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Multi-Level Monte Carlo method for pricing UE & path-dependent options

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European and path-dependent options pricing by MLMC

Objectives:

  • Implementation of Multi Level Monte Carlo method
  • Variance & computational cost comparison with crude MC
  • Test different discretization schemes/Add IS procedure

Output overview :

  • Closed-form formula:

MLMC_Results_1

  • Crude Monte Carlo:

MLMC_Results_2

  • Brownian Bridge method:

MLMC_Results_3

MLMC:

  • With time steps T/2**l:

MLMC

More paths for discrete approximations with lower time steps:

MLMC-RR2

After computation of the optimal parameters:

Example 1: comparison for pricing Call option

  • Using Euler scheme
  • MC: computational complexity = 300E6 & CPU time = 36.7 sec
  • MLMC: computational complexity = 51E6 & CPU time = 6.7 sec

Example 2: comparison for pricing Up-and-Out Call Barrier option

  • Using Milstein scheme
  • MC: computational complexity = 561E6 & CPU time = 25.5 sec
  • MLMC: computational complexity = 36E6 & CPU time = 6.3 sec

References:

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Multi-Level Monte Carlo method for pricing UE & path-dependent options

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