04/10/2023 Stuart J. McCall 1

Models of the Short RateVasicek Model

Cox, Ingersoll, Ross (CIR) ModelHull-White Model

Stuart J. McCallAdj. Prof. Finance

European School of Finance

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Flow of Presentation

• Overview of Concept• Mathematical Overview

– Non-tech Explanation of Theory• Instantaneous Short Rate• Equilibrium Models

– Vasicek Model– CIR Model (extension of Vasicek Model)

• No-arbitrage Models– Ho-Lee Model– Hull-White Model (extension of Ho-Lee Model)

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Overview of Concept

• Why do we look to solve for dr?– To get the best approximation of what r will be

in the future– Closer we can solve for dr at today better

approximation, and thus better pricing scenarios

– Solving for r is impossible• Time is continuously evolving• Newtonian Iteration would never allow it

– Thus, what is the best guess

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Non-tech Mathematical Overview

• All 3 models are based on a stochastic process, also known as Geometric Brownian Motion (GBM) with Mean Reversion (Drift)

• Can be continuous time or discrete time– This presentation is with continuous time– To implement with Excel, must use Discrete

Time• Advanced programs like MatLab can handle

continuous

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Pure GBM – following the Wiener

dtdz Random Variable, drawn from normal probability distribution

Infinitesimally small change in random variable.

• The random variable drawn follows the Central Limit Theorem, that an increasingly infinite number of events will follow a normal probability distribution – with a standard deviation σ of 1, and mean of 0

• Wiener process changes mean @ 0 per year and variance @ 1.0 per year• Thus, <1 year (e.g. 0.000001) will also be taken to the root…it’s a small

change• This variable can be drawn in Microsoft Excel by using =Rand()

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GBM w/ Mean Reversion

tt dzrbadv )(

Random Variable, drawn from normal probability distribution times σ

Mean Reversion Factor

Where a is the mean reversion rate, b is the equilibrium point of mean reversion, r is the short interest rate

When r<b, then a(b-r) will force the following rate downward toward equilibrium

When r>b, then a(b-r) will force the following rate upward toward the equilibrium

A high level of a will keep the range low, by increasing the speed that r reverts to b.

A low level of a will, conversely, imply longer periods away from the equilibrium b.

The σ determines the length of the range of possibilities, thus higher volatility will dictate higher ceilings and lower floors.

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GBM vs. GBM w/ Mean Reversion

GBM• Produces a random walk used to model prices under

assumption that price changes are independent of one another

• Historical price path leading to current is irrelevant for future– Markov process

GBM w/ Mean Reversion• A modification of random walk, where price changes are

not completely independent of one another but rather are related

• Prices will drift toward an equilibrium point, which is determined by historical values

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GBM vs. GBM w/ Mean Reversion

GBM• Imagine a drunk leaving a bar at night.• The future steps of the drunk are random, in both the

length and direction of strides.• Ultimately, no one knows where the drunk may wind up.

GBM w/ Mean Reversion• Same drunk, but walking with a guide dog that is on a

leash.• Still don’t know exactly what he will do, but his distance

from dog is determined by the leash, and the dog will eventually lead him on a familiar path home (hopefully).

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Instantaneous Short Rate

• The smallest number you can calculate– Not small enough

• The closer you drive the basis of your model to current t, the closer you get to perfection

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Short Rate Equilibrium Models

Model Formula

Vasicek

Cox, Ingersoll, and Ross

Where dr is the infinitesimally small change in r, a is the mean reversion rate, b is the equilibrium point of mean reversion, r is the short rate, dt is the infinitesimally small change in time t, σ is the standard deviation, dz is the stochastic random variable from a normal distribution, and the root of r increases efficiency to the model by reducing fluctuations in σ.

dzdtrbadr )(

dzrdtrbadr )(

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What is the Short-Rate?

• The r at an infinitesimally small period of time• The closer one gets to t0 the closer the

approximation is to realized r• Can never reach zero because two different

theories– Newtonian Iteration continues to approximate dividing

by 2, thus close but no cigar– Calculation is continuous, as is time, thus must be

made at the speed that the r can change, which is instantaneous• Maybe Windows 3007 will have that figured out

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Inputs to the Mean Reversion of the Model

• Volatility (σ) is the expected variability of r over time– High volatility will lead to larger range of

values of r. – Spikes in r will lead to higher σ, thus leading

to higher ranges• Is this fair?

dzrbadv )(

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Inputs to the Mean Reversion of the Model

Mean reversion level (b,= long-run equilibrium r)

• Historical spot price series

• Current market prices– Assumes that current

market prices represent time mean reversion levels

Mean reversion rate (a,= the speed that r reverts)

