Chapter 14
Pricing of Interest Rate Derivatives
In this chapter we consider the pricing of caplets, caps, and swaptions, using
change of numéraire and forward swap measures.
14.1 Forward Measures and Tenor Structure
The maturity dates are arranged according to a discrete tenor structure
{0 = T0 < T1 < T2 < · · · < Tn }.
An example of forward interest rate curve data is given in the table of Figure 14.1, which contains the values of (T1 , T2 , . . . , T23 ) and of {f (t, t + Ti , t +
Ti + δ)}i=1,2,...,23 , with t = 07/05/2003 and δ = six months.
2D
2.55
8Y
3.88
1W
2.53
9Y
4.02
1M
2.56
10Y
4.14
2M
2.52
11Y
4.23
3M
2.48
12Y
4.33
1Y
2.34
13Y
4.40
2Y
2.49
14Y
4.47
3Y
2.79
15Y
4.54
4Y
3.07
20Y
4.74
5Y
3.31
25Y
4.83
6Y 7Y
3.52 3.71
30Y
4.86
Fig. 14.1: Forward rates arranged according to a tenor structure.
Recall that by definition of P (t, Ti ) and absence of arbitrage the process
t 7−→ e−
rt
0
rs ds
P (t, Ti ),
0 ≤ t ≤ Ti ,
i = 1, 2, . . . , n,
is an Ft -martingale under P = P, and as a consequence (P (t, Ti ))t∈[0,Ti ] can
be taken as numéraire in the definition
∗
r Ti
dP̂i
1
=
e− 0 rs ds
dP
P (0, Ti )
"
(14.1)
N. Privault
of the forward measure P̂i . The following proposition will allow us to price
contingent claims using the forward measure P̂i , it is a direct consequence of
Proposition 12.2, noting that here we have P (Ti , Ti ) = 1.
Proposition 14.1. For all sufficiently integrable random variables F we
have
i
h
r Ti
ˆ i [F | Ft ], 0 ≤ t ≤ T, i = 1, 2, . . . , n.
IE∗ F e− t rs ds Ft = P (t, Ti )IE
(14.2)
Recall that for all Ti , Tj ≥ 0, the deflated process
t 7−→
P (t, Tj )
,
P (t, Ti )
0 ≤ t ≤ min(Ti , Tj ),
is an Ft -martingale under P̂i , cf. Proposition 12.3.
Dynamics under the forward measure
In order to apply Proposition 14.1 and to compute the price
i
h rT
ˆ i [F | Ft ],
IE∗ e− t rs ds F Ft = P (t, Ti )IE
it can be useful to determine the dynamics of the underlying processes rt ,
f (t, T, S), and P (t, T ) under the forward measure P̂i .
Let us assume that the dynamics of the bond price P (t, Ti ) is given by
dP (t, Ti )
= rt dt + ζi (t)dWt ,
P (t, Ti )
(14.3)
for i = 1, 2, . . . , n, where (Wt )t∈R+ is a standard Brownian motion under
P and (rt )t∈R+ and (ζi (t))t∈R+ are adapted processes with respect to the
filtration (Ft )t∈R+ generated by (Wt )t∈R+ , i.e.
w
wt
t
1wt
|ζi (s)|2 ds ,
P (t, Ti ) = P (0, Ti ) exp
rs ds +
ζi (s)dWs −
0
0
2 0
t ∈ [0, Ti ], i = 1, 2, . . . , n.
Forward Brownian motions
By the Girsanov Theorem 12.4,
wt
Ŵti := Wt −
ζi (s)ds,
0
0 ≤ t ≤ Ti ,
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(14.4)
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Pricing of Interest Rate Derivatives
is a standard Brownian motion under P̂i for all i = 1, 2, . . . , n, cf. e.g. (12.10),
hence we have
dŴti = dWt − ζi (t)dt,
i = 1, 2, . . . , n,
and
dŴtj = dWt − ζj (t)dt = dŴti + (ζi (t) − ζj (t))dt,
i, j = 1, 2, . . . , n,
(Ŵtj )t∈R+
which shows that
has drift (ζi (t) − ζj (t))t∈R+ under P̂i . Hence the
dynamics of t 7−→ P (t, Tj ) under P̂j is given by
dP (t, Tj )
= rt dt + |ζj (t)|2 dt + ζj (t)dŴtj ,
P (t, Tj )
i, j = 1, 2, . . . , n,
(14.5)
where (Ŵtj )t∈R+ is a standard Brownian motion under P̂j , and we have
w
wt
t
1wt
P (t, Tj ) = P (0, Tj ) exp
rs ds +
ζj (s)dŴsj +
|ζj (s)|2 ds
0
0
2 0
w
wt
wt
t
1wt
= P (0, Tj ) exp
rs ds +
ζj (s)dŴsi +
ζj (s)ζi (s)ds −
|ζj (s)|2 ds
0
0
0
2 0
w
wt
t
1wt
1wt
i
= P (0, Tj ) exp
rs ds +
ζj (s)dŴs −
|ζj (s) − ζi (s)|2 ds +
|ζi (s)|2 ds ,
0
0
2 0
2 0
t ∈ [0, Tj ], i, j = 1, 2, . . . , n. Consequently, the forward price P (t, Tj )/P (t, Ti )
can be written as
w
t
P (t, Tj )
P (0, Tj )
1wt
=
exp
(ζj (s) − ζi (s))dŴsj +
|ζj (s) − ζi (s)|2 ds
0
P (t, Ti )
P (0, Ti )
2 0
w
t
P (0, Tj )
1wt
i
=
|ζi (s) − ζj (s)|2 ds ,
exp
(ζj (s) − ζi (s))dŴs −
0
P (0, Ti )
2 0
t ∈ [0, Tj ],
i, j = 1, 2, . . . , n,
(14.6)
which also follows from Proposition 12.5.
In case the short rate process (rt )t∈R+ is given as the (Markovian) solution
to the stochastic differential equation
drt = µ(t, rt )dt + σ(t, rt )dWt ,
its dynamics will be given under P̂i by
drt = µ(t, rt )dt + σ(t, rt )ζi (t)dt + σ(t, rt )dŴti .
(14.7)
In the Vasicek case, by (13.15) we have
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N. Privault
drt = (a − brt )dt + σdWt ,
and
σ
ζi (t) = − (1 − e−b(Ti −t) ),
b
hence from (14.7) we have
drt = (a − brt )dt −
0 ≤ t ≤ Ti ,
σ2
(1 − e−b(Ti −t) )dt + σdŴti
b
(14.8)
and we obtain
dP (t, Ti )
σ2
σ
= rt dt + 2 (1 − e−b(Ti −t) )2 dt − (1 − e−b(Ti −t) )dŴti ,
P (t, Ti )
b
b
from (13.15).
14.2 Bond Options
The next proposition can be obtained as an application of the Margrabe formula (12.26) of Proposition 12.10 by taking Xt = P (t, Tj ), Nt = P (t, Ti ),
