Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.1
Interest Rate Derivatives,
The Standard Market Models
Chapter 22
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.2Why Interest Rate Derivatives are
Much More Difficult to Value Than
Stock Options
? We are dealing with the whole term
structure of interest rates; not a single
variable
? The probabilistic behavior of an
individual interest rate is more
complicated than that of a stock price
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.3Why Interest Rate Derivatives are
Much More Difficult to Value Than
Stock Options
? Volatilities of different points on
the term structure are different
? Interest rates are used for
discounting as well as for defining
the payoff
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.4
Main Approaches to Pricing
Interest Rate Options
? Use a variant of Black’s
model
? Use a no-arbitrage (yield
curve based) model
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.5
Black’s Model & Its Extensions
? Black’s model is similar to the
Black-Scholes model used for
valuing stock options
? It assumes that the value of an
interest rate,a bond price,or some
other variable at a particular time T
in the future has a lognormal
distribution
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.6Black’s Model & Its Extensions
(continued)
? The mean of the probability distribution is
the forward value of the variable
? The standard deviation of the probability
distribution of the log of the variable is
where s is the volatility
? The expected payoff is discounted at the
T-maturity rate observed today
s T
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.7
Black’s Model (Eqn 22.1 and 22.2,p 509)
Tdd
T
TKF
d
dNFdKNTPp
dKNdNFTPc
s??
s
s?
?
????
??
12
2
0
1
102
210;
2/)/ln (
)]()()[,0(
)]()()[,0(
? K, strike price
? F0, forward value of
variable
? T, option maturity
? s, volatility
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.8The Black’s Model,Payoff Later
Than Variable Being Observed
? K, strike price
? F0, forward value of
variable
? s, volatility
? T, time when
variable is observed
? T *, time of payoff
Tdd
T
TKF
d
dNFdKNTPp
dKNdNFTPc
s??
s
s?
?
????
??
12
2
0
1
102
*
210
*;
2/)/l n (
)]()()[,0(
)]()()[,0(
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.9
Validity of Black’s Model
Black’s model appears to make two
approximations:
1,The expected value of the underlying
variable is assumed to be its forward price
2,Interest rates are assumed to be constant
for discounting
We will see that these assumptions offset
each other
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.10
European Bond Options
? When valuing European bond options it is
usual to assume that the future bond price
is lognormal
? We can then use Black’s model (equations
22.1 and 22.2)
? Both the bond price and the strike price
should be cash prices not quoted prices
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.11
Yield Vols vs Price Vols
The change in forward bond price is related to
the change in forward bond yield by
where D is the (modified) duration of the
forward bond at option maturity
y
yDy
B
ByD
B
B ???????? or
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.12
Yield Vols vs Price Vols
continued
? This relationship implies the following
approximation
where sy is the yield volatility and s is the
price volatility,y0 is today’s forward yield
? Often sy is quoted with the understanding
that this relationship will be used to calculate
s
yDy s?s 0
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.13
Theoretical Justification for
Bond Option Model
m o d e l sB l a c k ' to l e a d s T h i s
A l s o
is p r i c e o p ti o n the ti m e at m a tu r i n g b o n d
c o u p o n-z e r o a w r tF R N is th a t w o r l da in W o r k i n g
0
][
)]0,[ m a x (),0(
,
FBE
KBETP
T
TT
TT
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.14Caps
? A cap is a portfolio of caplets
? Each caplet can be regarded as a call
option on a future interest rate with the
payoff occurring in arrears
? When using Black’s model we assume
that the interest rate underlying each
caplet is lognormal
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.15Black’s Model for Caps
(Equation 22.11,p,517)
? The value of a caplet,for period [tk,tk+1] is
? Fk, forward interest rate
for (tk,tk+1)
? sk, interest rate volatility
? L,principal
? RK, cap rate
? ?k=tk+1-tk
-= and w h e r e
k
kk
kkKk
Kkkk
tdd
t
tRF
d
dNRdNFtPL
s
s
s?
?
?? ?
12
2
1
211
2/)/l n (
)]()()[,0(
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.16
When Applying Black’s Model
To Caps We Must,..
? EITHER
– Use forward volatilities
– Volatility different for each caplet
? OR
– Use flat volatilities
– Volatility same for each caplet
within a particular cap but varies
according to life of cap
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.17
Theoretical Justification for Cap
Model
W or k i ng i n a w or l d t hat i s F R N w r t a
z er o - c oup on bon d mat ur i ng a t ti me
t he opt i o n pr i c e i s
A l s o
Thi s l ead s t o B l ac k ' s m ode l
t
P t E R R
E R F
k
k k k X
k k k
?
? ?
?
?
?
