## Cox-Ingersoll-Ross. CIR model. Parameter calibration & simulation

*in*Asset Liability Management

# CIR Model – Parameter Calibration and short rate simulation

The Cox-Ingersoll-Ross (CIR model) interest rate model is a one-factor, equilibrium interest rate model. One factor in that it models the short – term interest rate and equilibrium in that it uses assumptions about various economic variables (e.g. mean reversion) to derive a process for determining this rate.

We have already provided a detailed theoretical overview in the following posts of how CIR parameters may be calibrated and on how these resulting parameters may be used to simulate short-term interest rates and model longer term rates:

- Using CIR (Cox Ingersoll Ross) Model: Introduction
- Using CIR (Cox Ingersoll Ross) Model: Estimating Parameters & Calibrating the CIR Model
- Using CIR (Cox Ingersoll Ross) Model: Simulating the term structure of interest rates

In the following post we will consider a practical example. Specifically we will:

- Use the Daily Treasury Yield Curve Rates for the period 2-Jan-2009 to 27-Jul-2010 for our calibration exercise
- Calculate spreads between the short term rate as of 27-Jul-2010 with longer term rates for this date. These will be used later to model the longer term interest rates.
- Use the Simple Discretisation process to calibrate the parameters of the CIR model
- Use the derived parameters to simulate the short term rates for the next 360 month period
- Use the spreads derived in Step 2 along with the projected short term rates in Step 4 to determine the long-term interest rates

**STEP 1: Obtain necessary historical interest rate/ yield curve data for CIR model**

In order to calibrate the CIR Model to our data set we first need to obtain the necessary data set. For our illustration we have used the daily US Treasury Yield Curve Rates for the period 2-Jan-2009 to 27-Jul-2010 available at www.ustreas.gov.

**STEP 2: Calculate Spreads for CIR model**

Later on in this calculation we will be simulating future short term interest rates using the calibrated CIR interest rate model. As an off-shoot to that simulation exercise we will also be modeling longer term rates. These longer term rates will be a function of the projected short term rates and historical spreads. For simplicity we will be using spreads for a single day, i.e. 27-Jul-2010, but the user may choose to base his/her calculations on average spreads over the selected period instead.

For 27-Jul-2010 the interest rate term structure for US treasury rates was as follows:

Tenor | 27-Jul-10 (%) |

1 mo | 0.16 |

3 mo | 0.15 |

6 mo | 0.21 |

1 yr | 0.30 |

2 yr | 0.65 |

3 yr | 1.02 |

5 yr | 1.82 |

7 yr | 2.51 |

10 yr | 3.08 |

20 yr | 3.88 |

30 yr | 4.08 |

We first determine the interpolated rates for each month from 1 month to 360 months (i.e. 30 years). For instance the 2-month interest rate will be interpolated using the 1-month and 3-month rates.

2-month rate = [(2-1)*0.15%+(3-2)*0.16%]/[3-1] = 0.31%/2 =0.155%

Next we determine the spread between this interpolated rate and the short rate (i.e. the 1-month rate) = 2-month rate – short rate =0.155%-0.16%=-0.005%.

A snapshot of the resulting rates and spreads is given below:

**STEP 3: Calibration of CIR model parameters**

In the theoretical overview we presented the simple discretisation and covariance equivalent discretisation processes for calibration of parameters for the CIR model using a historical data set of interest rates.

For both processes a centered CIR model is used to determine the projected short rates:

where κ is the continuous drift and σ is the continuous volatility.

For the centered CIR model we need to determine a transformed short rate by subtracting the long run average (γ = AVERAGE (array of rates)) of the short rate from each rate observation (r_{t}*), i.e. the transformed short rate r_{t} is equal to r_{t}*- γ.

In our illustration we have used the simple discretisation process to determine the parameters of the model. Under this process the residual term is calculated at each data point as:

**Phi** , the discrete drift term, is first taken as a arbitrary number. The calibrated value, or least squares estimate for this term, will be determined after the sum of the residual terms (RSS) is minimized user the Solver function.

The RSS as mentioned above is the sum of the residual terms:

We next set up the solver function:

Go to the Data tab on the EXCEL worksheet and select Solver. Then

- Set Target Cell = the cell where residual terms are being summed, i.e. RSS (cell B5 in our illustration)
- Equal to: Min (because we are calibrating parameters by minimizing the sum of the residual terms)
- By Changing Cells = the cell for the value of the discrete drift term, i.e. Φ (cell B2 in our illustration)

Run the Solver function by clicking on “Solve” to find an optimal solution:

The next stage of the calibration process is to use the results obtained after running Solver to calculate the:

- Discrete volatility parameter, σ
_{a }= √(RSS/(N-1)), where N is the number of residual terms

- Continuous drift, κ = -ln Φ = ln(1/ Φ)

- And continuous volatility, σ = √[(2κσ
_{a}^{2})/(1-e^{-2κ})]

**STEP 4: Simulate short rates**

We use the calibrated and derived parameters together with the centered CIR model to simulate the short rate.

Recall the Centered CIR model is given as:

This may be restated for the actual short rate r_{t}* as:

**Δr _{t}* = κ(γ-r_{t-1}*)dt + σ√(r_{t-1}*)dz_{t}**

Given that r_{0}* = short rate as at 27-Jul-2010 = 0.16%

dt =1/360 (i.e. a daily time step assuming a 360 day year). Read the following post “CIR Interest Rate Model – Appropriate Time Step” to determine the correct time step to use in the interest rate model.

dz = NORMSINV(RAND())*√dt (normally scaled random number times the square root of the time step)

For ∆r_{1}* = κ(γ-r_{0}*)dt + σ√(r_{0}*)dz_{1 }=0.0019%

And Projected r_{1}* = r_{0}* + ∆r_{1}*= 0.16%+0.0019% =0.1619%

For r_{17}* = r_{16}* + ∆r_{17}* = 0.1569% + 0.0003% =0.1573%.

*Note: Clicking F9 will cause the series of simulated rates to change.*

Below is a depiction of one simulation run of the projected short rates over a 360 day period:

**STEP 5: Determining long term rates **

We had briefly talked about how to model longer term rates once we have projected the short rates in the following post:

In particular we had mentioned two specific ways of doing this by either assuming that:

1) Long term rates change by the same proportion as short term rates, or

2) Current spreads between the short rate and longer term rate are representative of future spreads

In the first case we calculate the projected long term rate at time step t as:

Long term rate_{t} =r_{t}/r_{0}*long term rate_{0}

e.g. 2-month rate = 0.18%/0.16% * 0.155% = 0.17438%

The following is a representation of the term structure of interest rates over the next five time steps using this proportional method to determine long term rates:

“27-Jul-2010” is the term structure today, “1” is the simulated term structure one month later, “2” is the simulated term structure two months down the line, and so on…

In the second instance we calculate the projected long term rate at time step t as:

Long term rate_{t} =r_{t} + (long term rate_{0}-r_{0})

e.g. 2-month rate = 0.18% + (-0.005%) =0.175%

The graph for the simulated term structures using a constant spread approach is given below:

Note that in both cases the long term rates are perfectly correlated with the short rates.