Title Paris-Princeton Lectures on Mathematical Finance 2003 English 2.0 MB 256
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Page 1

Lecture Notes in Mathematics 1847
Editors:
J.--M. Morel, Cachan
F. Takens, Groningen
B. Teissier, Paris

Page 2

Tomasz R. Bielecki Tomas Bj¤ork
Monique Jeanblanc Marek Rutkowski
Jos·e A. Scheinkman Wei Xiong

Paris-Princeton Lectures
on Mathematical Finance
2003

Editorial Committee:

R. A. Carmona, E. C‚ inlar,
I. Ekeland, E. Jouini,
J. A. Scheinkman, N. Touzi

123

Page 128

Hedging of Defaultable Claims 121

dXt = xt dt + x̂t dWt + x̃t dMt, (138)
dΘt = Θt−

(
ϑt dt + ϑ̂t dWt + ϑ̃t dMt

)
, (139)

dΨt = ψt dt + ψ̂t dWt + ψ̃t dMt, (140)

where the drifts xt, ϑt and ψt are yet to be determined. From Itô’s formula, we obtain
(recall that ξt = γt11{τ>t})

d(Vt −Xt)2 = 2(Vt −Xt)(πtσ − x̂t) dWt − 2(Vt − Xt−)x̃t dMt
+
[
(Vt −Xt− − x̃t)2 − (Vt −Xt−)2

]
dMt

+
(
2(Vt −Xt)(πtν − xt) + (πtσ − x̂t)2

+ ξt
[
(Vt −Xt − x̃t)2 − (Vt −Xt)2

])
dt,

where we denote Vt = V vt (π). The process J(π) is a martingale if and only if its
drift term k(t, πt, xt, ϑt, ψt) = 0 for every t ∈ [0, T ].
Straightforward calculations show that

k(t, πt, ϑt, xt, ψt) = ψt + Θt
[
ϑt(Vt −Xt)2

+ 2(Vt −Xt)
[
(πtν − xt) + ϑ̂t(πtσ − x̂t) + ξtx̃t

]
+ (πtσ − x̂t)2 + ξt(ϑ̃t + 1)

[
(Vt −Xt − x̃t)2 − (Vt −Xt)2

]]
.

In the first step, for any t ∈ [0, T ] we shall find π∗t such that the minimum of
k(t, πt, xt, ϑt, ψt) is attained. Subsequently, we shall choose the auxiliary processes
x = x∗, ϑ = ϑ∗ and ψ = ψ∗ in such a way that k(t, π∗t , x

t , ϑ

t , ψ

t ) = 0. This choice

will imply that k(t, πt, x∗t , ϑ

t , ψ

t ) ≥ 0 for any trading strategy π and any t ∈ [0, T ].

The strategy π∗, which minimizes k(t, πt, xt, ϑt, ψt), is the solution of the following
equation:

(V vt (π) −Xt)(ν + ϑ̂tσ) + σ(πtσ − x̂t) = 0, ∀ t ∈ [0, T ].

Hence, the strategy π∗ is implicitly given by

π∗t = σ
−1x̂t − σ−2(ν + ϑ̂tσ)(V vt (π

∗) −Xt) = At −Bt(V vt (π
∗) −Xt),

where we denote
At = σ

−1x̂t, Bt = σ
−2(ν + ϑ̂tσ).

After some computations, we see that the drift term of the process J admits the
following representation:

k(t, πt, ϑt, xt, ψt) = ψt + Θt(Vt −Xt)2(ϑt − σ2B2t )
+ 2Θt(Vt −Xt)

(
σ2AtBt − ϑ̂tx̂t − ϑ̃tx̃tξt − xt

)
+ Θtξt(ϑ̃t + 1)x̃

2
t .

Page 129

122 T.R. Bielecki, M. Jeanblanc, and M. Rutkowski

From now on, we shall assume that the auxiliary processes ϑ, x and ψ are chosen as
follows:

ϑt = ϑ

t = σ

2B2t ,

xt = x

t = σ

2AtBt − ϑ̂tx̂t − ϑ̃tx̃tξt,
ψt = ψ

t = −Θtξt(ϑ̃t + 1)x̃

2
t .

It is rather clear that if the drift coefficients ϑ, x, ψ in (138)-(140) are chosen as
above, then the drift term in dynamics of J is always non-negative, and it is equal to
0 for the strategy π∗, where π∗t = At −Bt(V vt (π∗) −Xt).
Our next goal is to solve equations (138)-(140). Let us first consider equation (139).
Since ϑt = σ2B2t , it suffices to find the three-dimensional process (Θ, ϑ̂, ϑ̃) which
is a solution to the following BSDE:

dΘt = Θt
(
σ−2(ν + ϑ̂tσ)

2 dt + ϑ̂t dWt + ϑ̃t dMt
)
, ΘT = 1.

It is obvious that the processes ϑ̂ = 0, ϑ̃ = 0 and Θ, given as

Θt = exp(−θ2(T − t)), ∀ t ∈ [0, T ], (141)

solve this equation.

In the next step, we search for a three-dimensional process (X, x̂, x̃), which solves
equation (138) with xt = x∗t = σ

2At(ν/σ2) = θx̂t. It is clear that (X, x̂, x̃) is the
unique solution to the linear BSDE

dXt = θx̂t dt + x̂t dWt + x̃t dMt, XT = X.

The unique solution to this equation is Xt = EQ(X | Gt), where Q is the risk-neutral
probability measure, so that dQ = ηt dP, where

dηt = −θηt dWt, η0 = 1.

The components x̂ and x̃ are given by the integral representation of the G-martingale
X with respect to WQ and M . Notice also that since ϑ̂ = 0, the optimal portfolio π∗

is given by the feedback formula

π∗t = σ
−1(x̂t − θ(V vt (π∗) −Xt)).

Finally, since ϑ̃ = 0, we have ψt = −ξtx̃2tΘt. Therefore, we can solve explicitly the
BSDE (140) for the process Ψ . Indeed, we are now looking for a three-dimensional
process (Ψ, ψ̂, ψ̃), which is the unique solution of the BSDE

dΨt = −Θtξtx̃2t dt + ψ̂t dWt + ψ̃t dMt, ΨT = 0.

Noting that the process

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