Download Derivatives

Derivatives
Fin 119
Spring 2009
Derivatives
Basic Definition
Any Asset whose value is based upon (or derived
from) an underlying asset.
The performance of the derivative is dependent upon
the performance of the underlying asset.
Risk Management
Since a derivatives performance is based on an
underlying asset they can often be used to decrease
the risk associated with changes in the spot price of
an asset.
Basic Types of Derivative Contracts
Forward Contracts
Agreement between two parties to purchase or sell something at a
later date at a price agreed upon today
Futures Contract
Same idea as a forward, but the contract trades on an exchange and
the counter party is not set.
Options
Buying or selling the right but not obligation to purchase or sell
something in the future at a price agreed upon today
Swaps
Exchange of Cash flow Streams based on a notional value.
Brief History
Organized Exchanges in US
Chicago Board of Trade
Established in 1848 to bring farmers and merchants
together. Futures Contracts were first traded on the
CBOT1865. Developed the first standard contract
Chicago Mercantile Exchange
Started as the Chicago Produce Exchange in 1874 for
trade in perishable agricultural products. In 1919 it
became the Chicago Mercantile Exchange (CME).
Introduced a contract for S&P 500 futures in 1982.
NYMEX 1872 KCBOT 1876
Other US Exchanges
NYBOT
Coffee Sugar and Cocca Exchange
New York Futures Exchange
Minneapolis Grain Exchange
Philadelphia Board of Trade
Payoff on Forward Contracts
Long Position
Agreeing to buy a specified amount (The Contract
Size) of a given commodity or asset at a set point in
time in the future (The Delivery Date) at a set price
(The Delivery Price)
Payoff
The payoff will depend upon the spot price at the
delivery date.
Payoff = Spot Price – Delivery Price
Example
Assume you have agreed to buy €1,000,000 in
3 months at a rate of €1 = $1.6196
Spot Rate
Spot – Delivery Price
Payoff
$1.65
$1.65-$1.6196=$0.0304 $30,400
$1.6169
$1.6196-$1.6196=0
0
$1.55
$1.55-$1.6196=$0.0696 -$69,600
Example Graphically
Payoff
.0304
1.55
-.0696
1.6196 1.650
Spot Price
Payoff: Short Position
Agreeing to sell a specified amount (The
Contract Size) of a given commodity or asset
at a point of time in the future (The Delivery
Date) at a set price (The Delivery Price).
Payoff on Short position
Since the position is profitable when the price
declines the payoff becomes:
Payoff = The Delivery Price – The Spot Price
Long vs. Short
For a long position to exist (someone agreeing to buy)
there must be an offsetting short position (someone
agreeing to sell).
Assume that you held the short position for the previous
example:
sell € 1,000,000 in 3 mos at a rate of €1 = $1.6196
Spot Rate
Spot – Delivery Price
Payoff
$1.65
$1.6196-$1.65=-$0.0304
-$30,400
$1.6169
$1.6196-$1.6196=0
0
$1.55
$1.6196-$1.55= $0.0696
$69,600
Example Graphically
Payoff
.0304
1.55
-.0696
1.6196 1.650
Spot Price
Contract Goals
The goal of the contract is to decrease risk,
assume that you had to pay €1,000,000 in 3
months for the shipment of an input. You are
afraid that the $ price will increase and you will
pay a higher price.
Similarly the other party may be afraid that the
$ price will decrease (maybe they are receiving
a payment in 3 months)
Determining the delivery price
The delivery price will be determined by the
participants expectations about the future price
and their willingness to enter into the contract.
(Today’s spot price most likely does not equal
the delivery price).
What else should be considered?
They should both also consider the time value of
money
Future and Forward contracts
Both Futures and Forward contracts are
contracts entered into by two parties who agree
to buy and sell a given commodity or asset (for
example a T- Bill) at a specified point of time in
the future at a set price.
Futures vs. Forwards
Future contracts are traded on an exchange,
Forward contracts are privately negotiated
over-the-counter arrangements between two
parties.
Both set a price to be paid in the future for a
specified contract.
Forward Contracts are subject to counter
party default risk, The futures exchange
attempts to limit or eliminate the amount of
counter party default risk.
Other Forward
Contract Risks
One goal of the negotiation is to specify exactly
the type, quantity, and means of delivery of the
underlying asset.
