Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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6.5 Some Computations on Bessel Bridges 377
C
EB
(x, T )=E
Q
exp
−
T
0
r
s
ds
P (T,T
∗
)1
{r
T
≤r
∗
}
− KE
Q
exp
−
T
0
r
s
ds
1
{r
T
≤r
∗
}
,
where B(r
∗
,T
∗
− T )=K,thatis
r
∗
= −
A(T
∗
− T )+lnK
G(T
∗
− T )
.
We observe now that the processes
L
1
(t)=(P (0,T))
−1
exp
−
t
0
r
s
ds
P (t, T )
L
2
(t)=(P (0,T
∗
))
−1
exp
−
t
0
r
s
ds
P (t, T
∗
)
are positive Q-martingales with expectations equal to 1, from the definition
of P (t, ·). Hence, using change of num´eraire techniques, we can define two
probabilities Q
1
and Q
2
by
Q
i
|
F
t
= L
i
(t) Q|
F
t
,i=1, 2.
Therefore,
C
EB
(x, T )=P (0,T
∗
)Q
2
(r
T
≤ r
∗
) − KP(0,T)Q
1
(r
T
≤ r
∗
) .
We characterize the law of the r.v. r
T
under Q
1
from its Laplace transform
E
Q
1
(e
−λr
T
)=E
Q
e
−λr
T
exp
−
T
0
r
s
ds
(P (0,T))
−1
.
From Corollary 6.3.4.3,
E
Q
1
(e
−λr
T
)=exp(−A
λ,1
+ A
0,1
− x(G
λ,1
− G
0,1
)) .
Let ˜c
1
=
σ
2
2
e
γT
− 1
D
with D = γ(e
γT
+1)+k(e
γT
− 1) . Then,
exp (−A
λ,1
+ A
0,1
)=
1
1+2λ˜c
1
2kθ/σ
2
.
Some computations and the equality γ
2
= k
2
+2σ
2
yield
G
λ,1
− G
0,1
=
λ
D
2
(1 + 2λ˜c
1
)
D(γ + k + e
γT
(γ − k)) − 2σ
2
(e
γT
− 1)
2
=
λ
D
2
(1 + 2λ˜c
1
)
γ
2
(e
γT
+1)
2
− k
2
(e
γT
− 1)
2
− 2σ
2
(e
γT
− 1)
2
= λ
4γ
2
e
γT
D(1 + 2λ˜c
1
)
= λ
˜c
1
˜x
1
(1 + 2λ˜c
1
)
378 6 A Special Family of Diffusions: Bessel Processes
where
˜x
1
=
8xγ
2
e
γT
σ
2
(e
γT
− 1)[γ(e
γT
+1)+k(e
γT
− 1)]
.
Hence,
E
Q
1
(e
−λr
T
)=
1
2λ˜c
1
+1
2kθ/σ
2
exp
−
λ˜c
1
˜x
1
2λ˜c
1
+1
and, from Corollary 6.3.4.4,ther.v.r
T
/˜c
1
follows, under Q
1
,aχ
2
law with
2kθ/σ
2
degrees of freedom and non-centrality ˜x
1
.
The same kind of computation establishes that the r.v. r
T
/˜c
2
follows, under
Q
2
,aχ
2
law with parameters ˜c
2
, ˜x
2
given by
˜c
2
=
σ
2
2
e
γT
− 1
γ(e
γT
+1)+(k + σ
2
G(T
∗
− T ))(e
γT
− 1)
,
˜x
2
=
8xγ
2
e
γT
σ
2
(e
γT
− 1) (γ(e
γT
+1)+(k + σ
2
G(T
∗
− T ))(e
γT
− 1))
.
Comment 6.5.3.2 Maghsoodi [615] presents a solution of bond option
valuation in the extended CIR term structure.
6.5.4 American Bond Options and the CIR Model
Consider the problem of the valuation of an American bond put option in the
context of the CIR model.
