Modelled Qty


Modelled Qty refers to List of possible types that assign a concrete interpretation to the modelled quantity as a function of time x(t) that is used to represent the curve.
A typical interpretation of x(t) is the
discount factor as a function of the maturity time t
Another possible interpretation is the
zero rate as a function of the maturity time t
An important constraint is that the knowledge of the function x(t) is sufficient to imply the discount factor at any time.
Concretely:
Given a specified set of n instruments and their market quotes q₁,q₂,...,qₙ, the curve building algorithm first assigns a grid of n times t₁,t₂,...,tₙ, referred as pillars.
Usually, these times match the maturities of the supplied instruments, but other choices are also possible.
The goal of curve construction is to find the n values x(t₁),x(t₂),...,x(tₙ) assumed by the modelled quantity x(t) on these pillar times.
An array x(t₁),x(t₂),...,x(tₙ) is considered a "solution" if it implies quotes for the given instruments that are close enough to the known market quotes.
But how any given array x(t₁),x(t₂),...,x(tₙ) may imply the quotes of the given instruments?
It works as follows:
While the array x(t₁),x(t₂),...,x(tₙ) conveys the values assumed by List of possible types that assign a concrete interpretation to the modelled quantity as a function of time x(t) that is used to represent the curve.
A typical interpretation of x(t) is the
discount factor as a function of the maturity time t
Another possible interpretation is the
zero rate as a function of the maturity time t
An important constraint is that the knowledge of the function x(t) is sufficient to imply the discount factor at any time.
Concretely:
Given a specified set of n instruments and their market quotes q₁,q₂,...,qₙ, the curve building algorithm first assigns a grid of n times t₁,t₂,...,tₙ, referred as pillars.
Usually, these times match the maturities of the supplied instruments, but other choices are also possible.
The goal of curve construction is to find the n values x(t₁),x(t₂),...,x(tₙ) assumed by the modelled quantity x(t) on these pillar times.
An array x(t₁),x(t₂),...,x(tₙ) is considered a "solution" if it implies quotes for the given instruments that are close enough to the known market quotes.
But how any given array x(t₁),x(t₂),...,x(tₙ) may imply the quotes of the given instruments?
It works as follows:
only on the discrete times
t₁,t₂,...,tₙ
, it is possible to also know the value
x(t)
at any arbitrary time
t
by applying an assumed interpolation method on
x
For example, if we assume a linear interpolation method, the value x(t) can be easily inferred by the one or two time points in t₁,t₂,...,tₙ that surround or are closest to t
As mentioned above, knowledge of x(t) implies knowledge of all discount factors.
It is also well known that any linear financial instruments (bonds, swaps) can be priced if we know the discount factors at certain instrument-specific times t₁,t₂,...,tₖ
For any given time horizon, their number k is usually much bigger than the number n of the curve pillars.
The notable exception is the zero bond instrument that has k = 1, since its price depends on the discount factor wrt only one time, the bond's maturity.
Whatever the k for each instrument might be, the array x(t₁),x(t₂),...,x(tₙ) leads through interpolation to discount factors P(t₁),P(t₂),...,P(tₖ) for each instrument, which then suffice to calculate that instrument's price, which price is finally translated to its corresponding implied quote.

If we collect all instrument-specific times t₁,t₂,...,tₖ into one single array, the result will be a sequence of m times t₁,t₂,...,tₘ on which the discount factor will be determined through the curve building process.
In the exceptional case where m = n, these discount factors will not depend on the choice of the modelled quantity x, since the number of unknown discount factors being equal to the number of market quotes dictates a unique solution for the discount factors P(t₁),P(t₂),...,P(tₖ)

This would be the case where all instruments were zero bonds.
But even in that exceptional case, the discount factors implied by the curve on time points off the grid t₁,t₂,...,tₖ would depend on both the choice of the modelled quantity x and the choice of the interpolation method.
Available Modelled Qty types:
Discount
Fwd Rate
Zero Yield
Zero Yield Annual