Vasicek and CIR Models - Marginal Qeraker.marginalq.com/fixedIncome/vasicek_CIR.pdf · · 2015-03-23The speed of mean reversion measures the average time it takes for a process

Vasicek and CIR Models

Bjørn Eraker

Wisconsin School of Business

March 20, 2015

Bjørn Eraker Vasicek Model

The Vasicek Model

The Vasicek model (Vasicek 1978) is one of the earliestno-arbitrage interest rate models.It

is based upon the idea of mean reverting interest ratesgives an explicit formula for the (zero coupon) yield curvegives explicit formulaes for derivatives such as bondoptionscan be used to create an interest rate tree


The Vasicek Model




The Vasicek Model




The Vasicek Model




AR(1) Processes

An autoregressive process of order 1 (AR(1)) is a stochasticprocess of the form

Yt = a + bYt−1 + εt

where a and b are constants, εt is a random error term, typically

εt ∼ N(0, σ2)


Mean Reversion

If a process is mean reverting, it tends to revert to aconstant, long run meanThe speed of mean reversion measures the average time ittakes for a process to revert to its long run meanMean reversion is reasonable for interest rates - randomwalk makes no sense because it is economicallyunreasonable to think that interest rates can "wander of toinfinity" or become arbitrarily large.The mean reversion in the AR(1) process is measured byb.So what is the typical speed of mean reversion in interestrates? Weekly data gives 0.9961 for 3m T-bills. This isextremely high persistence and indicates that interest ratesare close to random walk...


Mean Reversion



Mean Reversion



Mean Reversion



b close to 1 means slow mean reversionb = 1 means that Yt is a Random Walk. There is NO meanreversion!b close to zero means that there is rapid mean reversion.


The Short Rate

Many term structure models, including Vasicek’s, assume thatthe fundamental source of uncertainty in the economy is theshort rate.

The short rate is the annualized rate of return on a very shortterm investment (lets assume daily).


The Short Rate

Many term structure models, including Vasicek’s, assume thatthe fundamental source of uncertainty in the economy is theshort rate.

The short rate is the annualized rate of return on a very shortterm investment (lets assume daily).


The Vasicek Model

The Vasicek model simply assumes that the short rate, rtfollows an AR(1),

rt = a + brt−1 + εt . (1)

That is, Vasicek just assumes a purely statistical model for theevolution of interest rates.


The Vasicek Model


rt = a + brt−1 + εt . (1)



The Vasicek Model


rt = a + brt−1 + εt . (1)



Alternative Representations

The model is sometimes written,

4rt = κ(θ − rt−1)4t + σ4zt (2)

where4rt = rt − rt−1 (3)

and 4zt is a normally distributed error term.Using the definition of 4rt , and set 4t = 1, we have

rt = κθ + (1− κ)rt−1 + σ4zt (4)

which is an AR(1) if a = κθ and b = 1− κ.


We can also set 4t to something different from 1. In this casewe assume that

4z ∼ N(0,4t)


Differential Notation

Tuckman writes

drt = κ(θ − rt)dt + σdwt

This is a stochastic differential equation where dw is Brownianmotion. You should interpret it to mean

4rt = κ(θ − rt−1)4t + σ4zt (5)

In other words, we interpret drt to mean 4rt = rt − rt−1.Therefore, any time you see a the differential notation inTuckman, translate by equating to the first difference.


Differential Notation

Tuckman writes

drt = κ(θ − rt)dt + σdwt

This is a stochastic differential equation where dw is Brownianmotion. You should interpret it to mean

4rt = κ(θ − rt−1)4t + σ4zt (5)

In other words, we interpret drt to mean 4rt = rt − rt−1.Therefore, any time you see a the differential notation inTuckman, translate by equating to the first difference.


Vasicek Zero Coupon Curve

Let B(t ,T ) denote the time t price of a zero coupon bond withmaturity date T . It is

B(t ,T ) = exp(−A(t ,T )rt + D(t ,T )) (6)

A(t ,T ) =1− e−κ(T−t)

κ(7)

D(t ,T ) =

(θ − σ2

2κ

)[A(t ,T )− (T − t)]− σ2A(t ,T )2

4κ(8)

In other words, the Vasicek model gives us an exact expressionfor the value of a zero coupon. We can compare this to othermodels for zero coupon bond prices, such as the cubic splinemodel and Nelson-Siegel.