Proposition 6.5.4.1 Let us assume that
dr
t
= k(θ −r
t
)dt + σ
√
r
t
dW
t
,r
0
= x. (6.5.6)
The American put price decomposition reduces to
P
A
(r, T − t)=P
E
(r, T − t) (6.5.7)
+ KE
Q
T
t
exp
−
u
t
r
s
ds
r
u
1
{r
u
≥b(u)}
du
r
t
= r
where Q is the risk-neutral probability and b(·) is the put exercise boundary.
Proof: We give a decomposition of the American put price into two
components: the European put price given by Cox-Ingersoll-Ross, and the
American premium. We shall proceed along the lines used for the American
stock options. Let t be fixed and denote
R
t
u
=exp
−
u
t
r
v
dv
.
6.5 Some Computations on Bessel Bridges 379
We further denote by P
A
(r, T −u) the value at time u of an American put with
maturity T on a zero-coupon bond paying one monetary unit at time T
∗
with
T ≤ T
∗
. In a similar way, to the function F defined in the context of American
stock options, P
A
is differentiable and only piecewise twice differentiable in
the variable r.Itˆo’s formula leads to
R
t
T
P
A
(r
T
, 0) = P
A
(r
t
,T − t)+
T
t
R
t
u
L(P
A
)(r
u
,T − u) du
−
T
t
R
t
u
(r
u
P
A
(r
u
,T − u) − ∂
u
P
A
(r
u
,T − u))du
+
T
t
R
t
u
∂
r
P
A
(r
u
,T − u)σ
√
r
u
dW
u
(6.5.8)
where the infinitesimal generator L of the process solution of equation (6.5.6)
is defined by
L =
1
2
σ
2
r
∂
2
∂r
2
+ k(θ − r)
∂
∂r
.
Taking expectations and noting that the stochastic integral on the right-hand
side of (6.5.8) has 0 expectation, we obtain
E
Q
(R
t
T
P
A
(r
T
, 0)|F
t
)=P
A
(r
t
,T − t)+E
Q
T
t
R
t
u
L(P
A
)(r
u
,T − u) du|F
t
+ E
Q
T
t
R
t
u
(r
u
P
A
(r
u
,T − u) − ∂
u
P
A
(r
u
,T − u))du|F
t
. (6.5.9)
In the continuation region, P
A
satisfies the same partial differential equation
as the European put
A(r, T − u):=L(P
A
)(r, T − u)+rP
A
(r, T − u) − ∂
u
P
A
(r, T − u)) = 0,
∀u ∈ [t, T [, ∀r ∈]0,b(u)]
where b(u) is the level of the exercise boundary at time u:
b(u):=inf{α ∈ R
+
|P
A
(α, T −u)=K − B(α, T − u)}
Here, B(r, s)=exp(−A(s) − rG(s)) determines the price of the zero-coupon
bond with time to maturity s and current value of the spot interest rate r (see
Corollary 6.3.4.3). Therefore, the quantity A(r, T − u) is different from zero
only in the stopping region, i.e., when r
u
≥ b(u), or, since B is a decreasing
function of r (the function G is positive), when B(r
u
,T−u) ≤ B(b(u),T−u).
Equation (6.5.9) can be rewritten
E
Q
R
t
T
(K − B(r
T
,T
∗
− T ))
+
|F
t
= P
A
(r
t
,T − t)
+ E
Q
T
t
R
t
u
(−L(B(r
u
,T − u)) + r
u
(K − B(r
u
,T − u))
+ ∂
u
B(r
u
,T − u)) 1
{r
u
≥b(u)}
du|r
t
= r
. (6.5.10)
380 6 A Special Family of Diffusions: Bessel Processes
Another way to derive the latter equation from equation (6.5.9) makes use of
the martingale property of the process
R
t
u
P
A
(r
u
,T − u),u ∈ [t, T]
under the risk-adjusted probability Q, in the continuation region. Notice that
the bond value satisfies the same PDE as the bond option. Therefore
−L(B(r
u
,T − u)) + r
u
(K − B(r
u
,T − u)) + ∂
u
B(r
u
,T − u)=0.
Finally, equation (6.5.10) can be rewritten as follows
P
A
(r, T − t)=P
E
(r, T − t)+K
T
t
E
Q
R
t
u
r
u
1
{r
u
≥b(u)}
|F
t
du .