The chance that an asset different than
anticipated might be delivered should be
eliminated by the contract.
Futures contracts attempt to account for this
problem via standardization of the contract.
Futures Contracts
Long Position: Agreeing to purchase a
specified amount of a given commodity or
asset at a point in time in the future at a set
price (the futures price)
Short Position: Agreeing to sell a specified
amount of a given commodity or asset at a
point of time in the future for a set price (the
futures price).
Standardization of
Futures Contracts
To promote confidence in the system and
eliminate counter party default risk, future
contracts are highly standardized.
Specifications of
Futures Contract
The Asset
The Contract Size
Delivery Arrangements
Delivery Months
Price Quotes
Price Limits
Position Limits
Contract Specifications
Asset
Quality and type of asset are specified to
guarantee specific product is delivered.
Contract Size
The amount of asset that is to be delivered for
one contract
Delivery Arrangements
More important for commodities than financial
assets. Specify how delivery occurs and
location.
Contract Specifications
Delivery Months
When delivery will occur (and during what part
of the month delivery can occur)
Price Quotes
Contract must specify the units for the price
quote (1/32 of a dollar etc) Also implicitly
establishes the minimum fluctuation for the price
of the contract.
Contract Specifications
Price Limits
Designed to add stability to the market, limits on the
maximum fluctuation in price that can occur during a
trading day.
Position Limits
Limits the number of contracts that can be entered into
by a speculator.
Speculator –attempting to profit from a movement in
the market
Hedger – attempting to offset an underlying spot
position.
Does Delivery need to take place?
No – most contracts will be closed out.
Closing out a contract is simply taking the
opposite (short if you are long or vice versa)
position. The change in the futures price will
be your gain or loss.
With a futures contract your counter party does
not remain the same. It does not matter who
takes the opposite position. This is not the case
for a forward contract.
Summary
Forward Contracts
Private contract between
two parties
Not Standardized
Usually a single delivery date
Futures Contracts
Traded on
an exchange
Standardized
Range of delivery dates
Settled at the end of contract
Settled daily
Delivery or final cash
settlement usually takes place
Contract is usually closed
out prior to maturity
Important Terminology
Open Interest
The number of contracts that are currently open
(both a short and long position exist).
What happens to open interest if a new long
position is taken out?
It could Increase
It could decrease
It might not change.
The answer depends on whether both the long and
short positions are new, or closing out, or one of
each.
Margin Requirements
To limit counter party default risk, the futures exchange
requires participants to place funds in a margin account
when the contract is taken out.
Some Terminology:
Initial Margin: The original amount deposited in the
margin account
Maintenance margin: The amount that must
remain in the margin account
Margin call – Notice that the margin account has
dropped below the maintenance margin, more money
must be added to the account
Margin Example
Example:
An investor has taken a long position in gold
(agreed to buy gold at some date in the future).
Assume that the agreement is for 2 gold
contracts each contract consists of 100 ounces
of gold. The futures price is $400 per ounce.
This implies that the participant would need
200*400 = $80,000 to purchase gold at the
expiration of the contract.
Margin Example
If the futures price for gold decreases to $398,
the investor would suffer a loss if the contract is
closed out.
The loss would total (400 - 398)200 = $400.
The fear is that if at the expiration of the
contract the price is 398, the participant will not
honor the contract since it would result in a loss
of $400.
Margin Example
To counteract this the investor is ask to put a
sum of money into a margin account lets
assume $2,000 per contract or $4000 total.
When the futures price declines the loss of
$400 is taken from the margin account of the
investor and given to a participant that took a
short position.
Margin Example
The value of the contract is marked to market
each day, and the margin account is adjusted.
The margin is effectively guaranteeing that the
position is covered.
If the level of the account falls below the
maintenance margin the investor is required to
put more funds into the account this is known
as a margin call. The extra funds provided are
the variation margin, if they are not provided
the broker will close out the account.
Margin Account
Day Futures
Price
Daily
Changes
Cumulative Margin Margin
Change
Balance
Call
0
400
1
398
-2(200) = -400
-400
3600
2
395.5
-2.5(200)=-500
-900
3100
3
394.5
-1(200)=-200
-1100
2,900
4
395.5
(1)200=200
-900
4200
4000
1100
Note:
You can withdraw any amount above the initial
margin
Most accounts pay a money market rate of
interest
Some accounts allow deposit of securities, but
valued at less than face value. (treasures
valued at 90% other at 50%)
Role of Clearinghouse
The clearinghouse serves as an intermediary
that guarantees the contract.