and X̂t = Xt /Nt = P (t, Tj )/P (t, Ti ). In the Vasicek model, this formula has
been first obtained in [58].
We work with a standard Brownian motion (Wt )t∈R+ under P, generating
the filtration (Ft )t∈R+ , and an Ft -adapted short rate process (rt )t∈R+ .
Proposition 14.2. Assume that the dynamics of the bond prices P (t, Ti ),
P (t, Tj ) under P∗ are given by
dP (t, Ti )
= rt dt + ζi (t)dWt ,
P (t, Ti )
dP (t, Tj )
= rt dt + ζj (t)dWt ,
P (t, Tj )
where (ζi (t))t∈R+ and (ζj (t))t∈R+ are deterministic functions. Then the price
of a bond call option on P (Ti , Tj ) with payoff F = (P (Ti , Tj ) − κ)+ can be
written as
i
h r Ti
IE∗ e− t rs ds (P (Ti , Tj ) − κ)+ Ft
(14.9)
v
1
P (t, Tj )
v
1
P (t, Tj )
= P (t, Tj )Φ
+ log
− κP (t, Ti )Φ − + log
,
2 v
κP (t, Ti )
2 v
κP (t, Ti )
with v 2 =
w Ti
t
|ζj (s) − ζi (s)|2 ds.
Proof. First we note that using Nt = P (t, Ti ) as a numéraire the price of a
bond call option on P (Ti , Tj ) with payoff F = (P (Ti , Tj )−κ)+ can be written
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Pricing of Interest Rate Derivatives
from Proposition 12.2 using the forward measure P̂, or directly by (12.5), as
i
i
h
h r Ti
ˆ i (P (Ti , Tj ) − κ)+ Ft .
IE∗ e− t rs ds (P (Ti , Tj ) − κ)+ Ft = P (t, Ti )IE
Next, by (14.6) or by solving (12.12) in Proposition 12.5 we can write
P (Ti , Tj ) as the geometric Brownian motion
w
Ti
P (t, Tj )
1 w Ti
P (Ti , Tj ) =
(ζj (s) − ζi (s))dŴsi −
|ζi (s) − ζj (s)|2 ds ,
exp
t
P (t, Ti )
2 t
under the forward measure P̂. Since (ζi (s))s∈[0,Ti ] and (ζj (s))s∈[0,Tj ] in (14.3)
are deterministic functions, P (Ti , Tj ) is a lognormal random variable given
Ft under P̂ and we can use Lemma 6.6 to price the bond option by the
Black-Scholes formula
p
Bl κP (0, Ti ), P (t, Tj ), v/ Ti − t, 0, Ti − t
√
with strike price κP (0, Ti ), underlying P (t, Tj ), volatility parameter v/ Ti − t,
time to maturity Ti − t and zero interest rate, which yields (14.9).
Note that from Corollary 12.12 the decomposition (14.9) gives the selffinancing portfolio in the assets P (t, Ti ) and P (t, Tj ) for the claim with payoff
(P (Ti , Tj ) − κ)+ .
In the Vasicek case the above bond
option price could also be computed
rT
from the joint law of rT , t rs ds , which is Gaussian, or from the dynamics
(14.5)-(14.8) of P (t, T ) and rt under P̂i , cf. § 7.3 of [84], and [66] for the CIR
and other short rate models with correlated Brownian motions.
14.3 Caplet Pricing
The caplet on the spot forward rate f (Ti , Ti , Ti+1 ) with strike price κ is a
contract with payoff
(f (Ti , Ti , Ti+1 ) − κ)+ ,
priced at time t ∈ [0, Ti ] from Proposition 12.2 using the forward measure P̂i
as
i
h r Ti+1
rs ds
IE∗ e− t
(f (Ti , Ti , Ti+1 ) − κ)+ Ft
(14.10)
+
ˆ
= P (t, Ti+1 )IEi+1 (f (Ti , Ti , Ti+1 ) − κ) | Ft ,
by taking Nt = P (t, Ti+1 ) as a numéraire. Next, we consider the caplet with
payoff
(L(Ti , Ti , Ti+1 ) − κ)+
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N. Privault
on the LIBOR rate
L(t, Ti , Ti+1 ) =
1
Ti+1 − Ti
P (t, Ti )
−1 ,
P (t, Ti+1 )
0 ≤ t ≤ Ti < Ti+1 ,
which is a martingale under P̂i+1 defined in (14.1), from Proposition 12.3.
The next formula (14.12) is known as the Black caplet formula. We assume
that L(t, Ti , Ti+1 ) is modeled in the BGM model of Section 13.7.
Proposition 14.3. Assume that L(t, Ti , Ti+1 ) is modeled as
dL(t, Ti , Ti+1 )
= γi (t)dB̂ti+1 ,
L(t, Ti , Ti+1 )
(14.11)
0 ≤ t ≤ Ti , i = 1, 2, . . . , n − 1, where t 7−→ γi (t) is a deterministic function,
i = 1, 2, . . . , n − 1. The caplet on L(Ti , Ti , Ti+1 ) is priced as time t ∈ [0, Ti ]
as
i
h r Ti+1
rs ds
IE∗ e− t
(L(Ti , Ti , Ti+1 ) − κ)+ Ft
= P (t, Ti+1 )L(t, Ti , Ti+1 )Φ(d+ ) − κP (t, Ti+1 )Φ(d− ),
where
d+ =
log(L(t, Ti , Ti+1 )/κ) + σi2 (t)(Ti − t)/2
√
,
σi (t) Ti − t
d− =
log(L(t, Ti , Ti+1 )/κ) − σi2 (t)(Ti − t)/2
√
,
σi (t) Ti − t
and
and
|σi (t)|2 =
(14.12)
0 ≤ t ≤ Ti+1 ,
1 w Ti
|γi |2 (s)ds.
Ti − t t
Proof. Taking P (t, Ti+1 ) as a numéraire, the forward price
X̂t :=
P (t, Ti )
= 1 + (Ti+1 − Ti )L(Ti , Ti , Ti+1 )
P (t, Ti+1 )
is a martingale under P̂i+1 by Proposition 12.3. More precisely, by (14.11)
we have
w
Ti
1 w Ti
L(Ti , Ti , Ti+1 ) = L(t, Ti , Ti+1 ) exp
γi (s)dB̂si+1 −
|γi (s)|2 ds ,
t
2 t
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Pricing of Interest Rate Derivatives
0 ≤ t ≤ Ti , i.e. t 7−→ L(t, Ti , Ti+1 ) is a geometric Brownian motion with
volatility γi (t) under P̂i+1 . Hence by (14.10) we have
i
h r Ti+1
rs ds
IE∗ e− t
(L(Ti , Ti , Ti+1 ) − κ)+ Ft
= P (t, Ti+1 ) IEi+1 (L(Ti , Ti , Ti+1 ) − κ)+ | Ft
= P (t, Ti+1 )Bl(κ, L(t, Ti , Ti+1 ), σi (t), 0, Ti − t),
t ∈ [0, Ti ], where
Bl(κ, x, σ, r, τ ) = xΦ(d+ ) − κe−rτ Φ(d− )
is the Black-Scholes function with
|σi (t)|2 =
1 w Ti
