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

Remark. The way we impose boundary conditions, that is keeping them fixed or
setting them to Black-Scholes value, does make a difference when we look at extracted
Delta profile. Yet, in both cases, the approximation is still not accurate enough to
provide smooth surfaces on the Delta plot. Taking advantage of the fact that people
are mostly interested in the ∆’s in at-the-money or close to at-the-money states, the
disturbed boundaries were excluded from the plots above.
6.1.4 Greeks: Vega
∆ is just one of the several sensitivities that help explore exposure to risk in the short
position as a function of different model parameters. Other ’Greek letters’ include Θ
with respect to time changes and Γ as the second derivative with respect to changes in
underlying. In this part of the text we turn to vega, V, a sensitivity of a product with
respect to the volatility in one of the underlying forward rates:
V = ∂U
∂σ i
(6.2)
Information about V is not encapsulated in the solution explicitly. Here, whether full
and sparse, the solution has to be recomputed with a small shift in volatility value of
one of the underlying forward rates. Figure (6.8) below shows the sparse solutions
for vega with respect to the σ 1 on Level 8. Plots in Figure (6.9) illustrate slices of a
resulting vega profile with fixed L 2 rate.
Sparse Vega points. Level 8
Vega Value
0.03
0.025
0.015
0.02
0.005
0.01
-0.005 0
-0.01
0
0.02 0.04 0.06 0.08
0.1 0.12 0.14 0.16
0
Forward Rate L1
0.04
0.02
0.16
0.14
0.12
0.03
0.025
0.02
0.015
0.01
0.005
0
-0.005
-0.01
0.1
0.08
Forward Rate L2
0.06
Figure 6.8: 2D Chooser. Sparse vega surface on level 8.
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