The clearinghouse is an independent
corporation whose shareholders are comprised
of its member firms. Each member firm
maintains a margin account (similar to the
traders) with the clearinghouse.
In essence the clearinghouse guarantees the
long and the short trader that the other side will
honor the contract
Patterns of Futures Prices
Basis = Spot Price – Futures Price
The Basis moves toward zero as the spot price
matures.
This eliminates arbitrage possibilities.
If futures is greater than spot, you could enter short
in the futures market and make a profit by buying in
the spot and then delivering in futures
Since everyone will attempt this demand for short
positions increases and futures price decreases, also
spot price would increase….
Other Patterns
The Futures Price over time
Normal Market: The futures price increase as the time
to maturity increases
Inverted Market: the futures price is a decreasing
function of the time to maturity
Comparing the futures price to the expected future spot
price.
Normal Backwardation: The futures price is below
the expected future spot price.
Contango: The futures price is above the expected
futures price.
Theoretical Explanations of
Backwardation
Keynes and Hicks-- Speculators will only enter the market if
they expect to have a positive profit. If more speculators
are holding a long position, it implies that the futures price
is less than the expected spot price
A second explanation can be found by looking at the
relationship between risk and return in the market. If thee
is systematic risk involved with holding the security then the
investor should be compensated for accepting the risk
(nonsystematic risk can be diversified away).
Option Terminology
Call Option – the right to buy an asset at
some point in the future for a designated price.
Put Option – the right to sell an asset at some
point in the future at a given price
Review of Option Terminology
Expiration Date
The last day the option can be
exercised (American Option) also
called the strike date, maturity,
and exercise date
Exercise Price
The price specified in the
contract
American Option Can be exercised at any time up
to the expiration date
European Option Can be exercised only on the
expiration date
Review of Option Terminology
Long position: Buying an option
Long Call: Bought the right to buy the asset
Long Put: Bought the right to sell the asset
Short Position: Writing (Selling) the option
Short Call: Agreed to sell the other party the
right to buy the underlying asset, if the other
party exercises the option you deliver the asset.
Short Put: Agreed to buy the underlying asset
from the other party if they decide to exercise
the option.
Review of Terminology
In - the - money options
when the spot price of the underlying asset for a
call (put) is greater (less) than the exercise price
Out - of - the - money options
when the spot price of the underlying asset for a
call (put) is less (greater) than the exercise price
At - the - money options
when the exercise price and spot price are equal.
Interest Rate Options
Traded on Chicago Board of Options Exchange
(CBOE)
Interest rate Options are traded on 13 Week,
5 year, 10 year and 30 year treasury securities
Call Option Profit
Call option – as the price of the asset increases
the option is more profitable.
Once the price is above the exercise price
(strike price) the option will be exercised
If the price of the underlying asset is below the
exercise price it won’t be exercised – you only
loose the cost of the option.
The Profit earned is equal to the gain or loss on
the option minus the initial cost.
Profit Diagram Call Option
(Long Call Position)
Profit
S-X-C
S
Cost
X
Spot Price
Call Option Intrinsic Value
The intrinsic value of a call option is
equal to the current value of the
underlying asset minus the exercise price
if exercised or 0 if not exercised.
In other words, it is the payoff to the
investor at that point in time (ignoring the
initial cost)
the intrinsic value is equal to
max(0, S-X)
Payoff Diagram Call Option
Payoff
S-X
X
S
Spot
Price
Example: Naked Call Option
Assume that you can purchase a call option on
an 8% coupon bond with a par value of $100
and 20 years to maturity. The option expires in
one month and has an exercise price of $100.
Assume that the option is currently at the
money (the bond is selling at par) and selling
for $3.
What are the possible payoffs if you bought the
bond and held it until maturity of the option?
Five possible results
The price of the bond at maturity of the option
is $100. The buyer looses the entire purchase
price, no reason to exercise.
The price of the bond at maturity is less than
$100 (the YTM is > 8%). The buyer looses the
$3 option price and does not exercise the
option.