|γi |2 (s)ds.
Ti − t t
The formula (14.12) is also known as the Black (1976) formula when applied
to options on underlying futures or forward contracts on commodities, which
are modeled according to (14.11). In this case, the bond price P (t, Ti+1 ) can
be simply modeled as P (t, Ti+1 ) = e−r(Ti+1 −t) and (14.12) becomes
e−r(Ti+1 −t) L(t, Ti , Ti+1 )Φ(d+ ) − κe−r(Ti+1 −t) Φ(d− ),
where L(t, Ti , Ti+1 ) is the underlying future price.
In general we may also write (14.12) as
i
h r Ti+1
rs ds
(Ti+1 − Ti ) IE∗ e− t
(L(Ti , Ti , Ti+1 ) − κ)+ Ft
P (t, Ti )
= P (t, Ti+1 )
− 1 Φ(d+ ) − κ(Ti+1 − Ti )P (t, Ti+1 )Φ(d− )
P (t, Ti+1 )
= (P (t, Ti ) − P (t, Ti+1 ))Φ(d+ ) − κ(Ti+1 − Ti )P (t, Ti+1 )Φ(d− ),
and by Corollary 12.12 this gives the self-financing portfolio strategy
(Φ(d+ ), −Φ(d+ ) − κ(Ti+1 − Ti )Φ(d− ))
in the bonds (P (t, Ti ), P (t, Ti+1 )) with maturities Ti and Ti+1 , cf. Corollary 12.13 and [88].
Floorlets
Similarly, a floorlet on f (Ti , Ti , Ti+1 ) with strike price κ is a contract with
payoff (κ − f (Ti , Ti , Ti+1 ))+ , priced at time t ∈ [0, Ti ] as
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N. Privault
i
h r Ti+1
rs ds
IE∗ e− t
(κ − f (Ti , Ti , Ti+1 ))+ Ft
ˆ i+1 (κ − f (Ti , Ti , Ti+1 ))+ | Ft ,
= P (t, Ti+1 )IE
0 ≤ t ≤ Ti+1 .
Floorlets are analog to put options and can be similarly priced by the call/put
parity in the Black-Scholes formula.
Cap Pricing
More generally one can consider caps that are relative to a given tenor structure {T1 , . . . , Tn }, with discounted payoff
n−1
X
(Tk+1 − Tk )e−
r Tk+1
t
rs ds
(f (Tk , Tk , Tk+1 ) − κ)+ .
k=1
Pricing formulas for caps are easily deduced from analog formulas for caplets,
since the payoff of a cap can be decomposed into a sum of caplet payoffs. Thus
the price of a cap at time t ∈ [0, T1 ] is given by
"n−1
#
rT
X
∗
− t k+1 rs ds
+ IE
(Tk+1 − Tk )e
(f (Tk , Tk , Tk+1 ) − κ) Ft
k=1
=
n−1
X
rT
k+1
rs ds
(Tk+1 − Tk ) IE∗ e− t
(f (Tk , Tk , Tk+1 ) − κ)+ Ft
k=1
=
n−1
X
i
h
ˆ k+1 (f (Tk , Tk , Tk+1 ) − κ)+ Ft .
(Tk+1 − Tk )P (t, Tk+1 )IE
k=1
(14.13)
In the BGM model (14.11) the cap with payoff
n−1
X
(Tk+1 − Tk )(L(Tk , Tk , Tk+1 ) − κ)+
k=1
can be priced at time t ∈ [0, T1 ] by the Black formula
n−1
X
(Tk+1 − Tk )P (t, Tk+1 )Bl(κ, L(t, Tk , Tk+1 ), σk (t), 0, Tk − t).
k=1
14.4 Forward Swap Measures
In this section we introduce the forward measures to be used for the pricing
of swaptions, and we study their properties. We start with the definition of
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Pricing of Interest Rate Derivatives
the annuity numéraire
P (t, Ti , Tj ) =
j−1
X
(Tk+1 − Tk )P (t, Tk+1 ),
0 ≤ t ≤ Ti ,
(14.14)
k=i
with in particular
P (t, Ti , Ti+1 ) = (Ti+1 − Ti )P (t, Ti+1 ),
0 ≤ t ≤ Ti .
1 ≤ i < n. The annuity numéraire satisfies the following martingale
property,
rt
which can be proved by linearity and the fact that t 7−→ e− 0 rs ds P (t, Tk ) is
a martingale for all k = 1, 2, . . . , n, under Assumption (A).
Remark 14.4. The discounted annuity numéraire
t 7−→ e−
rt
0
rs ds
P (t, Ti , Tj ) = e−
rt
0
rs ds
j−1
X
(Tk+1 −Tk )P (t, Tk+1 ),
0 ≤ t ≤ Ti ,
k=i
is a martingale under P = P∗ .
The forward swap measure P̂i,j is defined by
r Ti
P (Ti , Ti , Tj )
dP̂i,j
= e− 0 rs ds
,
dP
P (0, Ti , Tj )
(14.15)
1 ≤ i < j ≤ n. We have
"
#
i
h r Ti
dP̂i,j 1
IE∗
IE∗ e− 0 rs ds P (Ti , Ti , Tj ) Ft
Ft =
dP
P (0, Ti , Tj )
=
P (t, Ti , Tj ) − r t rs ds
e 0
,
P (0, Ti , Tj )
0 ≤ t ≤ Ti , by Remark 14.4, and
r Ti
dP̂i,j|Ft
P (Ti , Ti , Tj )
= e− t rs ds
,
dP|Ft
P (t, Ti , Tj )
0 ≤ t ≤ Ti+1 ,
(14.16)
by Relation (12.3) in Lemma 12.1. We also know that the process
t 7−→ vki,j (t) :=
P (t, Tk )
,
P (t, Ti , Tj )
i, j, k = 1, 2, . . . , n,
is an Ft -martingale under P̂i,j by Proposition 12.3. It follows that the LIBOR
swap rate
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N. Privault
S(t, Ti , Tj ) :=
P (t, Ti ) − P (t, Tj )
= vii,j (t) − vji,j (t),
P (t, Ti , Tj )
0 ≤ t ≤ Ti ,
defined in Proposition 13.7 is also a martingale under P̂i,j .
Using the forward swap measure we obtain the following pricing formula
for a given integrable claim with payoff of the form P (Ti , Ti , Tj )F :
#
"
i
h r Ti
dP̂i,j|Ft IE∗ e− t rs ds P (Ti , Ti , Tj )F Ft = P (t, Ti , Tj ) IE∗ F
Ft
dP|Ft
h i
ˆ
= P (t, Ti , Tj )IEi,j F Ft ,
(14.17)
after applying (14.15) and (14.16) on the last line, or Proposition 12.2.
14.5 Swaption Pricing on the LIBOR
A swaption on the forward rate f (T1 , Tk , Tk+1 ) is a contract meant for protection against a risk based on an interest rate swap, and has payoff
j−1
rT
X
− k+1 rs ds
(Tk+1 − Tk )e Ti
(f (Ti , Tk , Tk+1 ) − κ)
!+
,
k=i
at time Ti . This swaption can be priced at time t ∈ [0, Ti ] under the riskneutral measure P∗ as


