Payoffs

abstract type Payoff end

A Payoff is a random variable X in our stochastic model.

It represents a market object that can be evaluated in conjunction with a given path.

We are interested in realisations of a payoff at a given path, i.e. X(omega). Moreover, for risk-neutral valuation we are also interested in discounted payoffs at a given path.

We implement a Payoff as a root of a computational graph. The nodes of the computational graph are itself Payoff objects which represent mathematical operations or use the path omega to determine its realisations.

abstract type Leaf <: Payoff end

A Leaf is a particular Payoff which has no outgoing links to other Payoff objects. A Leaf typically uses the path to determine its realisations.

We assume that a Leaf has a field obs_time.

abstract type UnaryNode <: Payoff end

A UnaryNode is a particular Payoff which has exactly one outgoing link to another Payoff object.

We assume that the reference to the outgoing Payoff object is a field denoted x.

A UnaryNode is typically a decorator of the linked Payoff.

abstract type BinaryNode <: Payoff end

A BinaryNode is a particular Payoff which has exactly two outgoing links to other Payoff objects.

We assume that the references to the outgoing Payoff objects are fields denoted x and y.

A BinaryNode is typically a mathematical operation.

obs_time(p::Payoff)

A Payoff is typically observed at a particular time. In that sense, a Payoff is an $F_t$-measurable random variable.

The observation time represents the time, after which the payoff is known.

obs_times(p::Payoff)

A payoff is typically linked to other payoffs and forms a DAG. This function returns all observation times associated with a given payoff.

This functionality is required to determine relevant simulation grid points.

at(p::Payoff, path::AbstractPath)

Evaluate a Payoff at a given path, X(omega).

Depending on the functionality associated with the path, this function typically returns a vector of realisations.

This function is invoked when using call operator on a Payoff,

(p::Payoff)(path::AbstractPath) = at(p::Payoff, path::AbstractPath)

struct Fixed <: Leaf
    value::ModelValue
end

A deterministic quantity.

struct ScalarValue <: Leaf
    value::ModelValue
end

A scalar deterministic quantity.

struct Pay <: UnaryNode
    x::Payoff
    obs_time::ModelTime
    test_times::Bool
end

A Pay payoff allows the user to modify the observation time of a given payoff. This is relevant for discounting.

Typically, we use Pay to specify the pay time for a payoff.

mutable struct Cache <: UnaryNode
    x::Payoff
    path::Union{AbstractPath, Nothing}
    value::Union{AbstractVector, Nothing}
end

A Cache payoff aims at avoiding repeated calculations of the same payoff.

If a Payoff object is referenced by several parent Payoff objects then each call of at() of the parent object triggers a call of at() of the child object that all return the same value(s).

A Cache payoff checks whether the payoff was already evaluated and if yes then returns a cached value.

struct Add <: BinaryNode
    x::Payoff
    y::Payoff
end

Addition of payoffs.

struct Sub <: BinaryNode
    x::Payoff
    y::Payoff
end

Subtraction of payoffs.

struct Mul <: BinaryNode
    x::Payoff
    y::Payoff
end

Multiplication of payoffs.

struct Div <: BinaryNode
    x::Payoff
    y::Payoff
end

Division of payoffs.

struct Logical <: BinaryNode
    x::Payoff
    y::Payoff
    op::String
end

Logical operations

struct Exp <: UnaryNode
    x::Payoff
end

Function exp() applied to payoff.

struct Log <: UnaryNode
    x::Payoff
end

Function log() applied to payoff.

struct Max <: BinaryNode
    x::Payoff
    y::Payoff
end

Path-wise maximum

struct Min <: BinaryNode
    x::Payoff
    y::Payoff
end

Path-wise minimum

obs_time(p::Payoff)

A Payoff is typically observed at a particular time. In that sense, a Payoff is an $F_t$-measurable random variable.

The observation time represents the time, after which the payoff is known.

obs_time(p::Leaf)

Return the observation time for a Leaf object.

obs_time(p::UnaryNode)

Return the observation time of the linked payoff.

obs_time(p::BinaryNode)

Derive the observation time from linked payoffs.

obs_time(p::Fixed)

Observation time for Fixed payoffs is zero because they are deterministic.

obs_time(p::ScalarValue)

Observation time for ScalarValue payoffs is zero because they are deterministic.

obs_time(p::Pay)

Return decorating observation time.

obs_time(p::CompoundedRate)

Calculate observation time for CompoundedRate payoff.

obs_time(p::Optionlet)

Return Optionlet observation time.

obs_time(p::Swaption)

Return Swaption observation time.

obs_time(p::AmcPayoff)

Return the AMC payoff observation time

obs_time(p::VanillaAssetOption)

Return VanillaAssetOption observation time.

obs_times(p::Payoff)

A payoff is typically linked to other payoffs and forms a DAG. This function returns all observation times associated with a given payoff.

This functionality is required to determine relevant simulation grid points.

obs_times(p::Leaf)

Derive the set of observation times from the single observation time of the Leaf object.

obs_times(p::UnaryNode)

Return all observation times of the linked payoff.

obs_times(p::BinaryNode)

Derive all observation times from linked payoff.

obs_times(p::Pay)

Return all observation times of the linked payoff.

obs_times(p::CompoundedRate)

Calculate all observation times (i.e. event times) for CompoundedRate payoff.

obs_times(p::Optionlet)

Return all Optionlet observation times.

obs_times(p::Swaption)

Return all Swaption observation times.

obs_times(p::AmcPayoff)

Return observation times of all referenced payoffs.

obs_times(p::VanillaAssetOption)

Return all VanillaAssetOption observation times.

at(p::Payoff, path::AbstractPath)

Evaluate a Payoff at a given path, X(omega).

