American Monte Carlo

abstract type AmcPayoff <: Payoff end

AmcPayoff is used to implement common methods for AMC payoffs. Concrete AMC payoffs are assumed to hold a fields links::AmcPayoffLinks and regr::AmcPayoffRegression.

An AMC payoff is special and does not fit into the structure of unary/binary nodes. Instead, we have several edges to other payoffs with observation times before (x and y) and after (z) its own observation time.

struct AmcPayoffLinks
    obs_time::ModelTime
    x::AbstractVector
    y::AbstractVector
    z::AbstractVector
    curve_key::String
end

An AmcPayoffLinks object holds common data fields for an American Monte Carlo (AMC) payoff.

Here, obs_time is the observation time of the AMC payoff and x, y, z are related Payoff vectors. The elements of x and y represent random variables with observation times after obs_time. The elements of z represent random variables with observation time (at or) before obs_time. z is the vector of regression variables.

The parameter curve_key is used to specify a discount curve for numeraire price calculation.

We calculate $X$ as the sum of discounted payoff values of x and $Y$ as the sum of discounted payoff values of y. $Z = [ Z_1, Z_2, ... ]$ represent the (undiscounted) values of the regression variables from the payoff vector z.

Then we estimate the trigger variable

$T = E[ X - Y | Z_1, Z_2, ... ]$

An actual AMC payoff uses $X$, $Y$, and $T$ to calculate its values.

mutable struct AmcPayoffRegression
    path::Union{AbstractPath, Nothing}
    make_regression::Union{Function, Nothing}
    regression::Any
end

AmcPayoffRegression holds the common data fields to regression and regression calibration for AMC payoffs. These data fields are supposed to be updated subsequent creation of the object. As a consequence, AmcPayoffRegression is declared mutable.

The element path is a Monte Carlo path. This element is typically linked to a simulation and a context mapping.

make_regression is a function/functor with signature

make_regression(C::AbstractMatrix, O::AbstractVector) -> obj.

This function is typically a lambda for polynomial_regression (or similar) where parameters like maximum polynomial degree are fixed.

The result of make_regression is stored in the regression field. For the result object regression we assume that a method

predict(regression, C)

is defined. The method predict is supposed to return a prediction for a matrix of controls C. See PolynomialRegression as an example.

struct AmcMax <: AmcPayoff
    links::AmcPayoffLinks
    regr::AmcPayoffRegression
end

An AmcMax payoff is used to model long call rights.

It calculates the expectation of maximum of (sum of) discounted payoffs x and (sum of) discounted payoffs y. Expectation is calculated conditional on information at obs_time. This is approximated by regression variable payoffs z.

AmcMax(
    obs_time::ModelTime,
    x::AbstractVector,
    y::AbstractVector,
    z::AbstractVector,
    path::Union{AbstractPath, Nothing},
    make_regression::Union{Function, Nothing},
    curve_key::String,
    )

Create an AmcMax payoff.

struct AmcMin <: AmcPayoff
    links::AmcPayoffLinks
    regr::AmcPayoffRegression
end

An AmcMin payoff is used to model short call rights.

It calculates the expectation of minimum of (sum of) discounted payoffs x and (sum of) discounted payoffs y. Expectation is calculated conditional on information at obs_time. This is approximated by regression variable payoffs z.

AmcMin(
    obs_time::ModelTime,
    x::AbstractVector,
    y::AbstractVector,
    z::AbstractVector,
    path::Union{AbstractPath, Nothing},
    make_regression::Union{Function, Nothing},
    curve_key::String,
    )

Create an AmcMin payoff.

struct AmcOne <: AmcPayoff
    links::AmcPayoffLinks
    regr::AmcPayoffRegression
end

An AmcOne payoff is used to model the indicator variable $1_{(X > Y)}$.

It calculates the expectation of maximum of (sum of) discounted payoffs x and (sum of) discounted payoffs y. Expectation is calculated conditional on information at obs_time. This is approximated by regression variable payoffs z.

AmcOne(
    obs_time::ModelTime,
    x::AbstractVector,
    y::AbstractVector,
    z::AbstractVector,
    path::Union{AbstractPath, Nothing},
    make_regression::Union{Function, Nothing},
    curve_key::String,
    )

Create an AmcOne payoff.

struct AmcSum <: AmcPayoff
    links::AmcPayoffLinks
    regr::AmcPayoffRegression
end

An AmcSum payoff is used to model general conditional expectations

$B(t) E[ X(T)/B(T) | Z(t) ]$

AmcSum payoffs are typically used to calculate future model prices in exposure simulation applications.

AmcSum(
    obs_time::ModelTime,
    x::AbstractVector,
    z::AbstractVector,
    path::Union{AbstractPath, Nothing},
    make_regression::Union{Function, Nothing},
    curve_key::String,
    )

Create an AmcSum payoff.

reset_regression!(
    p::AmcPayoff,
    path::Union{AbstractPath, Nothing} = nothing,
    make_regression::Union{Function, Nothing}  = nothing,
    )

Reset the regression properties for an AMC payoffs.

This method is used to allow setting and updating AMC regression after payoff creation.

reset_regression!(
    p::UnaryNode,
    path::Union{AbstractPath, Nothing} = nothing,
    make_regression::Union{Function, Nothing}  = nothing,
    )

Delegate resetting the regression properties to child payoff.

reset_regression!(
    p::BinaryNode,
    path::Union{AbstractPath, Nothing} = nothing,
    make_regression::Union{Function, Nothing}  = nothing,
    )

Delegate resetting the regression properties to child payoffs.

reset_regression!(
    p::Union{Leaf, CompoundedRate, Optionlet, Swaption},
    path::Union{AbstractPath, Nothing} = nothing,
    make_regression::Union{Function, Nothing}  = nothing,
    )

Ignore resetting the regression properties for Leaf and similar payoffs.

Note that some rates payoffs and rates options are no Leafs.

reset_regression!(
    p::Payoff,
    path::Union{AbstractPath, Nothing} = nothing,
    make_regression::Union{Function, Nothing}  = nothing,
    )

Throw an error if reset_regression! is not implemented for concrete payoff.

reset_regression!(
    leg::BermudanSwaptionLeg,
    path::Union{AbstractPath, Nothing} = nothing,
    make_regression::Union{Function, Nothing}  = nothing,
    )

Reset the regression properties for the AMC payoffs of the BermudanSwaptionLeg.

This method is used to allow setting and updating AMC regression after leg creation.

calibrate_regression(links::AmcPayoffLinks, regr::AmcPayoffRegression)

Calibrate the regression for an AMC payoff.

has_amc_payoff(p::AmcPayoff)

Determine whether a payoff is or contains an AMC payoff.

AMC payoffs require special treatment e.g. for sensitivity calculation.

has_amc_payoff(p::UnaryNode)

Determine whether a payoff is or contains an AMC payoff.

has_amc_payoff(p::BinaryNode)

Determine whether a payoff is or contains an AMC payoff.

has_amc_payoff(p::Union{Leaf, CompoundedRate, Optionlet, Swaption})

Determine whether a payoff is or contains an AMC payoff.

has_amc_payoff(p::Payoff)

Determine whether a payoff is or contains an AMC payoff.

has_amc_payoff(payoffs::AbstractVector)

Determine whether any payoff is or contains an AMC payoff.