!+
j−1
rT
r Ti
X

− k+1 rs ds
IE∗ e− t rs ds
(Tk+1 − Tk )e Ti
(f (Ti , Tk , Tk+1 ) − κ)
F
t .
k=i
(14.18)
In the sequel and in practice the price (14.18) of the swaption will be evaluated as


!+
j−1
rT
X

∗  − t i rs ds
IE e
(Tk+1 − Tk )P (Ti , Tk+1 )(f (Ti , Tk , Tk+1 ) − κ)
Ft ,
k=i
t ∈ [0, Ti ], where we approximate the discount factor e
ditional expectation P (Ti , Tk+1 ) given FTi .
−
r Tk+1
Ti
(14.19)
rs ds
by its con-
Using the inequality
+
+
(x1 + · · · + xm ) ≤ x+
1 + · · · xm ,
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x1 , . . . , xm ∈ R,
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Pricing of Interest Rate Derivatives
the above term (14.19) can be upper bounded by the cap price (14.13) written
as


!+
j−1
rT
X

∗  − t i rs ds
IE e
(Tk+1 − Tk )P (Ti , Tk+1 )(f (Ti , Tk , Tk+1 ) − κ)
Ft
k=i
"
≤ IE∗ e−
r Ti
t
rs ds
j−1
X
+
(Tk+1 − Tk )P (Ti , Tk+1 ) (f (Ti , Tk , Tk+1 ) − κ)
Ft
#
k=i
=
j−1
X
i
h r Ti
+ (Tk+1 − Tk ) IE∗ e− t rs ds P (Ti , Tk+1 ) (f (Ti , Tk , Tk+1 ) − κ) Ft
k=i
=
j−1
X
r
rT
Ti
− k+1 rs ds + (Tk+1 − Tk ) IE∗ e− t rs ds IE∗ e Ti
FTi (f (Ti , Tk , Tk+1 ) − κ) Ft
k=i
=
j−1
X
rT
k+1
+ rs ds
(Tk+1 − Tk ) IE∗ IE∗ e− t
(f (Ti , Tk , Tk+1 ) − κ) FTi Ft
k=i
=
j−1
X
rT
k+1
+ rs ds
(Tk+1 − Tk ) IE∗ e− t
(f (Ti , Tk , Tk+1 ) − κ) Ft
k=i
= IE∗
"j−1
X
(Tk+1 − Tk )e−
r Tk+1
t
rs ds
(f (Ti , Tk , Tk+1 ) − κ)
+
#
Ft .
k=i
In addition, when j = i + 1, the swaption price (14.19) coincides with the
price at time t of a caplet on [Ti , Ti+1 ] since
i
h r Ti
+ IE∗ e− t rs ds ((Ti+1 − Ti )P (Ti , Ti+1 )(f (Ti , Ti , Ti+1 ) − κ)) Ft
i
h r Ti
+ = (Ti+1 − Ti ) IE∗ e− t rs ds P (Ti , Ti+1 ) ((f (Ti , Ti , Ti+1 ) − κ)) Ft
r
rT
Ti
− i+1 rs ds +
= (Ti+1 − Ti ) IE∗ e− t rs ds IE∗ e Ti
FTi ((f (Ti , Ti , Ti+1 ) − κ))
r
rT
Ti
− i+1 rs ds
+ = (Ti+1 − Ti ) IE∗ IE∗ e− t rs ds e Ti
((f (Ti , Ti , Ti+1 ) − κ)) FTi
i
h r Ti+1
+ rs ds
= (Ti+1 − Ti ) IE∗ e− t
(f (Ti , Ti , Ti+1 ) − κ) Ft ,
0 ≤ t ≤ Ti , which coincides with the caplet price (14.10) up to the factor
Ti+1 − Ti .
In case we replace the forward rate f (t, T, S) with the LIBOR rate
L(t, T, S) defined in Proposition 13.7, the payoff of the swaption can be
rewritten as in the following lemma which is a direct consequence of the
definition of the swap rate S(Ti , Ti , Tj ).
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Ft
Ft
N. Privault
Lemma 14.5. The payoff of the swaption in (14.19) can be rewritten as
!+
j−1
X
(Tk+1 − Tk )P (Ti , Tk+1 )(L(Ti , Tk , Tk+1 ) − κ)
k=i
= (P (Ti , Ti ) − P (Ti , Tj ) − κP (Ti , Ti , Tj ))
+
+
= P (Ti , Ti , Tj ) (S(Ti , Ti , Tj ) − κ) .
(14.20)
(14.21)
Proof. The relation
j−1
X
(Tk+1 − Tk )P (t, Tk+1 )(L(t, Tk , Tk+1 ) − S(t, Ti , Tj )) = 0
k=i
that defines the forward swap rate S(t, Ti , Tj ) shows that
j−1
X
(Tk+1 − Tk )P (t, Tk+1 )L(t, Tk , Tk+1 )
k=i
= S(t, Ti , Tj )
j−1
X
(Tk+1 − Tk )P (t, Tk+1 )
k=i
= P (t, Ti , Tj )S(t, Ti , Tj )
= P (t, Ti ) − P (t, Tj ),
cf. Proposition 13.7, hence by the definition (14.14) of P (t, Ti , Tj ) we have
j−1
X
(Tk+1 − Tk )P (t, Tk+1 )(L(t, Tk , Tk+1 ) − κ)
k=i
= P (t, Ti ) − P (t, Tj ) − κP (t, Ti , Tj )
= P (t, Ti , Tj ) (S(t, Ti , Tj ) − κ) ,
and for t = Ti we get
!+
j−1
X
(Tk+1 − Tk )P (Ti , Tk+1 )(L(Ti , Tk , Tk+1 ) − κ)
k=i
+
= P (Ti , Ti , Tj ) (S(Ti , Ti , Tj ) − κ) .
The next proposition simply states that a swaption on the LIBOR rate can
be priced as a European call option on the swap rate S(Ti , Ti , Tj ) under the
forward swap measure P̂i,j .
Proposition 14.6. The price (14.19) of the European swaption with payoff
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!+
j−1
X
(Tk+1 − Tk )P (Ti , Tk+1 )(L(Ti , Tk , Tk+1 ) − κ)
(14.22)
k=i
on the LIBOR market can be written under the forward swap measure P̂i,j as
i
h
ˆ i,j (S(Ti , Ti , Tj ) − κ)+ Ft ,
0 ≤ t ≤ Ti .
P (t, Ti , Tj )IE
Proof. As a consequence of (14.17) and Lemma 14.5 we find


