The Standard Market Models
Financial Innovation & Product Design II
Dr. Helmut Elsinger
« Options, Futures and Other Derivatives »,
John Hull, Chapter 22
BIART Sébastien
Introduction
 What
 Why
are IR derivatives ?
are IR derivatives
important ?
IR derivatives : valuation

Black-Scholes collapses
1.
Volatility of underlying asset
constant
2.
Interest rate constant
IR derivatives : valuation
Why is it difficult ?




Dealing with the whole term structure
Complicated probabilistic behavior of
individual interest rates
Volatilities not constant in time
Interest rates are used for discounting as
well as for defining the payoff
Main Approaches to Pricing
Interest Rate Options

3 approaches:
1.
Stick to Black-Scholes
2.
Model term structure : Use a
variant of Black’s model
3.
Start from current term
structure: Use a no-arbitrage
(yield curve based) model
Black’s Model
The Black-Scholes formula for a European call on a stock
providing a continuous dividend yield can be written as:
ce
 rT
S e
0
 qT
e N (d1 )  KN (d 2 )
rT

But Se-qTerT is the forward price F of the underlying asset (variable)
 This is Black’s Model for pricing options :
c  e  rT F 0 N (d1 )  KN (d 2 )
p  e  rT  F 0 N (d1 )  KN (d 2 )
with :
d1 
ln( F 0 / K )
 0.5 T
 T
d 2  d1   T
Black’s Model
c  P(0, T )[ F0 N (d1 )  KN (d 2 )]
p  P(0, T )[ KN (d 2 )  F0 N (d1 )]
ln( F0 / K )   2T / 2
d1 
; d 2  d1   T
 T
•
•
K : strike price
F0 : forward value of variable
•

T : option maturity
 : volatility
The Black’s Model: Payoff Later
Than Variable Being Observed
c  P(0, T * )[ F0 N (d1 )  KN (d 2 )]
p  P(0, T * )[ KN (d 2 )  F0 N (d1 )]
ln( F0 / K )   2T / 2
d1 
;d 2  d1   T
 T
• K
• F0
•
: strike price
: forward value of
variable
 : volatility
•
•
T : time when
variable is observed
T * : time of payoff
Validity of Black’s Model
Black’s model appears to make two
approximations:


The expected value of the underlying
variable is assumed to be its forward price
Interest rates are assumed to be
constant for discounting
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
Example : Options on zerocoupons vs. Options on IR







Let us consider a 6-month call option on a 9month zero-coupon with face value 100
Current spot price of zero-coupon = 95.60
Exercise price of call option = 98
Payoff at maturity: Max(0, ST – 98)
The spot price of zero-coupon at the maturity of
the option depend on the 3-month interest rate
prevailing at that date.
ST = 100 / (1 + rT 0.25)
Exercise option if:


ST > 98
rT < 8.16%
Example : Options on zerocoupons vs. Options on IR


The exercise rate of the call option is R = 8.16%
With a little bit of algebra, the payoff of the option
can be written as:
98(8.16%  rT )0.25
Max (0,
)
1  rT 0.25


Interpretation: the payoff of an interest rate put
option
The owner of an IR put option:




Receives the difference (if positive) between a fixed
rate and a variable rate
Calculated on a notional amount
For an fixed length of time
At the beginning of the IR period
European options on interest rates
Options on zero-coupons


Face value: M(1+R)
Exercise price K
A call option
 Payoff:
Max(0, ST – K)
A put option
 Payoff:
Max(0, K – ST )
Option on interest rate

Exercise rate R
A put option
 Payoff:
Max[0, M (R-rT) /
(1+rT)]
A call option
 Payoff:
Max[0, M (rT -R) /
(1+rT)]
Yield Volatilities vs Price Volatilities
The change in forward bond price is related to
the change in forward bond yield by
B
B
  D y, or
B
B
  Dy
y
y
where D is the (modified) duration of the
forward bond at option maturity
Yield Volatilities vs Price Volatilities

