sparse grid method in the libor market model. option valuation and the

S 0 u
d
u S 0
S u 2
0
Su
3
0
S du 2
0
4
S0u
S
du 3
0
S0
S0
ud S 0
2
S0ud
S u d
2
0
2
T 0
T
1
S 0
d S 0
2
S0d
S ud 3
0
T
0
T
1
T
2
3
S0d
T3
T4
S d 4
0
Figure 2.1: From Single Period to Multi-Period
assets in a replicating portfolio have to be adjusted according to what is happening with
the prices of the assets. Intuitively, a replication strategy would be meaningful for the
writer of the claim only in case this strategy does not require any maintaining costs, i.e.
is self-financing. In other words, after spending a certain amount of cash to establish
positions in S 0 and B 0 , the replication procedure should involve only changing their
weights and exclude any additional cash inputs. Denote a vector of portfolio weights
as θ = (ζ, γ) t . A one-step example of the self-financing property for a replicating
portfolio V would be:
V i = ζ i S i + γ i B i = ζ i+1 S i + γ i+1 B i (2.4)
where ζ i+1 and γ i+1 are the quantities of S and B respectively, held over the period
t ∈ [ i∆t, (i + 1)∆t ) . A change in portfolio value over one-step period gives:
For an N-step recombinant tree:
V i+1 − V i = ζ i+1 (S i+1 − S i ) + γ i+1 (B i+1 − B i ) (2.5)
N−1
∑
N−1
∑
V N = V 0 + ζ i+1 (S i+1 − S i ) + γ i+1 (B i+1 − B i ) (2.6)
i=0
If we assume interest rates to be zero, then it can be easily shown that for V N B −1 =
Ṽ N :
N−1
∑
Ṽ N = Ṽ0 + ζ i+1 ( ˜S i+1 − ˜S i ) or ∆Ṽi = ζ i+1 ∆ ˜S i (2.7)
i=0
7
i=0