multilevel

criteria

Convergence criteria for the Multi-level Monte-Carlo

class ConvergenceCriteria(criteria: Callable[[float, array, float], bool], compute_mc_paths: Callable[[float, array, array], array])

Bases: object

Characterisation of the criteria convergence and the update of the number of paths after each pass

__init__(criteria: Callable[[float, array, float], bool], compute_mc_paths: Callable[[float, array, array], array])

Parameters

• criteria – criteria convergence function

• compute_mc_paths – function updating the number of paths for each level

compute_mc_paths_giles(rmse: float, vl: array, cl: array) → array

Same as in Giles papers :param rmse: root-mean square error :param vl: estimated variance of the correction terms \(|P_l - P_{l-1}|\) :param cl: cost of each level l :return: the updated number of Monte-Carlo paths for each level l

criteria_giles(alpha: float, ml: array, rmse: float) → bool

Same convergence criteria as in Giles papers

Parameters

• alpha – weak convergence rate

• ml – estimated mean of the correction terms \(|P_l - P_{l-1}|\)

• rmse – root-mean square error

Returns

true if the convergence criteria has been met

criteria_run_to_maximum_level(alpha: float, ml: array, rmse: float) → bool

Run the Multilevel Monte-Carlo until the last level :return: always false so that the algorithm runs until the maximum level

class GilesConvergenceCriteria

Bases: ConvergenceCriteria

__init__()

Parameters

• criteria – criteria convergence function

• compute_mc_paths – function updating the number of paths for each level

engine

Multilevel Monte-Carlo engine

Note

• the discounting is deterministic by design

• the implementation supports multiprocessing

helper_create_fun(this_cos_pricer, maturity)

class Engine(configuration: ConfigurationMultiLevel, coupling_process: Union[CouplingMarkovChain, CouplingSDE, CouplingProcessLevyCopula])

Bases: object

Multilevel Monte-Carlo engine

__init__(configuration: ConfigurationMultiLevel, coupling_process: Union[CouplingMarkovChain, CouplingSDE, CouplingProcessLevyCopula])

The Multilevel Monte-Carlo relies on the simulation of the fine and coarse processes linked by a coupling. :param configuration: Multilevel Monte-Carlo configuration :param coupling_process: stochastic coupling process of the fine and coarse processes

initialisation(product: Product) → None

Parameters

product – financial product to price

compute_level_l(level: int, current_mc_paths: int, extra_mc_paths: int, coupling_process: Union[CouplingMarkovChain, CouplingSDE, CouplingProcessLevyCopula], product: Product, df: float, statistics: MLMCStatistics) → None

Run the Multilevel Monte-Carlo for the level l :param level: current level :param current_mc_paths: number of Monte-Carlo paths applied to the level l :param extra_mc_paths: extra number of paths to be applied to the level l :param coupling_process: coupling process defining the coupling between the fine and coarse processes :param product: product to price :param df: discount factor function :param statistics: statistics object

price(product: Product, rmse: float) → MLMCStatistics

Pricing of the product by the Multilevel Monte-Carlo engine :param product: product to price :param rmse: root-mean square error

price_with_constant_mc_paths_and_level(product) → MLMCStatistics

Same as the price() function but with constant number of paths and a fixed number of levels :param product: product to price