Jamshidian [474] computed the early exercise premium given by the latter
equation, using the forward risk-adjusted probability measure. Under this
equivalent measure, the expected future spot rate is equal to the forward rate,
and the expected future bond price is equal to the future price. This enables
the discount factor to be pulled out of the expectation and the expression for
the early exercise premium to be simplified.
Here, we follow another direction (see Chesney et al. [171]) and show how
analytic expressions for the early exercise premium and the American put
price can be derived by relying on known properties of Bessel bridges [716].
Let us rewrite the American premium as follows
E
Q
T
t
R
t
u
r
u
1
{r
u
≥b(u)}
du
r
t
= r
= E
Q
T
t
du E
Q
R
t
u
|r
u
r
u
1
{r
u
≥b(u)}
r
t
= r
.
The probability density of the interest rate r
u
conditional on its value at
time t is known. Therefore, the problem of the valuation of the American
premium rests on the computation of the inner expectation. More generally
let us consider the following Laplace transform (as in Scott [777])
E
Q
R
t
u
r
t
,r
u
.
Defining the process Z by a simple change of time
Z
s
= r
4s/σ
2
, (6.5.11)
we obtain
E
Q
R
t
u
(r
t
,r
u
)
= E
Q
exp
−
4
σ
2
uσ
2
/4
tσ
2
/4
Z
v
dv
Z
tσ
2
4
,Z
uσ
2
4
.
6.6 Asian Options 381
In the risk-adjusted economy, the spot rate is given by the equation (6.3.2)
and from the time change (6.5.11), the process Z is a CIR process whose
volatility is equal to 2. Setting δ =
4kθ
σ
2
,κ= −
4k
σ
2
, we obtain
dZ
t
=(κZ
t
+ δ) dt +2
Z
t
dW
t
,
where (W
t
,t ≥ 0) is a Q-Brownian motion. We now use the absolute
continuity relationship (6.5.4) between CIR bridges and Bessel bridges, where
the notation
κ
Q
δ,T
x→y
is defined
E
R
t
u
r
t
= x, r
u
= y
=
κ
Q
δ,σ
2
(u−t)/4
x→y
exp
−
4
σ
2
uσ
2
/4
tσ
2
/4
Z
s
ds
=
Q
δ,σ
2
(u−t)/4
x→y
exp
−
4
σ
2
+
κ
2
2
uσ
2
/4
tσ
2
/4
Z
s
ds
Q
δ,σ
2
(u−t)/4
x→y
exp
−
κ
2
2
uσ
2
/4
tσ
2
/4
Z
s
ds
.
From the results (6.5.3) on Bessel bridges and some obvious simplifications
E
exp
−
u
t
r
s
ds
r
t
= x, r
u
= y
=
c sinh(κγ)exp
x + y
2γ
(1 − cγ coth(cγ))
I
ν
c
√
xy
sinh cγ
κ sinh(cγ)exp
x + y
2γ
(1 − κγ coth(κγ))
I
ν
κ
√
xy
sinh κγ
:=m(x, t, y, u)
where γ =
σ
2
(u−t)
4
, ν =
δ
2
− 1, c =
8
σ
2
+ κ
2
and I
ν
is the modified Bessel
function. We obtain the American put price
P
A
(r, T −t)=P
E
(r, T −t)+K
T
t
du
∞
b(u)
m(r, t, y, u) f
u−t
(r, y)dy , (6.5.12)
where f is the density defined in Proposition 6.3.2.1.
Note that, as with formula (6.5.7), formula (6.5.12) necessitates, in order
to be implemented, knowledge of the boundary b(u).
6.6 Asian Options
Asian options on the underlying S have a payoff, paid at maturity T ,equal
to (
1
T
T
0
S
u
du −K)
+
. The expectation of this quantity is usually difficult to
382 6 A Special Family of Diffusions: Bessel Processes
evaluate. The fact that the payoff is based on an average price is an attractive
feature, especially for commodities where price manipulations are possible.