!+
j−1
rT
X

∗  − t i rs ds
IE e
(Tk+1 − Tk )P (Ti , Tk+1 )(L(Ti , Tk , Tk+1 ) − κ)
Ft
k=i
=
=
=
=
i
h r Ti
+ IE e− t rs ds (P (Ti , Ti ) − P (Ti , Tj ) − κP (Ti , Ti , Tj )) Ft (14.23)
i
h r Ti
+ IE∗ e− t rs ds P (Ti , Ti , Tj ) (S(Ti , Ti , Tj ) − κ) Ft
#
"
1
+ ∗ dP̂i,j|Ft
IE
(S(Ti , Ti , Tj ) − κ) Ft
P (t, Ti , Tj )
dP|Ft
i
h
ˆ i,j (S(Ti , Ti , Tj ) − κ)+ Ft .
P (t, Ti , Tj )IE
(14.24)
∗
In the next proposition we price a swaption with payoff (14.22) or equivalently
(14.21).
Proposition 14.7. Assume that the LIBOR swap rate (13.34) is modeled as
a geometric Brownian motion under Pi,j , i.e.
dS(t, Ti , Tj ) = S(t, Ti , Tj )σ̂(t)dŴti,j ,
where (σ̂(t))t∈R+ is a deterministic function. Then the swaption with payoff
+
(P (T, Ti ) − P (T, Tj ) − κP (Ti , Ti , Tj ))+ = P (Ti , Ti , Tj ) (S(Ti , Ti , Tj ) − κ)
can be priced using the Black-Scholes formula as
i
h r Ti
+ IE∗ e− t rs ds P (Ti , Ti , Tj ) (S(Ti , Ti , Tj ) − κ) Ft
= (P (t, Ti ) − P (t, Tj ))Φ+ (t, S(t, Ti , Tj ))
−κΦ− (t, S(t, Ti , Tj ))
j−1
X
(Tk+1 − Tk )P (t, Tk+1 ),
k=i
where
d+ =
"
2
log(S(t, Ti , Tj )/κ) + σi,j
(t)(Ti − t)/2
√
,
σi,j (t) Ti − t
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N. Privault
and
d− =
2
log(S(t, Ti , Tj )/κ) − σi,j
(t)(Ti − t)/2
√
,
σi,j (t) Ti − t
and
|σi,j (t)|2 =
1 w Ti
(σ̂(s))2 ds.
Ti − t t
Proof. Since S(t, Ti , Tj ) is a geometric Brownian motion with volatility function (σ̂(t))t∈R+ under P̂i,j , by (14.20)-(14.21) and (14.23)-(14.24) we have
i
h r Ti
+ IE∗ e− t rs ds P (Ti , Ti , Tj ) (S(Ti , Ti , Tj ) − κ) Ft
i
h rT
= IE∗ e− t rs ds (P (T, Ti ) − P (T, Tj ) − κP (Ti , Ti , Tj ))+ Ft
i
h
ˆ i,j (S(Ti , Ti , Tj ) − κ)+ Ft
= P (t, Ti , Tj )IE
= P (t, Ti , Tj )Bl(κ, S(Ti , Ti , Tj ), σi,j (t), 0, Ti − t)
= P (t, Ti , Tj ) (S(t, Ti , Tj )Φ+ (t, S(t, Ti , Tj )) − κΦ− (t, S(t, Ti , Tj )))
= (P (t, Ti ) − P (t, Tj ))Φ+ (t, S(t, Ti , Tj )) − κP (t, Ti , Tj )Φ− (t, S(t, Ti , Tj ))
= (P (t, Ti ) − P (t, Tj ))Φ+ (t, S(t, Ti , Tj ))
−κΦ− (t, S(t, Ti , Tj ))
j−1
X
(Tk+1 − Tk )P (t, Tk+1 ).
k=i
In addition the hedging strategy
(Φ+ (t, S(t, Ti , Tj )), −κΦ− (t, S(t, Ti , Tj ))(Ti+1 − Ti ), . . .
. . . , −κΦ− (t, S(t, Ti , Tj ))(Tj−1 − Tj−2 ), −Φ+ (t, S(t, Ti , Tj )))
based on the assets (P (t, Ti ), . . . , P (t, Tj )) is self-financing by Corollary 12.13,
cf. also [88].
Swaption prices can also be computed by an approximation formula, from the
exact dynamics of the swap rate S(t, Ti , Tj ) under P̂i,j , based on the bond
price dynamics of the form (14.3), cf. [102], page 17.
Swaption volatilities can be estimated from swaption prices as implied
volatilities from the Black pricing formula:
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0.2
0.18
0.16
0.14
0.12
9
8
7
0.1
6
5
0.08
4
0
i
3
1
2
3
2
4
5
6
7
j
1
8
9
0
Fig. 14.2: Implied swaption volatilities.
Bermudan swaption pricing in Quantlib
The Bermudan swaption on the tenor structure {Ti , . . . , Tj } is priced as the
supremum


