Tutorat 3 de Mathématiques (2ème année)

St
St+δt = (1 + rδt +dBt)St,
δt r
dBt
St
±δb p+=p−= 1/2δb
δt
Xt= ln St
Xt+δt =Xt+dRt,
dRtδr±=
ln(1 + rδt ±δb)
P(Xt∈[x, x +dx]) Xt[x, x +dx]
P(Xt∈[x, x +dx]) = f(x, t)dx
f(x, t +δt) = 1
2(f(x−δr−, t) + f(x−δr+, t)) .
δt
∂f
∂t =1 2
∞
X
n=1 δrn
++δrn −
δt (−1)n n!∂nf ∂xn+O(δt).

δb
∂f
∂t =−r0∂f ∂x +σ2 2
∂2f
∂x2.
r0σ r δb Bt
g(x, t) = f(x+r0t, t)g
σ
g(x, 0) f(x, 0) g
g(x, t) = Zdx0f(x0,0)G(x−x0, t),
Gs0t= 0 f(x, 0)
f(x, t)St
St
t t =T
t= 0 K
t=T K > ST
K < ST
K ST−K
max(ST−K, 0) t=T
t= 0
V ST
V t = 0
V t = 0
max(ST−K, 0) t=T

t=T
Stt= 0
K
V(S0,0) t= 0 t=T
ST< K
STK < ST
Kmin(K, ST)
max(ST−K, 0) ST
V(S0,0)

STV(ST, T )V(ST, T ) = max(ST−K, 0)
t V (St, t)
t Stt= 0 V(S0,0)
V
St
St
∆ Π
t∆St+ Π ∆ Π
∆
Π
t
T=t+δt StST
t
V(ST, T )T∆(St, t)
Π(St, t)StSt+δt
t
V(St, t) = q+V(Steδr+, t +δt) + q−V(Steδr−, t +δt)
q+, q−q++q−= 1 q+eδr++q−eδr−= 1
St
t−δt
T=t+δt
t−δ St−δt t
St
0
T=Nδt N 1
δt V (S, t)
∂V
∂t +σ2S2 2
∂2V
∂S2= 0.

∆,Π
V δt →0
r σ
max(ST−K, 0)
C(St, t)
Φ(x) = 1
√2πZx
−∞
e−t2/2dt.
K=σ= 1
τ=T−t x = ln(S/K)−σ2/2τ
T
K
max(K−ST,0) P t C
C(St, t)−P(St, t) = St−K.