Furthermore, Asian options are often cheaper than the vanilla ones. Here, we
work in the Black and Scholes framework where the underlying asset follows
dS
t
= S
t
(rdt + σdW
t
) ,
or
S
t
= S
0
exp[σ(W
t
+ νt)] ,
where W is a BM under the e.m.m. The price of an Asian option is
C
Asian
(S
0
,K)=E
⎛
⎝
e
−rT
S
0
T
T
0
e
σ(W
s
+νs)
ds − K
+
⎞
⎠
.
For any real ν, we denote A
(ν)
T
=
T
0
exp[2(W
s
+ νs)]ds and A
T
= A
(0)
T
.
The scaling property of BM leads to S
s
= S
0
exp(σνs)exp(2
W
σ
2
s/4
),
where (
W
u
:=
σ
2
W
4u/σ
2
,u ≥ 0) is a Brownian motion. Using
W ,wesee
that
T
0
S
s
ds =
4
σ
2
S
0
σ
2
T/4
0
exp[2(
W
u
+ μu)] du
law
=
4
σ
2
S
0
A
(μ)
σ
2
T/4
with μ =
2ν
σ
.
6.6.1 Parity and Symmetry Formulae
Assume here that
dS
t
= S
t
((r − δ)dt + σdW
t
) .
We denote A
S
T
=
1
T
T
0
S
u
du and C
Asian
fi
= C
Asian
fi
(S
0
,K; r, δ) the price of
a call Asian option with a fixed-strike, whose payoff is (
1
T
A
S
T
− K)
+
and
C
Asian
f
= C
Asian
f
(S
0
,λ; r, δ) the price of a call Asian option with floating strike,
with payoff (λS
T
−
1
T
A
S
T
)
+
.
The payoff of a put Asian option with a fixed-strike is (K −
1
T
A
S
T
)
+
,
the price of this option is denoted by P
Asian
fi
= P
Asian
fi
(S
0
,K; r, δ). The price
of a put Asian option with floating strike, with payoff (
1
T
A
S
T
− λS
T
)
+
is
P
Asian
f
= P
Asian
f
(S
0
,λ; r, δ).
Proposition 6.6.1.1 The following parity relations hold
(i) P
Asian
f
= C
Asian
f
+
1
(r − δ)T
(e
−δT
− e
−rT
)S
0
− λS
0
e
−δT
,
(ii) P
Asian
fi
= C
Asian
fi
+
1
(r − δ)T
(e
−δT
− e
−rT
)S
0
− Ke
−rT
.
6.6 Asian Options 383
Proof: Obvious.
We present a symmetry result.
Proposition 6.6.1.2 The following symmetry relations hold
C
Asian
f
(S
0
,λ; r, δ)=P
Asian
fi
(S
0
,λS
0
; δ, r) ,
C
Asian
fi
(S
0
,K; r, δ)=P
Asian
f
(S
0
,K/S
0
; ,δ,r) .
Proof: This is a standard application of change of num´eraire techniques.
C
Asian
f
(S
0
,λ; r, δ)=E(e
−rT
(λS
T
−
1
T
A
S
T
)
+
)
= E
⎛
⎝
e
−rT
S
T
S
0
S
0
λ −
1
T
T
0
S
u
S
T
du
+
⎞
⎠
=
E
⎛
⎝
e
−δT
S
0
λ −
1
T
T
0
S
u
S
T
du
+
⎞
⎠
where
P|
F
T
= e
−(r−δ)T
S
T
S
0
P|
F
T
. From Cameron-Martin’s Theorem, the
process Z defined as Z
t
= W
t
− σt is a
P-Brownian motion. From
1
T
A
S
T
S
T
=
1
T
T
0
S
u
S
T
du =
1
T
T
0
e
(r−δ+
1
2
σ
2
)(u−T )+σ(Z
u
−Z
T
)
du ,
the law of
A
S
T
S
T
under
P is equal to the law of
1
T
T
0
e
(r−δ+
1
2
σ
2
)(u−T )−σZ
T −u
du
law
=
1
T
T
0
e
(δ−r−
1
2
σ
2
)s+σZ
s
ds .