!+
j−1
r
X

∗  − tτ rs ds
sup IE e
(Tk+1 − Tk )P (τ, Tk+1 )(L(τ, Tk , Tk+1 ) − κ)
Ft
τ ∈{Ti ,...,Tj−1 }
=
sup
∗
IE
h
e
−
k=τ
rτ
rs ds
t
(P (τ, τ ) − P (τ, Tj ) − κP (τ, τ, Tj ))
+
τ ∈{Ti ,...,Tj−1 }
=
sup
τ ∈{Ti ,...,Tj−1 }
i
Ft
i
h rτ
+ IE∗ e− t rs ds P (τ, τ, Tj ) (S(τ, τ, Tj ) − κ) Ft ,
where the supremum is over all stopping times τ taking values in {Ti , . . . , Tj }.
Bermudan swaptions can be priced using this code in (R)quantlib, with the
following output:
Summary of pricing results for Bermudan Swaption
Price (in bp) of Bermudan swaption is 24.92137
Stike is NULL (ATM strike is 0.05 )
Model used is: Hull-White using analytic formulas
Calibrated model parameters are:
a = 0.04641
sigma = 0.005869
This modified code can be used in particular the pricing of ordinary swaptions
with the output:
Summary of pricing results for Bermudan Swaption
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Price (in bp) of Bermudan swaption is 22.45436
Stike is NULL (ATM strike is 0.05 )
Model used is: Hull-White using analytic formulas
Calibrated model parameters are:
a = 0.07107
sigma = 0.006018
Exercises
Exercise 14.1 Consider two bonds with maturities T1 and T2 , T1 < T2 , which
follow the stochastic differential equations
dP (t, T1 ) = rt P (t, T1 )dt + ζ1 (t)P (t, T1 )dWt
and
dP (t, T2 ) = rt P (t, T2 )dt + ζ2 (t)P (t, T2 )dWt .
a) Using Itô calculus, show that the forward process P (t, T2 )/P (t, T1 ) is a
driftless geometric Brownian motion driven by dŴt := dWt −ζ1 (t)dt under
the T1 -forward measure
h P̂.r T1
i
b) Compute the price IE∗ e− 0 rs ds (K − P (T1 , T2 ))+ of a bond put option
using change of numeraire and the Black-Scholes formula.
Hint: Given X a centered Gaussian random variable with variance v 2 , we
have:
IE∗ [(κ−xeX−v
2
) ] = κΦ(v/2+(log(κ/x))/v)−xΦ(−v/2+(log(κ/x))/v).
/2 +
Exercise 14.2 Given two bonds with maturities T , S and prices P (t, T ),
P (t, S), consider the LIBOR rate
L(t, T, S) =
P (t, T ) − P (t, S)
(S − T )P (t, S)
at time t, modeled as
dL(t, T, S) = µt L(t, T, S)dt + σL(t, T, S)dWt ,
0 ≤ t ≤ T,
(14.25)
where (Wt )t∈[0,T ] is a standard Brownian motion under the risk-neutral measure P∗ , σ > 0 is a constant, and (µt )t∈[0,T ] is an adapted process. Let
i
h rS
Ft = IE∗ e− t rs ds (κ − L(T, T, S))+ Ft
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denote the price at time t of a floorlet option with strike price κ, maturity
T , and payment date S.
a) Rewrite the value of Ft using the forward measure P̂S with maturity S.
b) What is the dynamics of L(t, T, S) under the forward measure P̂S ?
c) Write down the value of Ft using the Black-Scholes formula.
Hint. Given X a centered Gaussian random variable with variance v 2 we
have
IE∗ [(κ − em+X )+ ] = κΦ(−(m − log κ)/v) − em+
v2
2
Φ(−v − (m − log κ)/v),
where Φ denotes the Gaussian cumulative distribution function.
Exercise 14.3 Consider a European swaption giving its holder the right to
enter into a 3-year annual pay swap in four years, where a fixed rate of 5% is
paid and the LIBOR rate is received. Assume that the yield curve is flat at
5% per annum with annual compounding and the volatility of the swap rate
is 20%. The notional principal is $10 millions and is quoted in basis points
(one basis point = .01%).
a) What are the key assumptions in order to apply Black’s formula to value
this swaption?
b) Compute the price of this swaption using Black’s formula as an application of Proposition 14.7, using the functions Φ(d+ ) and Φ(di ).
Exercise 14.4 We work in the short rate model
drt = σdBt ,
where (Bt )t∈R+ is a standard Brownian motion under P, and P2 is the forward
measure defined by
r T2
dP2
1
=
e− 0 rs ds .
dP
P (0, T2 )
a) State the expressions of ζ1 (t) and ζ2 (t) in
dP (t, Ti )
= rt dt + ζi (t)dBt ,
P (t, Ti )
i = 1, 2,
and the dynamics of the P (t, T1 )/P (t, T2 ) under P2 , where P (t, T1 ) and
P (t, T2 ) are bond prices with maturities T1 and T2 .
b) State the expression of the forward rate f (t, T1 , T2 ).
c) Compute the dynamics of f (t, T1 , T2 ) under the forward measure P2 with
r T2
1
dP2
=
e− 0 rs ds .
dP
P (0, T2 )
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d) Compute the price
i
h r T2
(T2 − T1 ) IE∗ e− t rs ds (f (T1 , T1 , T2 ) − κ)+ Ft
of a cap at time t ∈ [0, T1 ], using the expectation under the forward
measure P2 .
e) Compute the dynamics of the swap rate process
S(t, T1 , T2 ) =
P (t, T1 ) − P (t, T2 )
,
(T2 − T1 )P (t, T2 )
t ∈ [0, T1 ],
under P2 .
f) Compute the swaption price
i
h r T1
(T2 − T1 ) IE∗ e− t rs ds P (T1 , T2 )(S(T1 , T1 , T2 ) − κ)+ Ft
on the swap rate S(T1 , T1 , T2 ) using the expectation under the forward
swap measure P1,2 .
Exercise 14.5 Consider three zero-coupon bonds P (t, T1 ), P (t, T2 ) and
P (t, T3 ) with maturities T1 = δ, T2 = 2δ and T3 = 3δ respectively, and
the forward LIBOR L(t, T1 , T2 ) and L(t, T2 , T3 ) defined by
1
P (t, Ti )
L(t, Ti , Ti+1 ) =
−1 ,
i = 1, 2.
δ P (t, Ti+1 )
Assume that L(t, T1 , T2 ) and L(t, T2 , T3 ) are modeled in the BGM model by
dL(t, T1 , T2 )
= e−at dŴt2 ,
L(t, T1 , T2 )
0 ≤ t ≤ T1 ,
(14.26)
and L(t, T2 , T3 ) = b, 0 ≤ t ≤ T2 , for some constants a, b > 0, where Ŵt2 is a
standard Brownian motion under the forward rate measure P2 defined by
r T2
e− 0 rs ds
dP2
=
.
dP
P (0, T2 )
a) Compute L(t, T1 , T2 ), 0 ≤ t ≤ T2 by solving Equation (14.26).
b) Show that the price at time t of the caplet with strike price κ can be
written as
i
h r T2
∗
ˆ 2 (L(T1 , T1 , T2 ) − κ)+ | Ft ,
IE e− t rs ds (L(T1 , T1 , T2 ) − κ)+ Ft = P (t, T2 )IE
ˆ 2 denotes the expectation under the forward measure P2 .
where IE
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Pricing of Interest Rate Derivatives
c) Using the hint below, compute the price at time t of the caplet with strike
price κ on L(T1 , T1 , T2 ).
d) Compute
P (t, T1 )
,
P (t, T1 , T3 )
0 ≤ t ≤ T1 ,
and
P (t, T3 )
,
P (t, T1 , T3 )
0 ≤ t ≤ T2 ,
in terms of b and L(t, T1 , T2 ), where P (t, T1 , T3 ) is the annuity numéraire
P (t, T1 , T3 ) = δP (t, T2 ) + δP (t, T3 ),
0 ≤ t ≤ T2 .
e) Compute the dynamics of the swap rate
t 7−→ S(t, T1 , T3 ) =
P (t, T1 ) − P (t, T3 )
,
P (t, T1 , T3 )
0 ≤ t ≤ T1 ,
i.e. show that we have
dS(t, T1 , T3 ) = σ1.3 (t)S(t, T1 , T3 )dŴt2 ,
where σ1,3 (t) is a process to be determined.
f) Using the Black-Scholes formula, compute an approximation of the swaption price
i
h r T1
IE∗ e− t rs ds P (T1 , T1 , T3 )(S(T1 , T1 , T3 ) − κ)+ Ft
ˆ 2 (S(T1 , T1 , T3 ) − κ)+ | Ft ,
= P (t, T1 , T3 )IE
at time t ∈ [0, T1 ]. You will need to approximate σ1,3 (s), s ≥ t, by “freezing” all random terms at time t.
Hint. Given X a centered Gaussian random variable with variance v 2 we
have
v2
IE∗ (em+X − κ)+ = em+ 2 Φ(v + (m − log κ)/v) − κΦ((m − log κ)/v),
where Φ denotes the Gaussian cumulative distribution function.
Exercise 14.6 Consider a portfolio (ξtT , ξtS )t∈[0,T ] made of two bonds with
maturities T , S, and value
Vt = ξtT P (t, T ) + ξtS P (t, S),
0 ≤ t ≤ T,
at time t. We assume that the portfolio is self-financing, i.e.
dVt = ξtT dP (t, T ) + ξtS dP (t, S),
0 ≤ t ≤ T,
(14.27)
and that it hedges the claim (P (T, S) − κ)+ , so that