This relationship implies the
following approximation :   Dy 0 y
where sy is the yield volatility and s is the
price volatility, y0 is today’s forward yield

Often  y is quoted with the
understanding that this relationship
will be used to calculate 
Interest Rate Caps


A cap is a collection of call options on
interest rates (caplets).
When using Black’s model we assume
that the interest rate underlying each
caplet is lognormal
Interest Rate Caps
The cash flow for each caplet at time t is: Max[0, M (rt – R) ]

M is the principal amount of the cap

R is the cap rate

rt is the reference variable interest rate

is the tenor of the cap (the time period between
payments)
Used for hedging purpose by companies borrowing at variable
rate

If rate rt < R : CF from borrowing = – M rt 
If rate rT > R: CF from borrowing = – M rT  + M (rt – R)  =
–MR

Black’s Model for Caps

The value of a caplet, for period [tk, tk+1] is
L k P(0, tk 1 )[ Fk N (d1 )  RK N (d 2 )]
ln( Fk / RK )   2k tk / 2
where d1 
and d 2 = d1 -  tk
k tk
•
•
Fk : forward interest rate
for (tk, tk+1)
k : interest rate volatility
•
•
L: principal
RK : cap rate
 k=tk+1-tk
Example 22.3
•1-year cap on 3 month LIBOR
•Cap rate = 8% (quarterly compounding)
•Principal amount = $10,000
•Maturity
1
•Spot rate
6.39% 6.50%
•Discount factors
1.25
0.93810.9220
•Yield volatility = 20%
•Payoff at maturity (in 1 year) =
•Max{0, [10,000  (r – 8%)0.25]/(1+r  0.25)}
Example 22.3
The Cap as a portfolio of IR Options :
 Step 1 : Calculate 3-month forward in 1 year :


F = [(0.9381/0.9220)-1]  4 = 7% (with simple
compounding)
Step 2 : Use Black
7%
)
8
%
d1 
 0.5  0.20  1  0.5677  N (d1)  0.2851
0.20  1
ln(
d 2  0.5677  0.5  0.20  1  .7677  N (d 2 )  0.2213
Value of cap = 10,000  0.9220 [7%  0.2851 –
8%  0.2213]  0.25 = 5.19
cash flow takes place in 1.25 year
Example 22.3
The Cap as a portfolio of Bond Options :
1-year cap on 3 month LIBOR
Cap rate = 8%
Principal amount = 10,000
Maturity
1
1.25
Spot rate
6.39% 6.50%
Discount factors
0.938 0.9220
Yield volatility = 20%
1-year put on a 1.25 year zero-coupon
Face value = 10,200
[10,000 (1+8% * 0.25)]
Striking price = 10,000
Using Black’s model with:
Spot price of zero-coupon = 10,200 *
.9220 = 9,404
1-year forward price = 9,404 / 0.9381 =
10,025
Price volatility = (20%) * (6.94%) *
(0.25) = 0.35%
F = 10,025
K = 10,000
r = 6.39%
T=1
 = 0.35%
Put (cap) = 4.607
Delta = - 0.239
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
European 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
L
max ( sT  sK , 0)
m
where L is principal, m is payment frequency
and sT is market swap rate at time T
European Swaptions
The value of the swaption is
LA[ s0 N (d1 )  sK N (d 2 )]
ln( s0 / sK )  2T / 2
where d1 
; d 2  d1   T
 T
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
1 mn
A   P(0, ti )
m i 1
Relationship Between Swaptions and
Bond Options
1. Interest rate swap = the exchange of a fixedrate bond for a floating-rate bond
2. A swaption = option to exchange a fixed-rate
bond for a floating-rate bond
3. 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
Relationship Between Swaptions and
Bond Options
4. 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
5. When floating is paid and fixed is received,
it is a call option on the bond with a strike
price of par
… Thank you !