The second formula is obtained using the call-put parity.
Comment 6.6.1.3 The second symmetry formula of Proposition 6.6.1.2 is
extended to the exponential L´evy framework by Fajardo and Mordecki [339]
and by Eberlein and Papapantoleon [293]. The relation between floating and
fixed strike, due to Henderson and Wojakowski [432] is extended to the
exponential L´evy framework in Eberlein and Papapantoleon [292].
6.6.2 Laws of A
(ν)
Θ
and A
(ν)
t
Here, we follow Leblanc [571]andYor[863].
Let W be a Brownian motion, and A
t
=
t
0
e
2W
s
ds, A
†
t
=
t
0
e
W
s
ds.Let
f,g,h be Borel functions and
k(t)=E
f(A
t
) g(A
†
t
) h(e
W
t
)
.
384 6 A Special Family of Diffusions: Bessel Processes
Lemma 6.6.2.1 The Laplace transform of k is given by
∞
0
k(t)e
−θ
2
t/2
dt =
∞
0
du f (u)P
(θ)
1
h(R
u
)g(K
u
)
R
2+θ
u
where R is a Bessel process with index θ, starting from 1 and K
t
=
t
0
du
R
u
.
Proof: Let C be the inverse of the increasing process A.Wehaveseen,in
the proof of Lamperti’s theorem (applied here in the particular case ν =0)
that dC
u
=
1
R
2
u
du where R
u
=expW
C
u
is a Bessel process of index 0 (of
dimension 2) starting from 1. The change of variable t = C
u
leads to
A
C
u
= t, A
†
C
u
= K
u
=
u
0
ds
R
s
, exp W
C
u
= R
u
and
∞
0
k(t)e
−θ
2
t/2
dt = E
∞
0
dt e
−θ
2
t/2
f(A
t
) g(A
†
t
) h(e
W
t
)
= E
∞
0
du
R
2
u
e
−θ
2
C
u
/2
f(u) g(K
u
) h(R
u
)
.
The absolute continuity relationship (6.1.5)leadsto
∞
0
k(t)e
−θ
2
t/2
dt =
∞
0
du f (u) P
(θ)
1
h(R
u
)g(K
u
)
R
2+θ
u
.
As a first corollary, we give the joint law of (exp W
Θ
,A
Θ
), where Θ is an
exponential random variable with parameter θ
2
/2, independent of W .
Corollary 6.6.2.2
P(exp W
Θ
∈ dρ, A
Θ
∈ du)=
θ
2
2ρ
2+θ
p
(θ)
u
(1,ρ)1
{u>0}
1
{ρ>0}
dρdu
where p
(θ)
is the transition density of a BES
(θ)
.
Proof: It suffices to apply the result of Lemma 6.6.2.1 with g =1.
We give a closed form expression for P(A
(ν)
t
∈ du|B
t
+ νt = x). A
straightforward application of Cameron-Martin’s theorem proves that this
expression does not depend on ν and we shall denote it by a(t; x, u)du.
Proposition 6.6.2.3 If a(t; x, u)du = P(A
t
∈ du|B
t
= x), then
1
√
2πt
exp
−
x
2
2t
a(t; x, u)=
1
u
exp
−
1
2u
(1 + e
2x
)
Ψ
e
x
/u
(t) (6.6.1)
6.6 Asian Options 385
where
Ψ
r
(t)=
r
(2π
3
t)
1/2
exp
π
2
2t
Υ
r
(t) , (6.6.2)
Υ
r
(t)=
∞
0
dy exp(−y
2
/2t)exp[−r(cosh y)] sinh(y)sin
πy
t
. (6.6.3)
Proof: Consider two positive Borel functions f and g. On the one hand,
E
∞
0
dt exp
−
μ
2
t
2
f(exp(W
t
)) g(A
t
)
=
∞
0
dt exp
−
μ
2
t
2
∞
−∞
dx
√
2πt
f(e
x
)exp
−
x
2
2t
∞
0
du g(u)a(t; x, u) .
