## Discount and zero rates

The discount rate is the price at time of a unit cashflow paid at . The zero rate at time for maturity is the continuously compounded interest rate between and where:

Let denote the -tenor zero rate

## Spread Rates

A *spread rate* is an interest rate price factor that is defined relative to a *parent rate*. A
*base rate* is an interest rate price factor without a parent. The parent of a spread rate can be
another spread rate or a base rate.

A spread rate's ID has the form , where is the spread name and is the ID of the parent rate. The discount rate of is given by

where is the discount rate of the parent and is the spread. Examples of spreads are ZAR-SWAP.ZAR-USD-BASIS (ZAR-SWAP is the parent and ZAR-USD-BASIS is the spread).

## HullWhite1FactorInterestRateModel

The instantaneous spot rate (or short rate) which governs the evolution of the yield curve is modeled as:

Where:

- is a deterministic volatility curve
- is a constant mean reversion speed
- is a deterministic curve derived from the vol, mean reversion and initial discount factors
- is the quanto FX volatility and is the quanto FX correlation
- is the risk neutral Wiener process related to the real-world Wiener Process by where is the market price of risk (assumed to be constant)

Final form of the model is:

Where:

The simulation of the random increment (where represents the simulation grid) is normal with zero mean and variance

## HullWhite2FactorImpliedInterestRateModel

This is a generalization of the 1 factor Hull White model. There are 2 correlated risk factors where the factor has a volatility curve , constant reversion speed and market price of risk .

Final form of the model is:

Where:

- , , and are correlated Weiner Processes with correlation ( if else 1)

If the rate and base currencies match, and . Otherwise, is the volatility of the rate currency (in base currency) and is the correlation between the FX rate and the factor. The increment (where corresponds to the simulation grid) is gaussian with zero mean and covariance Matrix .

The cholesky decomposition of is The increment is simulated using where is a 2D vector of independent normals at time step .

## PCAInterestRateModel

The parameters of the model are: - a volatility curve for each tenor of the zero curve - a mean reversion parameter - eigenvalues and corresponding eigenvectors - optionally a historical yield curve for the long run mean of

The stochastic process for the rate at each tenor on the interest rate curve is specified as:

with a standard Ornstein-Uhlenbeck process and a Brownian motion. It can be shown that:

Currently, only the covarience matrix is used to define the eigenvectors with corresponding weight curves and normalized weight curve Final form of the model is

Where:

- is the zero rate with a tenor at time ( denotes the current rate at tenor )
- is the mean reversion level of zero rates
- is the OU process associated with Principle component

To simulate the mean rate (note that ), there are 2 choices:

**Drift To Forward** where the mean rate is the inital forward rate from to so that
**Drift To Blend** is a weighted average function of the current rate and a mean reversion level