1
1 1
1
0 0(,) [ m a x (,)]
[ ]
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.18European Swaptions
? When valuing European swap options it is
usual to assume that the swap rate is
lognormal
? Consider a swaption which gives the right to
pay sK on an n -year swap starting at time T,
The payoff on each swap payment date is
where L is principal,m is payment frequency
and sT is market swap rate at time T
m a x )0,( KT ssmL ?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.19European Swaptions continued
(Equation 22.13,page 545)
The value of the swaption is
s0 is the forward swap rate; s is the swap rate
volatility; ti is the time from today until the i th
swap payment; and
)]()([ 210 dNsdNsLA K?
A m P t i
i
m n
?
?
?1 0
1
(,)
Tdd
T
Tssd K s??
s
s??
12
2
0
1 ;
2/)/l n (w h e r e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.20Theoretical Justification for
Swap Option Model
m o d e l sB l a c k ' to l e a d s T h i s
A l s o
is p r i c e o p t i o n th e
s w a p,th e u n d e r l y i n ga n n u i ty th e
w r tF R N is th a t w o r l da in W o r k i n g
0
][
)]0,[ m a x (
ssE
ssL A E
TA
KTA
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.21Relationship Between Swaptions
and Bond Options
? An interest rate swap can be regarded as
the exchange of a fixed-rate bond for a
floating-rate bond
? A swaption or swap option is therefore an
option to exchange a fixed-rate bond for a
floating-rate bond
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.22Relationship Between Swaptions
and Bond Options (continued)
? At the start of the swap the floating-rate
bond is worth par so that the swaption can
be viewed as an option to exchange a fixed-
rate bond for par
? An option on a swap where fixed is paid and
floating is received is a put option on the
bond with a strike price of par
? When floating is paid and fixed is received,it
is a call option on the bond with a strike
price of par
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.23Convexity Adjustments
? We define the forward yield on a bond as the yield
calculated from the forward bond price
? There is a non-linear relation between bond yields and bond
prices
? It follows that when the forward bond price equals the
expected future bond price,the forward yield does not
necessarily equal the expected future yield
? What is known as a convexity adjustment may be necessary
to convert a forward yield to the appropriate expected future
yield
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.24Relationship Between Bond Yields
and Prices (Figure 22.4,page 525)
Bond
Price
YieldY3
B 1
Y1Y2
B 3
B 2
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.25Analytic Approximation for
Convexity Adjustment (Eqn 22.15,p,525)
? Suppose a derivative depends on a bond yield,yT
observed at time T, Define,
? G(yT), price of the bond as a function of its yield
y0, forward bond yield at time zero
sy, forward yield volatility
? The convexity adjustment that should be made to the
forward bond yield is
? ??
?
1
2 0
2 2 0
0
y T G y
G yy
s ( )
( )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.26
Convexity Adjustment for Swap
Rate
The same formula gives the convexity
adjustment for a forward swap rate,In
this case G(y) defines the relationship
between price and yield for a bond that
pays a coupon equal to the forward
swap rate
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.27
Example 22.5 (page 526)
? An instrument provides a payoff in 3 years equal
to the 1-year zero-coupon rate multiplied by
$1000
? Volatility is 20%
? Yield curve is flat at 10% (with annual
compounding)
? The convexity adjustment is 10.9 bps so that the
value of the instrument is 101.09/1.13 = 75.95
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.28
Example 22.6 (Page 527)
? An instrument provides a payoff in 3 years =
to the 3-year swap rate multiplied by $100
? Payments are made annually on the swap
? Volatility is 22%
? Yield curve is flat at 12% (with annual
compounding)
? The convexity adjustment is 36 bps so that
the value of the instrument is 12.36/1.123 =
8.80
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.29
Timing Adjustments
When a variable is observed at time T1 and the
resultant payoff occurs at time T2 rather than T1,the
growth rate of the variable should be increased by
where R is the forward interest rate between T1 and T2
expressed with a compounding frequency of m,sR is
the volatility of R,R0 is the value of R today,F is the
forward value of the variable for a contract maturing
at time T1,,,sF is the volatility of F,and r is the
correlation between R and F
mR
TTRRF
/1
)(
0
120
?
