# Forecasting Interest Rates — Mean Reverting Drift Term Structure Models

## How Mean Reverting Drift Term Structure Models Can Evolve Interest Rates

This article aims to introduce a number of mean-reverting short term interest rate models which can forecast and evolve interest rates. These models are known as term structure models. We need this knowledge before we can assess how machine learning algorithms can resolve the issues.

The article builds up on the knowledge that was acquired in the previous topics whereby I explained the basics of interest rate curves, outlined an overview of the key mathematical concepts and illustrated how basic interest rate models work.

It is crucial to understand how interest rates work as they impact our lives on daily basis.

One of the aims of my knowledge quest is to be able to forecast interest rates accurately by building a model that combines existing short term rate models along with the concepts we have learnt from Machine Learning advancements.

Although it is an extremely complex topic but what fun is there in simple concepts.

### Article Aim

I will start by explaining the key term: Mean Reverting. We always come across this term whenever we discuss term structure models.

Then I will explain how following term structure models work:

1. Vasicek Model

2. CIR

### Vasicek Model

Short rate is assumed to be stochastic/random/undertereminstic. Vasicek is a mean reverting short term interest rate model.

The fundamentals of the model are based on the assumption that the interest rates follow mean-reversion.

Larger the mean reversion, less the interest rates remain closer to their current levels.

Therefore, interest rates will drift towards their average mean faster over time. If the long run mean value is lower than the current short term rate then the drift adjustment will become negative. As a result, the short rate will end up being closer to the mean-reverting level.

This model is based on the assumptions of economic equilibrium factors. Mean reversion concept fails in economical stresses, high inflation and during crises. When the mean reversion parameter is large then the economic news is quickly incorporated into the security prices. Smaller mean reversion parameter implies that the impact will be in effect for longer.

If we know the long run value of the short term rate along with the current interest rate and mean reversion adjustment rate then we can calculate and evolve the interest rate using Vasicek model.

The model is based on following formula:

`Interest Rate at time t = Interest Rate At time t-1 + (A x [B — Interest Rate At time t-1] x [Size Of Timestep]) + (Volatility x Normally Distributed Error Term x Square Root Of Timestep)`
• A is the mean reversion adjustment rate
• B is the long run value of short term. It assumes risk neutrality

If A is a large value then the evolved interest rate will revert to the average mean faster. The parameters are calibrated from observed market prices. As a consequence, the term structure of volatility ends up declining. The long term volatility is overstated and subsequently the short term volatility is understated.

The model assumes that the volatility of interest rates is constant. The drift term is made up of the expected rate change and a risk premium. It is usually calculated in bps or % such as 0.2%.

The long run value of the short term rate is calculated as:

`Long Run Rate Of Interest + (Drift/Mean Reversion Adjustment Rate)`

Drift is replaced by:

Speed of reversion adjustment(long-run value of the short-term rate — Current interest rate level)

Vasicek model can help us estimate the expected interest rate in future. It calculates the rate as the weighted average between the current and the expected long term horizon value.

The mean-reversion rate is used to decay the weighting factor. Often, half-life factor is used.

Higher the mean reversion adjustment parameter, shorter the half-life factor

Error is assumed to be normally distributed. Formula is used to compute zero coupon yield curve and bond curve. Vasicek model does not produce parallel shifts from shocks as it is witnessed with no drift models.

Vasicek Model Issues

There are a number of limitations of the model:

1. Produces declining term structure of volatility
2. Short term volatility is understated and long term volatility is overstated.
3. Short term rate is impacted more by the long term rates.

Let’s consider that our portfolio contains a number of derivatives such as IR Caps and floors. The pay-off of these multi period products is based on the short term rate of next period. We can then use term structure models that incorporate time dependent volatility and drift.

### Cox Ingersoll Ross (CIR) And Lognormal Models

CIR Interest rate model is an improvement of Vasicek model. It has conditional volatility. CIR model assumes that the term structure increases with the rates and does not become negative. Therefore CIR avoids negative interest rates although it can lead to mis-pricing of the securities.

If we consider this formula:

dr=(α+βr)dt+σ(r^γ)dZ
• dr is the change of interest rates
• dt is the time interval
• σ is the volatility (variance) of the rate changes
• dZ is a random variable. This variable follows a normal distribution, has a mean of 0 and a standard deviation of square root of time interval.
• r is the risk-less interest rate

Change it slightly so that we get:

dr=(Time Dependent Drift)dt+σ(Time Dependent Volatility)dZ

Time dependent volatility could be e^-alpha x time

CIR model considers that the volatility and short term rates are positively correlated with each other. It assumes that the volatility increases proportional to Sqrt(r).

The model is based on following formula:

`Interest Rate at time t = Interest Rate At time t-1 + (A x [B — Interest Rate At time t-1] x [Size Of Timestep]) + (Volatility x Square Root (rate) Normally Distributed Error Term x Square Root Of Timestep)`
• A is the mean reversion adjustment rate
• B is the long run value of short term. It assumes risk neutrality

Drift: Time dependent

Volatility: Time dependent

Volatility increases proportional to the square root of the rate.

### Black-Karasinski model

It is also known as Model 4 Log-normal model. The model allows for time varying volatility and mean reversion. Natural log of short term rate is taken and it follows a normal distribution. As a result, the short term rates are not going to be negative.

The log normal model with mean reversion is:

`Natural log of short term rate = Mean Reversion Adjustment Rate At Time T x (Natural Log of Long Run Value — Natural Log of current short rate) x change in time + Volatility At Time T x Random Variable With Normal Distribution`

It transforms Vasicek model to a time varying model.

### Summary

This article summarised the three short term rate models: Vasicek, CIR and Black-Karasinski models.

My next article on the topic will discuss and outline SABR and Hull White models.

I want you to team up with me here and build the knowledge together with me. Therefore I encourage you to email me your suggestions and follow me as it will be a joint exercise amongst us.

I hope it helps.

Please let me know if you have any feedback.