums of 0.97 and 5.08.
• First is a synthetic forward purchase of a stock
  for 40 and the second is a synthetic forward
  sale of the stock for 45.

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Ex.3 Box Spread

• Payoff at expirations 45-405
• Cost of Strategy 5(1.0833)-.254.90
• Initial Cash flow (1.99-2.78)(.97-5.08)
• -4.90

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Ex.3 Box Spread Graph
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Ratio Spread

• Constructed by buying m calls at one strike and
  selling n calls at a different strike, with all
  options having the same time to maturity and the
  same underlying asset.
• Can also be constructed with puts.

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Collars

• The purchase of a put option and the sale of a
  call option with a higher strike price, with both
  options having the same underlying asset and
  having the same expiration date
• If the position is reversed, sale of put and
  purchase of a call, the collar is written.
• Collar Width- the difference between the call and
  the put strikes.

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Ex.4 Collars

• Sell 45 strike call with .97 premium and buy a
  40 strike put with 1.99 premium

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Ex.4 Collar Graph
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Collars

• Collars are used to implement insurance
  strategies.
• For example, buying a collar when we own the
  stock.
• A collared stock is buying the stock, buying a
  put, and selling a call

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Zero-Cost Collars

• Strike prices for a put and a call where the
  premiums exactly offset one another.
• An example is where you buy a stock and buy the
  40 strike put that has a premium of 1.99. Trial
  and error shows that a call with strike price of
  41.72 also has a premium of 1.99. So, you can
  buy a 40 strike put and sell a 41.72 strike call
  without paying any premium.

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Understanding Collars

• In the previous example of zero-cost collars it
  seems that we are getting free insurance, but we
  must take into account financing costs.
• Recall that if you pay 40 for a stock and sell
  it for 40 in 91 days, you have not broke even
  because you have forgone 40((1.0833).25-1).808
  in interest.

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Cost of Collars and the Forward Price

• Suppose you try to construct a zero-cost collar
  in which you set the strike of the put option at
  the stock price plus financing cost.
• If you try to insure against all losses on the
  stock, including interest, then a zero-cost
  collar has zero width.

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Part Four Speculating on Volatility

• Options used to create positions that are
  nondirectional with respect to the underlying
  asset.
• With nondirectional positions, the holder does
  not care if the stock goes up or down, but only
  how much it moves.

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Straddles

• Buying a call and a put with the same strike
  price and time to expiration
• If stock price rises, we will profit on the
  purchased call and if the stock price decreases
  we will profit on the purchased put.

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Straddles Advantages and Disadvantages

• The advantage is that we will profit with an
  increase or decrease in stock price.
• The disadvantage is that it has a high premium
  because you have to purchase two options.

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Ex.5 Straddle

• Buy a 40 strike call with premium of 2.78 and
  40 strike put option with premium of 1.99

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Ex.5 Straddle Graph
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Strangle

• To reduce premium you can buy out-of-the-money
  option rather than at-the-money option
• Ex. 6 Consider buying a 35 strike put and a 45
  strike call for premiums of 0.44 and 0.97 with
  future value of 1.44.
• These transactions reduce your maximum loss if
  the options expire with the stock near 40, but
  they also increase the stock-price move required
  for a profit.

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Ex.6 Strangle Graph
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Written Straddle

• Selling a call and a put with the same strike
  price and time to expiration.
• A purchased straddle is a bet that volatility is
  high, a written straddle is a bet that volatility
  is low, relative to the markets assessment.

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Ex.7 Written Straddle

• Sell a 40 strike call and a 40 strike put with
  premiums of 2.78 and 1.99

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Ex.7 Written Straddle Graph
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Butterfly

• Buy a call with strike price less than stock
  price, buy another call with strike price greater
  than stock price, and sell two calls with strike
  prices equal to stock price.

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Ex.8 Butterfly

• Buy a call with K equal to 35 with premium,
  6.13 and we buy a call with K equal to 45 with
  a premium of 0.97
• We also sell two calls with K equal to 40 and
  premiums of 2.78.

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Ex.8 Butterfly
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Asymmetric Butterfly

• Looks just like butterfly graph but it is
  asymmetrical.
• The peak is closer to the high strike price than
  to the low strike price

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Ex.9 Asymmetric Butterfly

• Created by buying two 35 strike calls, eight 45
  strike calls, and selling ten 43 strike calls.

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Ex. 9 Asymmetric Butterfly Graph
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• Questions?