• Linear regression to relate historical price changes to historical prices

• Calibrating instrument – which derives a “goodness-of-fit”

dzrbadr )(

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Disadvantage of Equilibrium

• They do not automatically fit today’s term structure of interest rates

• Must choose parameters to provide an approximate fit to many term structures

• Sometimes can find a good fit, sometimes not– Traders typically find this unacceptable– B/c they have low confidence in price when there is

low confidence in the r used in discounting– “A 1% error in the price of the bond may lead to a

25% error in the price of its option.” -Hull

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Equilibrium Models

Vasicek Model• The basic model for predicting the dr, • The term structure of interest rates is

the output• Drift of the short rate (coefficient dt)

not usually a function of time• Shape of zero curve can change with

passage of time• Shortcomings include:

– Spikes in r will skew the σ, thus giving false range of future prices

– High levels of mean reversion can push r<0…which doesn’t happen often in practice

CIR Model(extension of Vasicek) Extends the model

by incorporating the square of r,– Limits the effects of spikes in σ…

smoothing effect– Keeps r from drowning in negative –

which is, again, impossible in practice• The term structure of interest rates is

the output• Drift of the short rate (coefficient dt)

not usually a function of time• Shape of zero curve can change with

passage of time• Must integrate to track back the

function to t0

dzdtrbadr )( dzrdtrbadr )(

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Equilibrium Models

Mean Reversion Point b

High a

Low a

r<b ; a(b-r) revert positive

r

t

r>b ; a(b-r) revert negative

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Equilibrium Models

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Disadvantage of Equilibrium

• They do not automatically fit today’s term structure of interest rates

• Must choose parameters to provide an approximate fit to many term structures

• Sometimes can find a good fit, sometimes not– Traders typically find this unacceptable– B/c they have low confidence in price when there is

low confidence in the r used in discounting– “A 1% error in the price of the bond may lead to a

25% error in the price of its option.” -Hull

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Quick Question

• If the current short rate is 4% and standard deviation is 1% per annum, what happens to the standard deviation when the short rate increases to 8% in the – Vasicek Model– CIR Model

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Quick Answer

• Vasicek: Standard Deviation stays at 1%

• CIR: The Standard Deviation of the short rate is proportional to the square root of the short rate.– Thus, when the short rate increases from 4%

to 8% the standard deviation of the short rate increases from 1% to 1.414%

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What is a No-Arbitrage Model?

• Designed to be exactly consistent with today’s term structure of interest rates

• The major difference between No-Arbitrage and Equilibrium Models is:– Equilibrium: today’s term structure of interest

rates = output– No-arbitrage: today’s term structure of interest

rates = input• How?

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Short Rate No-Arbitrage Models

Model Formula

Ho-Lee

Hull-White

Time-dependent element

•Where the element θ(t) is the time-dependent function, which is determined by requiring that r at time t0 be matched to the forward price Ft(0,t) – which is a partial derivative with respect to subset t.•The average direction that the short rate will be moving in the future is approximately equal to the slope of the instantaneous forward curve.

dzdttdr )(

ttFt t2),0()(

dzdtra

tadr

)(

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No-arbitrage Models

Ho-Lee Model• The today’s term structure of interest rates

is the input. • Drift of the short rate (coefficient dt) a

function of time• B/c the shape of the initial zero curve

governs the average path taken by the short rate in the future

• The direction is given by the partial derivative θ(t), which sets parameters for quantifying the possibilities directions based on a normal distribution

• Because the volatility (σ) is the root of the variance (σ2) it can be ±.

– The root of 9 is either +3 or -3

Hull-White (extension of Vasicek) Model• Extends the Ho-Lee model by incorporating

the mean reversion at a• a and σ are constants throughout model• AKA, Vasicek model with a time-dependent

reversion level, θ(t)• At time t the short rate reverts to θ(t)/a at

the rate of a– makes sense doesn’t it?

• Must integrate to track back the function to t0

dzdttdr )( dzdtra

tadr

)(

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Equilibrium Modelsθ(t) drives the sloper

tt1 t2 t3t0

Normally Distributed

Initial Forward Curve

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Equilibrium Modelsθ(t) drives the sloper

tt1 t2 t3t0

Normally Distributed

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Where is our drunk now?

Equilibrium Model• Tells us which bar the drunk would likely start from (the

output) and how the dog is likely to take him home• CIR tells us that random shocks (like falling down) will

not keep him from making it home (back to mean reversion)

• All thanks to the dog – named Drift

No-arbitrage Model• Here we use the bar assumed as his starting point,

based on where he has been drinking in the past at the time when he leaves (r at time t)

• Once we know which bar he starts from, we know which way he is likely to go, on average and guided by the dog

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Questions

If your brain is not fried; fire up the question engine!

?

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Thanks…If you are still awake!

• Danke• Merci• Grazzi• Gracias• Xsi Xsi