Business F723

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Business F723

• Fixed Income Analysis
• Week 11
• Options and Swaps

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Option Basics

• Options are based on buying or selling an asset
  in the future at a fixed price
• This transaction is not guaranteed to take place
• With an option one party to the option decides
  whether or not the transaction will be completed
  on the specified date

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Option Basics

• The option has given one party the right but not
  an obligation to buy or sell the asset at the
  fixed price
• As you might guess, this party has to pay the
  other party for this privilege
• The payment is called the option premium

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Options on Physical

• Few options on physical are still traded in the
  fixed income segment, US treasury is most active
  on CBOE
• mainly replaced by futures options
• Some OTC markets remain
• Institutional investors hedging specific
  securities are the most common, so the lack of
  liquidity is not a problem

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Option Terminology

• Call option the right to buy an asset
• Put option the right to sell an asset
• Exercise deciding to buy or sell the asset
• Exercise price the predetermined price
• Strike price the predetermined price
• Expiration date the last day that the option can
  be exercised

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Option Terminology

• American option can be exercised at any time on
  or before the expiration date
• European option can only be exercised on the
  expiration date
• Note although most options are American, most
  pricing models assume European options
• This assumption is acceptable if the option is
  not on a dividend paying share or similar asset

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Option Jargon

• In the money an option that would yield a profit
  if exercised today
• Out of the money an option that can not be
  profitably exercised at the current market prices
• At the money an option where the exercise price
  is the same as the current market price

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Basic Option Positions

• Buy a call option
• Pay a premium to buy at specified price
• Write a call option
• Sell the right to buy from you
• Buy a put option
• Pay a premium to sell at specified price
• Write a put option
• Sell the right to sell to you

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Option Payoffs

• For the buyer, if the price moves in their
  favour, they will gain on exercise
• If the price moves in the other direction, they
  will allow the option to expire unused, so the
  payoff will be zero

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Payoff Example

• Given the following options, expiring today, find
  the payoff

• Option A allows the owner to purchase the asset
  for 25
• Without the option she can purchase that asset
  for 24
• The option should not be exercised so its payoff
  is zero

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Payoff Example

• Given the following options, expiring today, find
  the payoff

• With option B the investor can buy the asset for
  30 and immediately sell it for 37
• This is a payoff of 7
• Option C allows the investor to sell the asset
  for 20, she can replace it for 18
• Payoff 2
• Option D is worthless
• Payoff 0

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Option Profits

• For the option holder profit equals the payoff
  minus the price paid for the option
• For the writer the profit equals the premium
  received minus the payoff that the holder gets at
  exercise, if any
• Breakeven is the market price where the payoff
  equals the initial premium

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Option Value

• Intrinsic value how much of a profit you could
  make if you exercised the option today
• Minimum of 0 since exercise is not mandatory
• Time value the difference between the current
  price of an option and its intrinsic value

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Logical Limits

• What is the maximum price at which a call option
  should trade?
• Since it gives you the right to buy the asset at
  a fixed price the maximum price of a call option
  should be the current price of the asset
• What is the minimum price?
• The option should sell for at least its intrinsic
  value, below that price it is better to exercise
  the option than to sell the option

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Call Option Price
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Time Value

• An option which can be profitably exercised today
  will have a positive intrinsic value
• Should you exercise that option?
• In most cases the answer is no
• The reason is that the intrinsic value is the
  minimum price of the option and the option is
  likely selling for more than the minimum

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Time Value

• The difference, between the intrinsic value and
  price is the time value of the option
• As the time to expiry approaches zero, the time
  value of the option approaches zero
• Exercising an option before expiry yields only
  the intrinsic value, the time value of the option
  is lost

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When to Exercise Early

• The main reason to consider exercising an option
  early is if the option is on an asset that has
  periodic cash flows
• If the option is on a bond that will pay a coupon
  tomorrow, and the coupon is larger than the
  remaining time value of the option, it would make
  sense to exercise the option to get that coupon

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Futures Options

• The right to enter into a futures contract at a
  pre-specified price
• Call option the right to take a long position
• Put option the right to take a short position
• If exercised, the futures contract is written at
  the specified price, and immediately marked to
  market by the exchange

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Exercise Example

• Call option on futures contract at 85
• Current futures price 90
• On exercise writer agrees to sell for 85, call
  owner agrees to buy for 85
• Immediately marked to market
• writer pays the exchange 5
• call owner gets 5 from the exchange

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Margin Requirements

• Buyer of option faces no chance of losing money
  beyond what was paid for the option so they post
  no margin
• Writer of option must post a margin equal to the
  margin requirement of the underlying contract
  plus the option premium
• Writer can be subject to margin calls

