Transcript Com 4FJ3 - McMaster University

Business F723
Fixed Income Analysis
Week 11
Options and Swaps
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
Call Option Payoff Profile
Payoff
Exercise Price
Market
Price
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Payoff Example
• Given the following
options, expiring
today, find the payoff
Option
A
B
C
D
Type
Call
Call
Put
Put
Strike
Price
$25
$30
$20
$15
Market
Price
$24
$37
$18
$18
• 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
Option
A
B
C
D
Type
Call
Call
Put
Put
Strike
Price
$25
$30
$20
$15
Market
Price
$24
$37
$18
$18
• 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
Value
option price
intrinsic value
Exercise Price
Market
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 don’t 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
C  SN d1   XN d 2 e
Rf t
N(d1) means the area under
the cumulative standard
normal distribution curve
with respect to d1.
1 
S 
ln    R f  s 2   t
X
2 
d1    
s t
1 
S 
ln    R f  s 2   t
X
2 
d2    
 d1  s t
s t
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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
R t
C  FN d1   XN d 2 e
R t
P  XN  d 2   FN  d1 e
f
f
F 1 2
ln   s t
X
2
d1   
s t
d 2  d1  s t
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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
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