Five Possible Results continued
The price of the bond at maturity is greater
than $100, but less than $103. The buyer will
exercise the option and recover a portion of the
option cost.
The price of the bond is equal to $103. The
buyer will exercise the option and recover the
cost of the option.
The price of the bond is greater than $103.
The buyer will make a profit of S-$100-$3.
Profit Diagram Call Option
(Long Call Position)
Profit
S-100-3
103
-3
100
S
Spot Price
Price vs. Rate
Note buying a call on the price of the bond is
equivalent to buying a put on the interest rate
paid by the bond.
As the rate decreases, the price increases
because of the time value of money.
Profit Diagram Call Option
(Short Call Position)
Profit
X
C+X-S
S
Spot Price
Put option payoffs
The writer of the put option will profit if the
option is not exercised or if it is exercised and
the spot price is less than the exercise price
plus cost of the option.
In the previous example the writer will profit as
long as the spot price is less than $103.
What if the spot price is equal to $103?
Put Option Profits
Put option – as the price of the asset decreases
the option is more profitable.
Once the price is below the exercise price
(strike price) the option will be exercised
If the price of the underlying asset is above the
exercise price it won’t be exercised – you only
loose the cost of the option.
Profit Diagram Put Option
Profit
X-S-C
S
Cost
Spot Price
X
Put Option Intrinsic Value
The intrinsic value of a put option is equal
to exercise price minus the current value
of the underlying asset if exercised or 0 if
not exercised.
In other words, it is the payoff to the
investor at that point in time (ignoring the
initial cost)
the intrinsic value is equal to
max(X-S, 0)
Payoff Diagram Put Option
Profit
X-S
S
Cost
X
Spot Price
Profit Diagram Put Option
Short Put
Profit
S
S-X+C
X
Spot Price
Pricing an Option
Arbitrage arguments
Black Scholes
Binomial Tree Models
PV and FV in continuous time
e = 2.71828 y = lnx x = ey
FV = PV (1+k)n for yearly compounding
FV = PV(1+k/m)nm for m compounding periods
per year
As m increases this becomes
FV = PVern =PVert
let t =n
rearranging for PV
PV = FVe-rt
Black Scholes
The basic starting point for the actual pricing of
an European option is the model developed by
Fisher Black, Myron Scholes, and Robert
Merton.
Black Scholes Assumptions
1)
2)
3)
4)
5)
6)
7)
Stock prices follow a lognormal distribution with m and
s constant.
There are no transaction costs or taxes and all
securities are perfectly divisible
There are no dividend on the asset during the life of
the option
There are no riskless arbitrage opportunities
Security trading is continuous
Investors can borrow and lend at the same risk free
rate
The short term risk free rate is constant
Inputs you will need
S = Current value of underlying asset
X = Exercise price
t = life until expiration of option
r = riskless rate
s2 = variance
Black Scholes
Value of Call Option = SN(d1)-Xe-rtN(d2)
S = Current value of underlying asset
X = Exercise price
t = life until expiration of option
r = riskless rate
s2 = variance
N(d ) = the cumulative normal distribution
(the probability that a variable with a
standard normal distribution will be less than
d)
Black Scholes (Intuition)
Value of Call Option
SN(d1)
The expected
Value of S
if S > X
-
Xe-rt
N(d2)
PV of cost
Risk Neutral
of investment Probability of
S>X
Black Scholes
Value of Call Option = SN(d1)-Xe-rtN(d2)
Where:
S
s
ln(
)  (r 
)t
X
2
d1 
s t
2
d 2  d1  s
t
Payoff Diagram and Call Price
The payoff diagram is effectively the price (or
value of the option) at the expiration of the
contract. (the price should be the intrinsic
value)
At any given time, the current price is a
function the amount of time remaining prior to
expiration.
The price or value will generally be above the
intrinsic value for a call option.
Value of Call Option
Payoff
Value of option given
time to expiration
S-X
X
S
Spot
Price
Delta of an option
The delta of the option shows how the
theoretical price of the option will change with a
small change in the underlying asset.
change in price of call option
delta 
change in price of underlying bond
Value of Call Option
Payoff
S-X
Value
of option
X
S
Spot
Price
Delta is the slope of the tangent line
at the given stock price
Delta of an option
Intuitively a higher stock price should lead to a
higher call price. The relationship between the
call price and the stock price is expressed by a
single variable, delta.