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Why Futures Options

• No Accrued interest payments
• No concern about a delivery squeeze
• you dont need to deliver a physical asset
• there is no cost to enter a futures contract
• Pricing of the option requires knowing the value
  of the underlying asset at all times, easy with
  futures contracts

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Traded Futures Options

• Active market for all futures contracts listed in
  the previous lecture
• T-bond and T-note on CBOT
• Eurodollar CD on IMM
• All contracts are American
• CBOT competing with OTC by introducing the
  flexible treasury futures options

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Option Pricing

• Six factors affect the price of an option
• Current price on underlying (S)
• Strike or exercise price (X)
• Time to expiry (t)
• Risk free rate (Rf)
• Expected price volatility (s)
• Coupon rate on bond higher coupon rates will
  make owning the bond better than the option

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Black-Scholes

• Most popular option pricing model

N(d1) means the area under the cumulative
standard normal distribution curve with respect
to d1.
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Put Option Pricing

• How much is a put option worth?
• If we buy an asset and a put option on that asset
  and we sell a call option that has the same
  strike price (both European), how much will this
  portfolio be worth at the end of the time period?

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Put-Call Parity

• Whether the market value is above or below the
  strike price, the value of the portfolio will be
  the strike price of the options.
• S P - C X /(1 Rf)
• P X /(1 Rf) C - S
• This is known as Put-Call Parity.

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Problems with Bond Options

• The maximum value of a bond is the undiscounted
  future cash flows
• Black-Scholes can give a positive value to a call
  option with a strike price greater than the
  undiscounted future cash flows
• Price volatility varies with time, interest rate
  volatility is more appropriate as an input

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Arbitrage-Free Binomial Model

• A popular model with dealer firms is the
  Black-Derman-Toy model
• Uses the interest rate ladder method shown in
  lecture 9
• Also assumes European options
• Only the volatility component is not observable

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Valuing Futures Options

• Most popular model is Fischer Black
• Similar concerns to Black-Scholes

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Option Strategy

• Most common hedge strategy is the protective put
  to guard against a large decrease in value
• Covered call strategy attempts to enhance yield,
  but limits price appreciation
• Which is best depends on the goals of the
  portfolio management

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Interest Rate Swaps

• Main idea is to trade fixed rate interest
  payments (receipts) for floating rate payments
  (receipts)
• Swaps have counterparty risks since they are not
  traded on organized exchanges
• May involve a securities firm or commercial bank
  as a broker or dealer

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Swap Example

• Company Z has a bond issue outstanding with a
  face 50 m and a coupon of 9
• The firm would prefer a floating rate
• The firm enters into a swap arrangement to pay
  LIBOR on 50 m, and in return receives 2.25 m
  every six months in return
• Typically only the difference is paid

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Interpreting a Swap

• There are 2 ways of looking at a swap
• A package of forward contracts
• A package of cash market instruments
• Buy a 9 fixed coupon 50m bond
• Finance by borrowing 50m at LIBOR

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Valuing a Swap

• At inception, the swap contract will have a value
  of zero, the present value of the traded cash
  flows should be the same or one of the parties
  will not enter the contract
• As interest rates change, the swap can increase
  or decrease in value

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Beyond Plain Vanilla

• Varying principal swaps the principal on which
  interest is calculated changes over time often
  for amortizing securities
• Basis swaps exchanging floating rate payments
  based on different reference rates
• Constant Maturity Swap one of the reference
  rates is the constant maturity treasury (CMT)
  rate published by the federal reserve

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Beyond Plain Vanilla

• Swaptions an option to enter into a swap
  contract at a point in the future
• Forward start swap a swap contract were the
  start date of the swap is in the future

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Interest Rate Agreements

• Similar to a series of interest rate options or
  insurance policies
• For an upfront payment, one party agrees to pay
  compensation for unfavourable interest rate
  movements, paid periodically over the life of the
  agreement
• Also called caps or floors

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Caps and Floors

• Interest rate agreements include
• The reference rate
• The strike rate (cap or floor)
• The length of the agreement
• The frequency of settlement
• The notional principal

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Cap Example

• A 9 cap is sold on LIBOR for a 5 year period
  with semi-annual settlement on a notional
  principal of 5 m
• If in the next period LIBOR is 8.75, there is no
  payment since it is under the cap
• If LIBOR is 9.25 half a year later, there is a
  payment of 6,250
• (0.0925-0.09)/2 x 5,000,000

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Valuing Caps and Floors

• Done using the binomial interest rate lattice
  method in chapter 24
• The value of each individual possible payment
  date (caplet or floorlet) is found independently
  and summed
• The value at any node is either zero if the
  cap/floor has not been violated, or the amount of
  payment that is required

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Collars

• The simultaneous buying of a cap and selling a
  floor
• Often offered to the floating rate payer by a
  swap dealer, the effect is to restrict the
  floating rate to a certain range