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

6.3 Discontinuous Payoff. 2D Digital option.
The third and final task case that we have considered is the case of a discontinuous
payoff, characterictic of so-called digital options. Figures (6.16) and (6.5) show the
digital payoff that describes our specific product:
{
V Digital 0.02 L1 > K OR L
(T ) =
2 > K
0 otherwise
(6.5)
With full grid of level 6 and K=5.5%, such payoff can be demonstrated visually:
Discontinuous Payoff
0.02
0.015
0.01
0.005
0
0.16
0.14
0.12
0.1
0.08
Forward Rate L2 0.06
0.04
0.02
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Forward Rate L1
0.02
0.015
0.01
0.005
0
Figure 6.16: 2D Digital Option. Terminal payoff discretized on a L6 full grid.
The stripe of different color in the corner hints that in such setting the strike rate K
is an off-grid point. Indeed, in case of level 6 discretization, K falls roughly between
values 0.05391 and 0.05625.
6.3.1 Valuation Results.
Level of Discretization
7 7.5 8 8.5 9 9.5 10
21.2
MC Simulation Value+Std.Error
Full-Grid FD value
21
20.8
Option price
20.6
20.4
20.2
20
19.8
19.6
500000 1e+006 1.5e+006 2e+006 2.5e+006 3e+006
Number of MC simulations
Figure 6.17: 2D Digital Option. Convergence of Monte Carlo and full grid FD.
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