The delta is the change in the call price for a
very small change it the price of the underlying
asset.
Delta
Delta can be found from the call price equation as:
c

 N (d1 )
S
Using delta hedging for a short position in a European call
option would require keeping a long position of N(d1) shares
at any given time. (and vice versa).
Delta explanation
Delta will be between 0 and 1.
A 1 cent change in the price of the underlying
asset leads to a change of delta cents in the
price of the option.
Delta
For deep in the money call options the delta will
be close to 1.
For deep out of the money call options the delta
will be close to zero.
Gamma
Gamma measures the curvature of the
theoretical call option price line.
change in price of option
gamma 
Change in price of underlying bond
Gamma of an Option
The change in delta for a small change in the
stock price is called the options gamma:
Call gamma =
e
 d 12 / 2
Ss 2T
Other Measures
change in th e price of option
theta 
decrease in time to expiration
Change in option Price
kappa 
Change in expected
Combining Positions
A firm can use two or more option positions in
combination to hedge a risk
Allows you to create “synthetic positions”
Hedging
Assume you are afraid that the asset you own
will decline in value.
As the price declines you would like to have the
loss offset with a gain
Buying a put option will allow this to happen.
Profit Diagram Put Option
Profit
X-S-C
S
Cost
Spot Price
X
Spot Position
You also owned the stock so you must consider
the combined impact of the gain on the option
and the value of the stock itself.
Diagramming the spot
The spot position could be represented by a
straight line that represents the corresponding
change in price
The line will also slope up to the right. As the
price increases there is improvement.
Profit Diagram Spot
Profit
Spot Price
Cost
Combined position
Assume the exercise price on the option is a
little bit below the current price of the asset.
What happens to the combined payoff of the
spot position and the option?
Profit Diagram Put Option
X
Combined
Position
Other possibilities
By selling an option and buying another option
you can offset the cost of the option.
This creates a spread which limits gains and
losses.
Profit Diagram
Put Options
Short Put
Spot Price
Long Put
Profit Diagram
Bear Spread
Spot price
Bear Spread
Adding the spot position
What if you own the asset in the previous
example?
Assume that the current price slightly below the
lower of the two strike prices.
Profit Diagram
Bear Spread
Spot price
Hedged
Position
Bear Spread
Another Strategy
To avoid the downside of the previous example
you could combine a short call and a long put
Profit Diagram Call Option
Profit
Long Put
Short Call
Profit Diagram Costless Collar
Profit
Long Put
Combined
Profit
Short
Call
Combined with Spot
By using options selling at the same price the net cost is
zero.
At prices above the higher strike price (below the lower
yield) the gain is offset by a loss in the option position.
At prices below the lower strike price (above the higher
yield) the loss is offset by gains in the option.
You have limited both the gain and loss.
Assume that the current price is exactly between the two
strike prices
Profit Diagram
Costless Collar (Fence)
Profit
Costless
Collar
Swaps Introduction
An agreement between two parties to exchange cash
flows in the future.
The agreement specifies the dates that the cash flows
are to be paid and the way that they are to be
calculated.
A forward contract is an example of a simple swap. With
a forward contract, the result is an exchange of cash
flows at a single given date in the future.
In the case of a swap the cash flows occur at several
dates in the future. In other words, you can think of a
swap as a portfolio of forward contracts.
Mechanics of Swaps
The most common used swap agreement is an
exchange of cash flows based upon a fixed and
floating rate.
Often referred to a “plain vanilla” swap, the
agreement consists of one party paying a fixed
interest rate on a notional principal amount in
exchange for the other party paying a floating rate
on the same notional principal amount for a set
period of time.
In this case the currency of the agreement is the
same for both parties.
Notional Principal
The term notional principal implies that the
principal itself is not exchanged. If it was
exchanged at the end of the swap, the exact
same cash flows would result.
An Example
Company B agrees to pay A 5% per annum on
a notional principal of $100 million
Company A Agrees to pay B the 6 month LIBOR
rate prevailing 6 months prior to each payment
date, on $100 million. (generally the floating
rate is set at the beginning of the period for
which it is to be paid)
The Fixed Side
We assume that the exchange of cash flows
should occur each six months (using a fixed
rate of 5% compounded semi annually).