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h rT
+
Vt = IE∗ e− t rs ds (P (T, S) − κ)
h
+ = P (t, T ) IET (P (T, S) − κ) i
Ft
i
Ft ,
0 ≤ t ≤ T.
a) Show that
i
h rT
+ IE∗ e− t rs ds (P (T, S) − K) Ft
h
i wt
wt
+
= P (0, T ) IET (P (T, S) − K) +
ξsT dP (s, T ) +
ξsS dP (s, S).
0
0
b) Show that under the
self-financing condition (14.27), the discounted portrt
folio price Vet = e− 0 rs ds Vt satisfies
dVet = ξtT dP̃ (t, T ) + ξtS dP̃ (t, S),
rt
rt
where P̃ (t, T ) = e− 0 rs ds P (t, T ) and P̃ (t, S) = e− 0 rs ds P (t, S) denote
the discounted bond prices.
c) Show that
h
i
h
i
+
+
IET (P (T, S) − K) |Ft = IET (P (T, S) − K)
w t ∂C
+
(Xu , T − u, v(u, T ))dXu .
0 ∂x
Hint: use the martingale property and the Itô formula.
d) Show that the discounted portfolio price V̂t = Vt /P (t, T ) satisfies
∂C
(Xt , T − t, v(t, T ))dXt
∂x
P (t, S) ∂C
=
(Xt , T − t, v(t, T ))(σtS − σtT )dB̂tT .
P (t, T ) ∂x
dV̂t =
e) Show that
dVt = P (t, S)
∂C
(Xt , T − t, v(t, T ))(σtS − σtT )dBt + V̂t dP (t, T ).
∂x
f) Show that
dVet = P̃ (t, S)
∂C
(Xt , T − t, v(t, T ))(σtS − σtT )dBt + V̂t dP̃ (t, T ).
∂x
g) Compute the hedging strategy (ξtT , ξtS )t∈[0,T ] of the bond option.
h) Show that
∂C
log(x/K) + τ v 2 /2
√
(x, τ, v) = Φ
.
∂x
τv
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Exercise 14.7 Consider a LIBOR rate L(t, T, S), t ∈ [0, T ], modeled as
dL(t, T, S) = µt L(t, T, S)dt + σ(t)L(t, T, S)dWt , 0 ≤ t ≤ T , where (Wt )t∈[0,T ]
is a standard Brownian motion under the risk-neutral measure P∗ , (µt )t∈[0,T ]
is an adapted process, and σ(t) > 0 is a deterministic function.
a) What is the dynamics of L(t, T, S) under the forward measure P̂ with
numeraire Nt := P (t, S) ?
b) Rewrite the price
h rS
i
IE∗ e− t rs ds φ(L(T, T, S))Ft
(14.28)
at time t ∈ [0, T ] of an option with payoff function φ using the forward
measure P̂.
c) Write down the value of the above option price (14.28) using an integral.
Exercise 14.8 Given n bonds
with maturities T1 , . . . , Tn , consider the annuity
Pj−1
numeraire P (t, Ti , Tj ) = k=i (Tk+1 − Tk )P (t, Tk+1 ) and the swap rate
S(t, Ti , Tj ) =
P (t, Ti ) − P (t, Tj )
P (t, Ti , Tj )
at time t ∈ [0, Ti ], modeled as
dS(t, Ti , Tj ) = µt S(t, Ti , Tj )dt + σS(t, Ti , Tj )dWt ,
0 ≤ t ≤ Ti , (14.29)
where (Wt )t∈[0,Ti ] is a standard Brownian motion under the risk-neutral measure P∗ , (µt )t∈[0,T ] is an adapted process and σ > 0 is a constant. Let
i
h r Ti
IE∗ e− t rs ds P (Ti , Ti , Tj )φ(S(Ti , Ti , Tj ))Ft
(14.30)
at time t ∈ [0, Ti ] of an option with payoff function φ.
a) Rewrite the option price (14.30) at time t ∈ [0, Ti ] using the forward swap
measure P̂i,j defined from the annuity numeraire P (t, Ti , Tj ).
b) What is the dynamics of S(t, Ti , Tj ) under the forward swap measure P̂i,j ?
c) Write down the value of the above option price (14.28) using a Gaussian
integral.
d) Apply the above to the computation at time t ∈ [0, Ti ] of the put swaption
price
i
h r Ti
IE∗ e− t rs ds P (Ti , Ti , Tj )(κ − S(Ti , Ti , Tj ))+ Ft
with strike price κ, using the Black-Scholes formula.
Hint. Given X a centered Gaussian random variable with variance v 2 we
have
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IE[(κ − em+X )+ ] = κΦ(−(m − log κ)/v) − em+
v2
2
Φ(−v − (m − log κ)/v),
where Φ denotes the Gaussian cumulative distribution function.
Exercise 14.9 Consider a bond market with two bonds with maturities T1 ,
T2 , whose prices P (t, T1 ), P (t, T2 ) at time t are given by
dP (t, T1 )
= rt dt + ζ1 (t)dBt ,
P (t, T1 )
dP (t, T2 )
= rt dt + ζ2 (t)dBt ,
P (t, T2 )
where (rt )t∈R+ is a short term interest rate process, (Bt )t∈R+ is a standard
Brownian motion generating a filtration (Ft )t∈R+ , and ζ1 (t), ζ2 (t) are volatility processes. The LIBOR rate L(t, T1 , T2 ) is defined by
L(t, T1 , T2 ) =
P (t, T1 ) − P (t, T2 )
.
P (t, T2 )
Recall that a caplet on the LIBOR market can be priced at time t ∈ [0, T1 ]
as
i
i
h r T2
h
+ ˆ (L(T1 , T1 , T2 ) − κ)+ Ft ,
IE e− t rs ds (L(T1 , T1 , T2 ) − κ) Ft = P (t, T2 )IE
(14.31)
under the forward measure P̂ defined by
r T1
P (T1 , T2 )
dP̂
= e− 0 rs ds
,
dP
P (0, T2 )
under which
B̂t := Bt −
wt
0
ζ2 (s)ds,
t ∈ R+ ,
(14.32)
is a standard Brownian motion.
In the sequel we let Lt = L(t, T1 , T2 ) for simplicity of notation.
a) Using Itô calculus, show that the LIBOR rate satisfies
dLt = Lt σ(t)dB̂t ,
0 ≤ t ≤ T1 ,
(14.33)
where the LIBOR rate volatility is given by
σ(t) =
P (t, T1 )(ζ1 (t) − ζ2 (t))
.
P (t, T1 ) − P (t, T2 )
b) Solve the equation (14.33) on the interval [t, T1 ], and compute LT1 from
the initial condition Lt .
c) Assuming that σ(t) in (14.33) is a deterministic function, show that the
price
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Pricing of Interest Rate Derivatives
i
h
ˆ (LT − κ)+ Ft
P (t, T2 )IE
1
of the caplet can be written as P (t, T2 )C(Lt , v(t, T1 )), where v 2 (t, T1 ) =
w T1
|σ(s)|2 ds, and C(t, v(t, T1 )) is a function of Lt and v(t, T1 ).
t
d) Consider a portfolio (ξt1 , ξt2 )t∈[0,T1 ] made of bonds with maturities T1 , T2
and value
Vt = ξt1 P (t, T1 ) + ξt2 P (t, T2 ),
at time t ∈ [0, T1 ]. We assume that the portfolio is self-financing, i.e.
dVt = ξt1 dP (t, T1 ) + ξt2 dP (t, T2 ),
0 ≤ t ≤ T1 ,
(14.34)
and that it hedges the claim (LT1 − κ) , so that
i
i
h r T1
h
+ ˆ (LT − κ)+ Ft ,
Vt = IE e− t rs ds (P (T1 , T2 )(LT1 − κ)) Ft = P (t, T2 )IE
1
+
0 ≤ t ≤ T1 . Show that we have
i
h r T1
+ IE e− t rs ds (P (T1 , T2 )(LT1 − κ)) Ft
h
i wt
wt
ˆ (LT − κ)+ +
= P (0, T2 )IE
ξs1 dP (s, T1 ) +
ξs2 dP (s, T1 ),
1
0
0
0 ≤ t ≤ T1 .
e) Show that under the
self-financing condition (14.34), the discounted portrt
folio price Vet = e− 0 rs ds Vt satisfies
dVet = ξt1 dP̃ (t, T1 ) + ξt2 dP̃ (t, T2 ),
where P̃ (t, Tk ) = e−
prices.
f) Show that
rt
0
rs ds
P (t, Tk ), k = 1, 2, denote the discounted bond
h
h
i w t ∂C
i
ˆ (LT − κ)+ Ft = IE
ˆ (LT − κ)+ +
(Lu , v(u, T1 ))dLu ,
IE
1
1
0 ∂x
and that the discounted portfolio price V̂t = Vt /P (t, T2 ) satisfies
dV̂t =
∂C
∂C
(Lt , v(t, T1 ))dLt = Lt
(Lt , v(t, T1 ))σ(t)dB̂t .
∂x
∂x
Hint: use the martingale property and the Itô formula.
g) Show that
dVt = (P (t, T1 ) − P (t, T2 ))
∂C
(Lt , v(t, T1 ))σ(t)dBt + V̂t dP (t, T2 ).
∂x
h) Show that
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N. Privault
dVet =
∂C
(Lt , v(t, T1 ))d(P̃ (t, T1 ) − P̃ (t, T2 ))
∂x
∂C
+ V̂t − Lt
(Lt , v(t, T1 )) dP̃ (t, T2 ),
∂x
and deduce the values of the hedging portfolio (ξt1 , ξt2 )t∈R+ .
Exercise 14.10 Consider a bond market with tenor structure {Ti , . . . , Tj } and
bonds with maturities Ti , . . . , Tj , whose prices P (t, Ti ), . . . P (t, Tj ) at time t
are given by
dP (t, Tk )
= rt dt + ζk (t)dBt ,
P (t, Tk )
k = i, . . . , j,
where (rt )t∈R+ is a short term interest rate process and (Bt )t∈R+ denotes a standard Brownian motion generating a filtration (Ft )t∈R+ , and
ζi (t), . . . , ζj (t) are volatility processes.
The swap rate S(t, Ti , Tj ) is defined by
S(t, Ti , Tj ) =
where
P (t, Ti , Tj ) =
P (t, Ti ) − P (t, Tj )
,
P (t, Ti , Tj )
j−1
X
(Tk+1 − Tk )P (t, Tk+1 )
k=i
is the annuity numéraire. Recall that a swaption on the LIBOR market can
be priced at time t ∈ [0, Ti ] as


