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Comptes Rendus Mathématique
Volume 354, n° 11
pages 1132-1138 (novembre 2016)
Doi : 10.1016/j.crma.2016.09.010
Received : 23 May 2016 ;  accepted : 27 September 2016
The parareal algorithm for American options
La méthode pararéelle pour les options américaines
 

Fig. 1




Fig. 1 : 

Black–Scholes case: errors on the payoff versus Δt on the left for several values of n and versus n on the right for several values of Δt . Both graphs are for Algorithm 3.2 in log–log scales and indicate a general behavior of the error ϵ not incompatible with (3.1).


Fig. 2




Fig. 2 : 

Constant elasticity case: same legend as in Fig. 1.


Fig. 3




Fig. 3 : 

Comparison between a standard LSMC solution and the parareal solution for the same number of time intervals at the finest level. The 4 points have respectively 1,2,3,4 levels; the first data point has one level and 4 intervals, the second has 2 levels and 16 intervals, the third one 3 levels and 64 intervals, the fourth one 4 levels and 256 intervals. The total number of time steps is on the horizontal axis, in log scale, and the error at n =2 is on the vertical axis, in log scale as well.


Fig. 4




Fig. 4 : 

Speed-up versus the number of processors, i.e. the parareal CPU time on a parallel machine divided by the parareal CPU time on the same machine but running on one processor. There are two levels only; the parameters are K =1,2..,32, n =2 and J =100 so as to keep each processor fully busy.

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