Company B will pay:
($100M)(.025) = $2.5 Million
to Firm A each 6 months
Summary of Cash Flows
for Firm B
Date
3-1-98
9-1-98
3-1-99
9-1-99
3-1-00
9-1-00
3-1-01
LIBOR
4.2%
4.8%
5.3%
5.5%
5.6%
5.9%
6.4%
Cash Flow
Received
2.10
2.40
2.65
2.75
2.80
2.95
Cash Flow
Net
Paid
Cash Flow
2.5
2.5
2.5
2.5
2.5
2.5
-0.4
-0.1
0.15
0.25
0.30
0.45
Swap Diagram
LIBOR
Company A
Company B
5%
Offsetting Spot Position
Assume that A has a commitment to borrow at a fixed rate of
5.2% and that B has a commitment to borrow at a rate of
LIBOR + .8%
Company A
Borrows (pays)
5.2%
Pays
LIBOR
Receives
5%
Net
LIBOR+.2%
Company B
Borrows (pays) LIBOR+.8%
Receives
LIBOR
Pays
5%
Net
5.8%
Swap Diagram
5.2%
Company A
LIBOR +.2%
LIBOR
5%
Company B
LIBOR+.8%
5.8%
The swap in effect transforms a fixed rate liability
or asset to a floating rate liability or asset (and
vice versa) for the firms respectively.
Role of Intermediary
Usually a financial intermediary works to
establish the swap by bring the two parties
together.
The intermediary then earns .03 to .04% per
annum in exchange for arranging the swap.
The financial institution is actually entering into
two offsetting swap transactions, one with each
company.
Swap Diagram
5.2%
LIBOR
Co A
4.985%
LIBOR
FI
5.015%
Co B
A pays LIBOR+.215%
B pays 5.815%
The FI makes .03%
LIBOR+.8%
Role of the Intermediary
It is unlikely that a financial intermediary will be
contacted by parties on both side of a swap at
the same time.
The intermediary must enter into the swap
without the counter party. The intermediary
then hedges the interest rate risk using interest
rate instruments while waiting for a counter
party to emerge.
This practice is referred to as warehousing
swaps.
Why enter into a swap?
The Comparative Advantage Argument
Fixed
Floating
A
10%
6 mo LIBOR+.3
B
11.2%
6 mo LIBOR +
1.0%
Difference between fixed rates = 1.2%
Difference between floating rates = 0.7%
B Has an advantage in the floating rate.
Swap Diagram
10%
LIBOR
Co A
9.935%
LIBOR
FI
9.965%
Co B
LIBOR+1%
A pays LIBOR+.065% instead of LIBOR+.3%
B pays 10.965% instead of 11.2%
The FI makes .03%
Spread Differentials
Why do spread differentials exist?
Differences in business lines, credit history,
asset and liabilities, etc…
Why Enter Into A Swap?
Managing Cash Flows
Assume that an insurance firm sold an annuity
lasting 5 years and paying 5 Million each year.
To offset the cash outflows they invest in a 10
year security that pays $6 million each year.
The firm runs a reinvestment risk when they
stop paying the cash outflows on the annuity –
a combination of swaps could eliminate this risk
(on board in class)
Valuation of Interest Rate Swaps
After the swap is entered into it can be valued
as either:
A long position in one bond combined with a
short position in another bond or
A portfolio of forward rate agreements.
Relationship of Swaps to Bonds
In the examples above the same relationship
could have been written as
Company B lent company A $100 million at the
six month LIBOR rate
Company A lent company B $100 million at a
fixed 5% per annum
Bond Valuation
Given the same floating rates as before the
cash flow would be the same as in the swap
example.
The value of the swap would then be the
difference between the value of the fixed rate
bond and the floating rate bond.
Fixed portion
The value of either bond can be found by
discounting the cash flows from the bond (as
always). The fixed rate value is straight
forward it is given as:
n
B fix 

k /(1  rt ) t  NP /(1  r ) n
t 1
where NP is the notional principal and k is the
fixed interest payment
Floating rate valuation
The floating rate is based on the fact that it is a
series of short term six months loans.