!+
j−1
r Ti
X

IE∗ e− t rs ds
(Tk+1 − Tk )P (Ti , Tk+1 )(S(Ti , Tk , Tk+1 ) − κ)
F
t
k=i
i
h
+ = P (t, Ti , Tj ) IEi,j (S(Ti , Ti , Tj ) − κ) Ft ,
(14.35)
under the forward swap measure Pi,j defined by
r Ti
P (Ti , Ti , Tj )
dPi,j
= e− 0 rs ds
,
dP
P (0, Ti , Tj )
1 ≤ i < j ≤ n,
under which
B̂ti,j := Bt −
j−1
X
P (t, Tk+1 )
(Tk+1 − Tk )
ζk+1 (t)dt
P (t, Ti , Tj )
(14.36)
k=i
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Pricing of Interest Rate Derivatives
is a standard Brownian motion. Recall that the swap rate can be modeled as
dS(t, Ti , Tj ) = S(t, Ti , Tj )σi,j (t)dB̂ti,j ,
0 ≤ t ≤ Ti ,
(14.37)
where the swap rate volatilities are given by
j−1
X
P (t, Tj )
P (t, Tl+1 )
(Tl+1 −Tl )
(ζi (t)−ζl+1 (t))+
(ζi (t)−ζj (t))
P (t, Ti , Tj )
P (t, Ti ) − P (t, Tj )
l=i
(14.38)
1 ≤ i, j ≤ n, cf. e.g. Proposition 10.8 of [83]. In the sequel we denote St =
S(t, Ti , Tj ) for simplicity of notation.
σi,j (t) =
a) Solve the equation (14.37) on the interval [t, Ti ], and compute S(Ti , Ti , Tj )
from the initial condition S(t, Ti , Tj ).
b) Assuming that σi,j (t) is a deterministic function of t for 1 ≤ i, j ≤ n, show
that the price (14.24) of the swaption can be written as
P (t, Ti , Tj )C(St , v(t, Ti )),
where
v 2 (t, Ti ) =
w Ti
t
|σi,j (s)|2 ds,
and C(x, v) is a function to be specified using the Black-Scholes formula
Bl(K, x, σ, r, τ ), with
IE[(xem+X − K)+ ] = Φ(v + (m + log(x/K))/v) − KΦ((m + log(x/K))/v),
where X is a centered Gaussian random variable with mean m = rτ −v 2 /2
and variance v 2 .
c) Consider a portfolio (ξti , . . . , ξtj )t∈[0,Ti ] made of bonds with maturities
Ti , . . . , Tj and value
j
X
Vt =
ξtk P (t, Tk ),
k=i
at time t ∈ [0, Ti ]. We assume that the portfolio is self-financing, i.e.
dVt =
j
X
ξtk dP (t, Tk ),
0 ≤ t ≤ Ti ,
(14.39)
k=i
and that it hedges the claim (S(Ti , Ti , Tj ) − κ)+ , so that


!+
j−1
rT
X

∗  − t i rs ds
Vt = IE e
(Tk+1 − Tk )P (Ti , Tk+1 )(L(Ti , Tk , Tk+1 ) − κ)
Ft
k=i
i
h
+ = P (t, Ti , Tj ) IEi,j (S(Ti , Ti , Tj ) − κ) Ft ,
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N. Privault
0 ≤ t ≤ Ti . Show that


!+
j−1
rT
X

∗  − t i rs ds
(Tk+1 − Tk )P (Ti , Tk+1 )(L(Ti , Tk , Tk+1 ) − κ)
IE e
Ft
k=i
j w
h
i X
t
+
= P (0, Ti , Tj ) IEi,j (S(Ti , Ti , Tj ) − κ) +
ξsk dP (s, Ti ),
k=i
0
0 ≤ t ≤ Ti .
d) Show that under the
self-financing condition (14.39), the discounted portrt
folio price Vet = e− 0 rs ds Vt satisfies
dVet =
j
X
ξtk dP̃ (t, Tk ),
k=i
rt
where P̃ (t, Tk ) = e− 0 rs ds P (t, Tk ), k = i, . . . , j, denote the discounted
bond prices.
e) Show that
i
h
+ IEi,j (S(Ti , Ti , Tj ) − κ) Ft
h
i w t ∂C
+
= IEi,j (S(Ti , Ti , Tj ) − κ) +
(Su , v(u, Ti ))dSu .
0 ∂x
Hint: use the martingale property and the Itô formula.
f) Show that the discounted portfolio price V̂t = Vt /P (t, Ti , Tj ) satisfies
dV̂t =
∂C
∂C
(St , v(t, Ti ))dSt = St
(St , v(t, Ti ))σti,j dB̂ti,j .
∂x
∂x
g) Show that
dVt = (P (t, Ti ) − P (t, Tj ))
∂C
(St , v(t, Ti ))σti,j dBt + V̂t dP (t, Ti , Tj ).
∂x
h) Show that
j−1
dVt = St ζi (t)
X
∂C
(St , v(t, Ti ))
(Tk+1 − Tk )P (t, Tk+1 )dBt
∂x
k=i
j−1
X
∂C
+(V̂t − St
(St , v(t, Ti )))
(Tk+1 − Tk )P (t, Tk+1 )ζk+1 (t)dBt
∂x
k=i
+
∂C
(St , v(t, Ti ))P (t, Tj )(ζi (t) − ζj (t))dBt .
∂x
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i) Show that
dVet =
∂C
(St , v(t, Ti ))d(P̃ (t, Ti ) − P̃ (t, Tj ))
∂x
∂C
+(V̂t − St
(St , v(t, Ti )))dP̃ (t, Ti , Tj ).
∂x
j) Show that
∂C
(x, v(t, Ti )) = Φ
∂x
log(x/K) v(t, Ti )
+
v(t, Ti )
2
.
k) Show that we have
log(St /K) v(t, Ti )
+
d(P̃ (t, Ti ) − P̃ (t, Tj ))
dVet = Φ
v(t, Ti )
2
log(St /K) v(t, Ti )
−κΦ
−
dP̃ (t, Ti , Tj ).
v(t, Ti )
2
l) Show that the hedging strategy is given by
log(St /K) v(t, Ti )
ξti = Φ
+
,
v(t, Ti )
2
log(St /K) v(t, Ti )
log(St /K) v(t, Ti )
+
−
ξtj = −Φ
−κ(Tj −Tj−1 )Φ
,
v(t, Ti )
2
v(t, Ti )
2
and
ξtk = −κ(Tk+1 − Tk )Φ
"
log(St /K) v(t, Ti )
−
v(t, Ti )
2
,
i ≤ k ≤ j − 2.
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