?srs?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.30
When is a Convexity or Timing
Adjustment Necessary
? A convexity or timing adjustment is
necessary when the payoff from a
derivative does not incorporate the
natural time lags between an interest
rate being set and the interest payments
being made
? They are not necessary for a vanilla
swap,a cap or a swap option
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.31
Deltas of Interest Rate Derivatives
Alternatives:
? Calculate a DV01 (the impact of a 1bps parallel shift
in the zero curve)
? Calculate impact of small change in the quote for
each instrument used to calculate the zero curve
? Divide zero curve (or forward curve) into buckets and
calculate the impact of a shift in each bucket
? Carry out a principal components analysis,Calculate
delta with respect to each of the first few factors
factors
22.1
Interest Rate Derivatives,
The Standard Market Models
Chapter 22
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.2Why Interest Rate Derivatives are
Much More Difficult to Value Than
Stock Options
? We are dealing with the whole term
structure of interest rates; not a single
variable
? The probabilistic behavior of an
individual interest rate is more
complicated than that of a stock price
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.3Why Interest Rate Derivatives are
Much More Difficult to Value Than
Stock Options
? Volatilities of different points on
the term structure are different
? Interest rates are used for
discounting as well as for defining
the payoff
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.4
Main Approaches to Pricing
Interest Rate Options
? Use a variant of Black’s
model
? Use a no-arbitrage (yield
curve based) model
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.5
Black’s Model & Its Extensions
? Black’s model is similar to the
Black-Scholes model used for
valuing stock options
? It assumes that the value of an
interest rate,a bond price,or some
other variable at a particular time T
in the future has a lognormal
distribution
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.6Black’s Model & Its Extensions
(continued)
? The mean of the probability distribution is
the forward value of the variable
? The standard deviation of the probability
distribution of the log of the variable is
where s is the volatility
? The expected payoff is discounted at the
T-maturity rate observed today
s T
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.7
Black’s Model (Eqn 22.1 and 22.2,p 509)
Tdd
T
TKF
d
dNFdKNTPp
dKNdNFTPc
s??
s
s?
?
????
??
12
2
0
1
102
210;
2/)/ln (
)]()()[,0(
)]()()[,0(
? K, strike price
? F0, forward value of
variable
? T, option maturity
? s, volatility
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.8The Black’s Model,Payoff Later
Than Variable Being Observed
? K, strike price
? F0, forward value of
variable
? s, volatility
? T, time when
variable is observed
? T *, time of payoff
Tdd
T
TKF
d
dNFdKNTPp
dKNdNFTPc
s??
s
s?
?
????
??
12
2
0
1
102
*
210
*;
2/)/l n (
)]()()[,0(
)]()()[,0(
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.9
Validity of Black’s Model
Black’s model appears to make two
approximations:
1,The expected value of the underlying
variable is assumed to be its forward price
2,Interest rates are assumed to be constant
for discounting
We will see that these assumptions offset
each other
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.10
European Bond Options
? When valuing European bond options it is
usual to assume that the future bond price
is lognormal
? We can then use Black’s model (equations
22.1 and 22.2)
? Both the bond price and the strike price
should be cash prices not quoted prices
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.11
Yield Vols vs Price Vols
The change in forward bond price is related to
the change in forward bond yield by
where D is the (modified) duration of the
forward bond at option maturity
y
yDy
B
ByD
B
B ???????? or
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.12
Yield Vols vs Price Vols
continued
? This relationship implies the following
approximation
where sy is the yield volatility and s is the
price volatility,y0 is today’s forward yield
? Often sy is quoted with the understanding
that this relationship will be used to calculate
s
yDy s?s 0
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.13
Theoretical Justification for
Bond Option Model
m o d e l sB l a c k ' to l e a d s T h i s
A l s o
is p r i c e o p ti o n the ti m e at m a tu r i n g b o n d
c o u p o n-z e r o a w r tF R N is th a t w o r l da in W o r k i n g
0
][
)]0,[ m a x (),0(
,
FBE
KBETP
T
TT
TT
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.14Caps
? A cap is a portfolio of caplets
? Each caplet can be regarded as a call
option on a future interest rate with the
payoff occurring in arrears
? When using Black’s model we assume
that the interest rate underlying each
caplet is lognormal
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.15Black’s Model for Caps
(Equation 22.11,p,517)
? The value of a caplet,for period [tk,tk+1] is
? Fk, forward interest rate
for (tk,tk+1)
? sk, interest rate volatility
? L,principal
? RK, cap rate
? ?k=tk+1-tk
-= and w h e r e
k
kk
kkKk
Kkkk
tdd
t
tRF
d
dNRdNFtPL
s
s
s?
?
?? ?
12
2
1
211
2/)/l n (
)]()()[,0(
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.16
When Applying Black’s Model
To Caps We Must,..
? EITHER
– Use forward volatilities
– Volatility different for each caplet
? OR
– Use flat volatilities
– Volatility same for each caplet
within a particular cap but varies
according to life of cap
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.17
Theoretical Justification for Cap
Model
W or k i ng i n a w or l d t hat i s F R N w r t a
z er o - c oup on bon d mat ur i ng a t ti me
t he opt i o n pr i c e i s
A l s o
Thi s l ead s t o B l ac k ' s m ode l
t
P t E R R
E R F
k
k k k X
k k k
?
? ?
?
?
?