Properties of the Vasicek term structure

Lets take some example parameter values to see how theVasicek term structure behaves. Reasonable values are

κ = (1− 0.9961)× 52 ≈ 0.2,

σ = 0.0025×√

52 ≈ 0.018,

(180 basis p/ year) and

θ = 0.05

(the average risk free rate).


Effects of mean reversion

The smaller the mean reversion in interest rates, the morewe expect interest rates to remain close to their currentlevels.If we expect interest rates to change very slowly, longdated bonds tend to move in parallel to short.Thus, higher (lower) speed of mean reversion (κ = 1− b)lead to a steeper (less steep) term structure because theterm structure depends on future expected short rates(illustrated on next slide)








Vasicek Bond Options

Consider an option on a zero coupon. The underlying bond hasmaturity T and the option matures at T0 (we assume T0 < Twhy?).The value of a call option with strike K is

C = B(t ,T )N(d1)− KB(t ,T0)N(d2) (9)

where d1 and d2 have expressions that mimic those in theBlack-Scholes model.



Cox-Ingersoll-Ross

One of the features of the Vasicek model is that interest ratechanges have constant volatility, σ, no matter what happens inthe economy.It can be useful to relax this assumption for two reasons

We have empirical evidence suggesting that interest ratechanges are more volatile when the level of interest ratesare highAs we will show, interest rates will remain positive..


The model is

drt = κ(θ − rt)dt + σ∗√

rtdWt

Thus, the CIR model has the same drift term as the Vasicekmodel, but the volatility is

Std(drt) = σ∗√

rt

This is an example of a conditional volatility: when the shortrate rt is high the volatility of interest rate changes is high andvise versa.The parameter σ∗ no longer represents the volatility of interestrate changes. It is just a parameter...


Choosing σ∗

Last lecture we argued that the volatility of short rates changesare some 180 basis points per year. In the CIR model, volatilityis time varying.It is reasonable therefore to choose σ∗ such that the averagevolatility under the CIR model is approximately equal to 180basis point (0.018).We can do this by solving

0.0182 = E(Var(drt)) = σ∗2E(rt) = (σ∗)20.05

or

σ∗ =

√0.0182

0.05≈ 0.08


Random Simulation

Suppose we draw the random shocks, et out of a hat fort = 1, ..,T . All the numbers in the hat are normally distributedwith mean zero and variance one.We can now use these numbers to simulate what wouldhappen to future interest rates. We use the equation

rt = rt−1 + κ(θ − rt−1)dt + σ√

dtet (10)

for some dt .In the last lecture I argued that κ = 0.2, θ = 0.05 and σ = 0.018represent reasonable values describing the year-to-yearbehavior of US interest rates.


Simulation in Excel

Get normally distributed random numbers usingNORMSINV(RAND())Use equation (??) to compute all future interest rates asfunctions of κ, θ, σ and the time step dt .

See thatWhen interest rates are high (low), the subsequent ratestend to decrease (increase) toward the long run averagerate, θ.When κ is high, rates mean revert more quicklyInterest rates can become negative in the Vasicek model


Figure : Simulated Vasicek and CIR interest rates. Shocks are thesame.


Notice that the simulated CIR path is positive while becomingnegative for the Vasicek model.This is not a coincidence: In theory, all CIR paths shouldremain positive....Our excel sheet for simulation may produce negative values stillbecause our simulations are approximations to the continuoustime model in CIR....


CIR term structure

Prices of zero coupons with maturity T are given by

P(T ) = A(T )e−B(T )r0 (11)

where

A(T ) =

[2he(κ+h)T/2

2h + (κ+ h)(ehT − 1)

] 2κθσ∗2

(12)

B(T ) =2(ehT − 1)

2h + (κ+ h)(ehT − 1)(13)

h =√κ2 + 2σ∗2 (14)


Documents

Vasicek and CIR Models - Marginal Qeraker.marginalq.com/fixedIncome/vasicek_CIR.pdf · · 2015-03-23The speed of mean reversion measures the average time it takes for a process