Immediately after a payment date Bfl is equal
to the notional principal Q. Allowing the time
until the next payment to equal t1
B fl  Q /(1  r1 )  k /(1  r1 )
t1
*
t1
where k* is the known next payment
Current Swap Value
If the financial institution is paying fixed and
receiving floating the value of the swap is
Vswap = Bfl-Bfix
The other party will have a value of
Vswap = Bfix-Bfl
Example
Assume you previously entered into a swap with the
following conditions:
Pay 6 mo LIBOR & receive 8% on 100M
6 mo 10%
12 mo 10.5%
18 mo 11%
Bfix = 4/(1+.05)+4/(1+(.105/2))2
+104/(1+(.11/2))3
=96.00M
Bfloat = (100+5)/(1.05) = 100
=-100M
-4 M
A better valuation
Relationship of Swap value to Forward Rte
Agreements
Since the swap could be valued as a forward
rate agreement (FRA) it is also possible to value
the swap under the assumption that the
forward rates are realized.
To do this you would need to:
Calculate the forward rates for each of the
LIBOR rates that will determine swap cash flows
Calculate swap cash flows using the forward
rates for the floating portion on the assumption
that the LIBOR rates will equal the forward
rates
Set the swap value equal to the present value
of these cash flows.
Swap Rate
This works after you know the fixed rate.
When entering into the swap the value of
the swap should be 0.
This implies that the PV of each of the two
series of cash flows is equal. Each party is then
willing to exchange the cash flows since they
have the same value.
The rate that makes the PV equal when used
for the fixed payments is the swap rate.
Example
Assume that you are considering a swap where
the party with the floating rate will pay the
three month LIBOR on the $50 Million in
principal.
The parties will swap quarterly payments each
quarter for the next year.
Both the fixed and floating rates are to be paid
on an actual/360 day basis.
First floating payment
Assume that the current 3 month LIBOR rate is
3.80% and that there are 93 days in the first
period.
The first floating payment would then be
 93 
.038
50,000,000  490,833.3333
 360 
Second floating payment
Assume that the three month futures price on
the Eurodollar futures is 96.05 implying a
forward rate of 100-96.05 = 3.95
Given that there are 91 days in the period.
The second floating payment would then be
 91 
.0395
50,000,000  499,236.1111
 360 
Example Floating side
Period
Day
Count
Futures
Price
91
Fwd
Rate
Floating
Cash flow
3.80
1
93
96.05
3.95
490,833.3333
2
91
95.55
4.45
499,236.1111
3
90
95.28
4.72
556,250.0000
4
91
596,555.5555
PV of Floating cash flows
The PV of the floating cash flows is then
calculated using the same forward rates.
The first cash flow will have a PV of:
490,833.3333
 486,061.8263

 93  
1  .038

 360  

PV of Floating cash flows
The PV of the floating cash flows is then
calculated using the same forward rates.
The second cash flow will have a PV of:
499,236.1111
 489,495.4412

 93  
 91  
1  .038
 1  .0395

 360  
 360  

Example Floating side
Day
Period
Count
Fwd
Rate
Floating
Cash flow
PV of Floating CF
91
3.80
1
93
3.95
490,833.3333
486,061.8263
2
91
4.45
499,236.1111
489,495.4412
3
90
4.72
556,250.0000
539,396.1423
4
91
596,555.5555
525,668.5915
PV of floating
The total PV of the floating cash flows is then
the sum of the four PV’s:
$2,040,622.0013
Swap rate
The fixed rate is then the rate that using the
same procedure will cause the PV of the fixed
cash flows to have a PV equal to the same
amount.
The fixed cash flows are discounted by the
same rates as the floating rates.
Note: the fixed cash flows are not the same
each time due to the changes in the number of
days in each period.
The resulting rate is 4.1294686
Example: Swap Cash Flows
Day
Period
Count
Fwd
Rate
Floating
Cash flow
Fixed CF
91
3.80
1
93
3.95
490,833.3333
533,389.7003
2
91
4.45
499,236.1111
521,918.9541
3
90
4.72
556,250.0000
516,183.5810
4
91
596,555.5555
521,918.9541
Swap Spread
The swap spread would then be the difference
between the swap rate and the on the run
treasury of the same maturity.
Swap valuation revisited
The value of the swap will change over time.
After the first payments are made, the futures
prices and corresponding interest rates have
likely changed.
The actual second payment will be based upon
the 3 month LIBOR at the end of the first
period.
Therefore the value of the swap is recalculated.