1
1 1
1
0 0(,) [ m a x (,)]
[ ]
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.18European Swaptions
? When valuing European swap options it is
usual to assume that the swap rate is
lognormal
? Consider a swaption which gives the right to
pay sK on an n -year swap starting at time T,
The payoff on each swap payment date is
where L is principal,m is payment frequency
and sT is market swap rate at time T
m a x )0,( KT ssmL ?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.19European Swaptions continued
(Equation 22.13,page 545)
The value of the swaption is
s0 is the forward swap rate; s is the swap rate
volatility; ti is the time from today until the i th
swap payment; and
)]()([ 210 dNsdNsLA K?
A m P t i
i
m n
?
?
?1 0
1
(,)
Tdd
T
Tssd K s??
s
s??
12
2
0
1 ;
2/)/l n (w h e r e
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.20Theoretical Justification for
Swap Option Model
m o d e l sB l a c k ' to l e a d s T h i s
A l s o
is p r i c e o p t i o n th e
s w a p,th e u n d e r l y i n ga n n u i ty th e
w r tF R N is th a t w o r l da in W o r k i n g
0
][
)]0,[ m a x (
ssE
ssL A E
TA
KTA
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.21Relationship Between Swaptions
and Bond Options
? An interest rate swap can be regarded as
the exchange of a fixed-rate bond for a
floating-rate bond
? A swaption or swap option is therefore an
option to exchange a fixed-rate bond for a
floating-rate bond
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.22Relationship Between Swaptions
and Bond Options (continued)
? At the start of the swap the floating-rate
bond is worth par so that the swaption can
be viewed as an option to exchange a fixed-
rate bond for par
? An option on a swap where fixed is paid and
floating is received is a put option on the
bond with a strike price of par
? When floating is paid and fixed is received,it
is a call option on the bond with a strike
price of par
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.23Convexity Adjustments
? We define the forward yield on a bond as the yield
calculated from the forward bond price
? There is a non-linear relation between bond yields and bond
prices
? It follows that when the forward bond price equals the
expected future bond price,the forward yield does not
necessarily equal the expected future yield
? What is known as a convexity adjustment may be necessary
to convert a forward yield to the appropriate expected future
yield
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.24Relationship Between Bond Yields
and Prices (Figure 22.4,page 525)
Bond
Price
YieldY3
B 1
Y1Y2
B 3
B 2
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.25Analytic Approximation for
Convexity Adjustment (Eqn 22.15,p,525)
? Suppose a derivative depends on a bond yield,yT
observed at time T, Define,
? G(yT), price of the bond as a function of its yield
y0, forward bond yield at time zero
sy, forward yield volatility
? The convexity adjustment that should be made to the
forward bond yield is
? ??
?
1
2 0
2 2 0
0
y T G y
G yy
s ( )
( )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.26
Convexity Adjustment for Swap
Rate
The same formula gives the convexity
adjustment for a forward swap rate,In
this case G(y) defines the relationship
between price and yield for a bond that
pays a coupon equal to the forward
swap rate
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.27
Example 22.5 (page 526)
? An instrument provides a payoff in 3 years equal
to the 1-year zero-coupon rate multiplied by
$1000
? Volatility is 20%
? Yield curve is flat at 10% (with annual
compounding)
? The convexity adjustment is 10.9 bps so that the
value of the instrument is 101.09/1.13 = 75.95
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.28
Example 22.6 (Page 527)
? An instrument provides a payoff in 3 years =
to the 3-year swap rate multiplied by $100
? Payments are made annually on the swap
? Volatility is 22%
? Yield curve is flat at 12% (with annual
compounding)
? The convexity adjustment is 36 bps so that
the value of the instrument is 12.36/1.123 =
8.80
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.29
Timing Adjustments
When a variable is observed at time T1 and the
resultant payoff occurs at time T2 rather than T1,the
growth rate of the variable should be increased by
where R is the forward interest rate between T1 and T2
expressed with a compounding frequency of m,sR is
the volatility of R,R0 is the value of R today,F is the
forward value of the variable for a contract maturing
at time T1,,,sF is the volatility of F,and r is the
correlation between R and F
mR
TTRRF
/1
)(
0
120
?
?srs?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.30
When is a Convexity or Timing
Adjustment Necessary
? A convexity or timing adjustment is
necessary when the payoff from a
derivative does not incorporate the
natural time lags between an interest
rate being set and the interest payments
being made
? They are not necessary for a vanilla
swap,a cap or a swap option
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
22.31
Deltas of Interest Rate Derivatives
Alternatives:
? Calculate a DV01 (the impact of a 1bps parallel shift
in the zero curve)
? Calculate impact of small change in the quote for
each instrument used to calculate the zero curve
? Divide zero curve (or forward curve) into buckets and
calculate the impact of a shift in each bucket
? Carry out a principal components analysis,Calculate
delta with respect to each of the first few factors
factors
