## Modelled Qty

**refers to List of possible types that assign a concrete interpretation to the**

*Modelled Qty***as a function of time**

*modelled quantity***that is used to represent the curve.**

*x(t)*A typical interpretation of

**is the discount factor as a function of the maturity time**

*x(t)*

*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

**is sufficient to imply the discount factor at any time.**

*x(t)*Concretely:

Given a specified set of

**instruments and their market quotes**

*n***, the curve building algorithm first assigns a grid of**

*q₁,q₂,...,qₙ***times**

*n***, referred as**

*t₁,t₂,...,tₙ***.**

*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

**values**

*n***assumed by the modelled quantity**

*x(t₁),x(t₂),...,x(tₙ)***on these pillar times.**

*x(t)*An array

**is considered a "solution" if it implies quotes for the given instruments that are close enough to the known market quotes.**

*x(t₁),x(t₂),...,x(tₙ)*But how any given array

**may imply the quotes of the given instruments?**

*x(t₁),x(t₂),...,x(tₙ)*It works as follows:

While the array

**conveys the values assumed by List of possible types that assign a concrete interpretation to the**

*x(t₁),x(t₂),...,x(tₙ)***as a function of time**

*modelled quantity***that is used to represent the curve.**

*x(t)*A typical interpretation of

**is the discount factor as a function of the maturity time**

*x(t)*

*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

**is sufficient to imply the discount factor at any time.**

*x(t)*Concretely:

Given a specified set of

**instruments and their market quotes**

*n***, the curve building algorithm first assigns a grid of**

*q₁,q₂,...,qₙ***times**

*n***, referred as**

*t₁,t₂,...,tₙ***.**

*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

**values**

*n***assumed by the modelled quantity**

*x(t₁),x(t₂),...,x(tₙ)***on these pillar times.**

*x(t)*An array

**is considered a "solution" if it implies quotes for the given instruments that are close enough to the known market quotes.**

*x(t₁),x(t₂),...,x(tₙ)*But how any given array

**may imply the quotes of the given instruments?**

*x(t₁),x(t₂),...,x(tₙ)*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

**can be easily inferred by the one or two time points in**

*x(t)***that surround or are closest to**

*t₁,t₂,...,tₙ*

*t*As mentioned above, knowledge of

**implies knowledge of all discount factors.**

*x(t)*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

**is usually much bigger than the number**

*k***of the curve pillars.**

*n*The notable exception is the zero bond instrument that has

**, since its price depends on the discount factor wrt only one time, the bond's maturity.**

*k = 1*Whatever the

**for each instrument might be, the array**

*k***leads through interpolation to discount factors**

*x(t₁),x(t₂),...,x(tₙ)***for each instrument, which then suffice to calculate that instrument's price, which price is finally translated to its corresponding implied quote.**

*P(t₁),P(t₂),...,P(tₖ)*If we collect all instrument-specific times

**into one single array, the result will be a sequence of**

*t₁,t₂,...,tₖ***times**

*m***on which the discount factor will be determined through the curve building process.**

*t₁,t₂,...,tₘ*In the exceptional case where

**, these discount factors will not depend on the choice of the**

*m = n*

*modelled quantity***, since the number of unknown discount factors being equal to the number of market quotes dictates a unique solution for the discount factors**

*x*

*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

**would depend on both the choice of the**

*t₁,t₂,...,tₖ*

*modelled quantity***and the choice of the interpolation method.**

*x*Available

**types:**

*Modelled Qty*Discount

Fwd Rate

Zero Yield

Zero Yield Annual