On the other hand, from Proposition 6.6.2.2, this quantity equals
∞
0
du g(u)
∞
0
dρ
ρ
μ+2
f(ρ) p
(μ)
u
(1,ρ)
=
∞
−∞
dx exp[−(μ +1)x] f (e
x
)
∞
0
du g(u)p
(μ)
u
(1,e
x
) .
We obtain
1
√
2πt
∞
0
dt exp
−
1
2
μ
2
t +
x
2
t
a(t; x, u)=exp(−(μ +1)x) p
(μ)
u
(1,e
x
) .
(6.6.4)
Using the equality
I
ν
(r)=
∞
0
e
−
ν
2
u
2
Ψ
r
(u)du ,
the explicit form of the density p
(μ)
u
and the definition of Ψ , we write the
right-hand side of (6.6.4)as
1
u
exp
−
1
2u
1+e
2x
∞
0
dt Ψ
e
x
/u
(t)exp
−
μ
2
t
2
.
Corollary 6.6.2.4 The law of A
(ν)
t
is P(A
(ν)
t
∈ du)=ϕ(t, u)du,where
ϕ(t, u)=u
ν−1
1
(2π
3
t)
1/2
exp
π
2
2t
−
1
2u
−
ν
2
t
2
∞
0
dy y
ν
exp(−
1
2
uy
2
)Υ
y
(t)
(6.6.5)
where Υ is defined in (6.6.3).
386 6 A Special Family of Diffusions: Bessel Processes
Proof: From the previous proposition
P(A
(ν)
t
∈ du)=
R
f(t, x)a(t, x, u)dx du = ϕ(t, u)du
where
f(t, x)dx = P(B
t
+ νt ∈ dx)=
1
√
2πt
exp
−
1
2t
(x − νt)
2
dx .
Using the expression of a(t, x, u), some obvious simplifications and a change
of variable, we get
ϕ(t, u)=
1
u
√
2πt
∞
−∞
dx e
−(x−νt)
2
/2t
exp
−
1+e
2x
2u
Ψ
e
x
/u
(t)
√
2πt e
x
2
/(2t)
=
1
u(2π
3
t)
1/2
exp
π
2
2t
−
1
2u
−
ν
2
t
2
∞
−∞
dx
e
x(ν+1)
u
exp
−
e
2x
2u
Υ
e
x
/u
(t)
=
1
u(2π
3
t)
1/2
exp
π
2
2t
−
1
2u
−
ν
2
t
2
∞
0
dy (yu)
ν
exp
−
1
2
uy
2
Υ
y
(t) .
One can invert the Laplace transform of the pair (e
W
t
,A
t
) given in
Corollary 6.6.2.2 and we obtain:
Corollary 6.6.2.5 Let W be a Brownian motion and A
t
=
t
0
e
2W
s
ds.Then,
for any positive Borel function f,
E(f(e
W
t
,A
t
)) =
1
(2π
3
t)
1/2
∞
0
dy
∞
0
dv f(y,
1
v
) e
−v(1+y
2
)/2
Υ
yv
(t) (6.6.6)
where the function Υ was defined in (6.6.3).
Exercise 6.6.2.6 Prove that the density of the pair (exp(W
Θ
+ νΘ),A
(ν)
Θ
),
where A
(ν)
t
=
t
0
e
2(W
s
+νs)
ds,is
θ
2
2x
2+λ
x
ν
p
(λ)
a
(1,x)1
{x>0}
1
{a>0}
dxda
with λ
2
= θ
2
+ ν
2
.
Proposition 6.6.2.7 Let W be a BM, A
t
=
t
0
e
2W
s
ds and A
†
t
=
t
0
e
W
s
ds.
The Laplace transform of (W
Θ
,A
Θ
,A
†
Θ
),whereΘ is an exponential random
variable with parameter θ
2
/2, independent of W ,is
H
a,b,c
(θ)=E
exp(−aW
Θ
−
b
2
2
A
Θ
− cA
†
Θ
)
=
θ
2
2
4
1+θ
Γ (1 + θ)
∞
0
dte
−ct
∞
0
dy J
y+a/4,b/2, 2θ
4
(t) y
θ