Depending on the functionality associated with the path, this function typically returns a vector of realisations.

This function is invoked when using call operator on a Payoff,

(p::Payoff)(path::AbstractPath) = at(p::Payoff, path::AbstractPath)

at(p::Numeraire, path::AbstractPath)

Derive the numeraire price at a given path.

at(p::BankAccount, path::AbstractPath)

Derive the bank account price at a given path.

at(p::ZeroBond, path::AbstractPath)

Derive the zero bond price at a given path.

at(p::Asset, path::AbstractPath)

Derive the asset price at a given path.

at(p::ForwardAsset, path::AbstractPath)

Derive the asset price at a given path.

at(p::Fixing, path::AbstractPath)

Derive the fixing value at a given path.

at(p::Fixed, path::AbstractPath)

Return the deterministic value broadcasted to the length of the path.

at(p::ScalarValue, path::AbstractPath)

Return the deterministic scalar value.

This aims at avoiding some unnecessary allocations.

at(p::AssetConvexityAdjustment, path::AbstractPath)

Derive the YoY payoff convexity adjustment at a given path.

at(p::ForwardIndex, path::AbstractPath)

Derive forward index value at a given path.

at(p::IndexConvexityAdjustment, path::AbstractPath)

Derive the YoY payoff convexity adjustment at a given path.

at(p::FutureIndex, path::AbstractPath)

Derive forward index value at a given path.

at(p::Pay, path::AbstractPath)

Derive payoff of the child payoff.

at(p::Cache, path::AbstractPath)

Derive payoff of the child payoff only if not yet calculated.

at(p::Exp, path::AbstractPath)

Evaluate exp-function.

at(p::Log, path::AbstractPath)

Evaluate log-function.

Addition.

Subtraction.

Multiplication.

Division.

Maximum

Minimum

Logical

at(p::LiborRate, path::AbstractPath)

Derive the forward Libor rate at a given path.

at(p::CompoundedRate, path::AbstractPath)

Derive the compounded backward looking rate at a given path.

at(p::Optionlet, path::AbstractPath)

Evaluate a Optionlet at a given path, X(omega).

at(p::Swaption, path::AbstractPath)

Evaluate a Swaption at a given path, X(omega).

at(links::AmcPayoffLinks, regr::AmcPayoffRegression, path::AbstractPath)

Calculate the common components of AMC payoffs for a given valuation path.

at(p::AmcMax, path::AbstractPath)

Evaluate an AmcMax payoff at a given path.

at(p::AmcMin, path::AbstractPath)

Evaluate an AmcMin payoff at a given path.

at(p::AmcOne, path::AbstractPath)

Evaluate an AmcOne payoff at a given path.

at(p::AmcSum, path::AbstractPath)

Evaluate an AmcSum payoff at a given path.

at(p::VanillaAssetOption, path::AbstractPath)

Evaluate a VanillaAssetOption at a given path, X(omega).

string(p::Numeraire)

Formatted (and shortened) output for Numeraire payoff.

string(p::BankAccount)

Formatted (and shortened) output for BankAccount payoff.

string(p::ZeroBond)

Formatted (and shortened) output for ZeroBond payoff.

string(p::Asset)

Formatted (and shortened) output for Asset payoff.

string(p::ForwardAsset)

Formatted (and shortened) output for ForwardAsset payoff.

string(p::Fixing)

Formatted (and shortened) output for Fixing payoff.

string(p::Fixed)

Formatted (and shortened) output for deterministic payoff.

string(p::ScalarValue)

Formatted (and shortened) output for deterministic payoff.

string(p::AssetConvexityAdjustment)

Formatted (and shortened) output for AssetConvexityAdjustment payoff.

string(p::ForwardIndex)

Formatted (and shortened) output for ForwardIndex payoff.

string(p::IndexConvexityAdjustment)

Formatted (and shortened) output for IndexConvexityAdjustment payoff.

string(p::FutureIndex)

Formatted (and shortened) output for FutureIndex payoff.

string(p::Pay)

Formatted (and shortened) output for Pay payoff.

string(p::Cache)

Formatted (and shortened) output for Cache payoff.

string(p::Exp)

Formatted output for Exp payoff.

string(p::Log)

Formatted output for Log payoff.

Formatted addition.

Formatted subtraction.

Formatted multiplication.

Formatted division.

Formatted maximum.

Formatted minimum.

Formatted logical.

string(p::LiborRate)

Formatted (and shortened) output for LiborRate payoff.

string(p::CompoundedRate)

Formatted (and shortened) output for CompoundedRate payoff.

string(p::Optionlet)

Formatted (and shortened) output for Optionlet payoff.

string(p::Swaption)

Formatted (and shortened) output for Swaption payoff.

string(links::AmcPayoffLinks)

Formatted (and shortened) output for AMC payoff links.

string(p::AmcMax)

Formatted (and shortened) output for AmcMax payoff.

string(p::AmcMin)

Formatted (and shortened) output for AmcMin payoff.

string(p::AmcOne)

Formatted (and shortened) output for AmcOne payoff.

string(p::AmcSum)

Formatted (and shortened) output for AmcSum payoff.

string(p::VanillaAssetOption)

Formatted (and shortened) output for VanillaAssetOption payoff.