Interest Rate Modelling: Autumn 2 0 0 9
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A U T U M N 2 0 0 9
INTEREST RATE MODELLING
MSc
COMPUTATIONAL FINANCE
Numerical Methods in Finance (Implementing Market Models)
©Finbarr Murphy 2007
Lecture Objectives Interest Rate Modelling Traditional Interest Rate Models Vasicek 77
MSc
COMPUTATIONAL FINANCE
CIR 95
©Finbarr Murphy 2007
Agenda Page
Interest Rate Modelling
1
2 2
Traditional Term Structure Models
10
MSc
COMPUTATIONAL FINANCE
3
3
©Finbarr Murphy 2007
Interest Rate Modelling Definition: Interest Rate The percentage of an amount of money that's paid
for its use over a specified time period Very few market instruments do not use interest rates
in their effective valuation
MSc
COMPUTATIONAL FINANCE
Definition: The Term Structure of Interest Rates The term structure of interest rates is a graphic
representation of how interest rates vary with maturity; it shows the relationship between the yield from a financial instrument and its maturity. 4
©Finbarr Murphy 2007
Interest Rate Modelling Yields across different outstanding maturities can be
plotted to create a yield curve The term structure of interest rates is better known
as the yield curve.
MSc
COMPUTATIONAL FINANCE
Yield curves come in different shapes but they always
refer to instruments of a homogeneous (the same) nature. As interest rates change - or as expectations of future
interest rates change - investors will typically switch between maturities to try to achieve capital gains (or at least to avoid capital losses). 5
©Finbarr Murphy 2007
Interest Rate Modelling In this lecture, we are primarily concerned with how
one might model an interest rate/term structure In the same way that the modelling of stock price
MSc
COMPUTATIONAL FINANCE
behaviour gave us models to calculate option prices, the ability to model the term structure of interest rates allows us to model any number of fixed income derivatives There are two basic approaches: See slide:
6
©Finbarr Murphy 2007
Interest Rate Modelling For the moment, we will look at modelling interest
rates using traditional term structure models We begin with a look at a pure discount bond
MSc
COMPUTATIONAL FINANCE
This is denoted by s ˆ P( t , s ) = Et exp − ∫ r (τ ) dτ t
Where r(τ) is the short rate path between t and s And Êt is the risk neutral expectation at time = t
7
©Finbarr Murphy 2007
Interest Rate Modelling Consider a constant rate of 6% (the green line) The PV discount bond, in this case, can be calculated
MSc
COMPUTATIONAL FINANCE
as exp(-r(s-t))
8
©Finbarr Murphy 2007
Interest Rate Modelling But the actual value will be the expected short path To see this, consider the integral of the blue line,
MSc
COMPUTATIONAL FINANCE
discretised into 10 Δt steps
Δt
9
©Finbarr Murphy 2007
Interest Rate Modelling Early term structure models supposed that the entire
term structure was driven by one source of uncertainty So called, 1-factor models This single source of uncertainty is the short rate
MSc
COMPUTATIONAL FINANCE
We start with the Vasicek (1977) model
10
©Finbarr Murphy 2007
Agenda Page
Interest Rate Modelling
1
2 2
Traditional Term Structure Models
9
MSc
COMPUTATIONAL FINANCE
3
11
©Finbarr Murphy 2007
Traditional Term Structure Models The Vasicek (1977) model The SDE representing the uncertainty in the short
rate is give by the equation
dr = α ( r − r ) dt + σdz
Where
MSc
COMPUTATIONAL FINANCE
r(t) is the short rate at time t dz is a Weiner increment r is the long term mean rate of r The volatility of the short rate is assumed to be a constant σ α is the rate at which r reverts to r
12
©Finbarr Murphy 2007
Traditional Term Structure Models The short rate is assumed to follow the Ornstein-
Uhlenbeck process r, α and σ can be calculated using regression
techniques Under Vasicek, the discount rate is given by
MSc
COMPUTATIONAL FINANCE
P(t , s ) = A(t , s )e
− rB ( t , s )
The equivalent continuously compounded yield is
given by
ln A(t , s ) B(t , s ) R(t , s ) = − + r s −t s −t 13
©Finbarr Murphy 2007
Traditional Term Structure Models Where
(
1 B(t , s ) = 1 − e −α ( s −t ) α
)
And 2 R∞ σ ln A( t , s ) = 1 − e −α ( s −t ) − ( s − t ) R∞ − 3 1 − e −α ( s −t ) α 4α
MSc
COMPUTATIONAL FINANCE
(
and
)
(
1σ R∞ = lim R( t ,τ ) = r − τ →∞ 2 α2
2
14
)
2
©Finbarr Murphy 2007
Traditional Term Structure Models These functions can be easily coded in MatLab Try it!
The spot rate volatility structure is determined by
two parameters σ and α
MSc
COMPUTATIONAL FINANCE
σ −α ( s −t ) ( ) ( ) σ R t, s = 1− e α(s − t)
15
©Finbarr Murphy 2007
Traditional Term Structure Models
MSc
COMPUTATIONAL FINANCE
Here is the Vasicek code for the discount bond
16
©Finbarr Murphy 2007
Traditional Term Structure Models Cox-Ingersoll-Ross The Vasicek model has two major short falls Interest rates can be negative in the model The negative correlation between interest rates and interest
rate sensitivity is ignored
MSc
COMPUTATIONAL FINANCE
CIR (1995) introduce a model where the volatility of
the short rate increases with the square root of the level of the rate This precludes negative rates Displays more variability in high rate environments
and visa versa 17
©Finbarr Murphy 2007
Traditional Term Structure Models The CIR SDE is given by
dr = α ( r − r ) dt + σ r dz
As with Vasicek 77, the discount rates and
MSc
COMPUTATIONAL FINANCE
continuously compounded yields are given by
P(t , s ) = A(t , s )e
− rB ( t , s )
ln A(t , s ) B(t , s ) R(t , s ) = − + r s −t s −t 18
©Finbarr Murphy 2007
Traditional Term Structure Models With
MSc
COMPUTATIONAL FINANCE
A( t , s ) = φ2 B( t , s ) = φ2
( (
φ3
φ1 ( s −t ) e − 1 + φ1 φ1 ( s −t ) e −1 φ1 ( s −t ) e − 1 + φ1
φ1e
φ2 ( s −t )
)
)
And where
φ1 = α + 2σ 2
2αr φ3 = 2 σ
2
α + φ1 φ2 = 2 19
©Finbarr Murphy 2007
Traditional Term Structure Models
MSc
COMPUTATIONAL FINANCE
Here is the CIR Code
20
©Finbarr Murphy 2007
Traditional Term Structure Models The CIR volatility term structure of spot rates is given
by
MSc
COMPUTATIONAL FINANCE
σ r σ R (t , s ) = B (t , s ) s −t
21
©Finbarr Murphy 2007
Traditional Term Structure Models Here is a graph of 100 random movement in the short
MSc
COMPUTATIONAL FINANCE
rate. Note the convergence to the mean
22
©Finbarr Murphy 2007
Recommended Texts Required/Recommended Clewlow, L. and Strickland, C. (1996) Implementing derivative
models, 1st ed., John Wiley and Sons Ltd. — Chapter 6, 7
Additional/Useful Hull, J. (2005) Options, futures and other derivatives, 6th ed.,
Prentice Hall
MSc
COMPUTATIONAL FINANCE
— Chapters 4
23
INTEREST RATE MODELLING
MSc
COMPUTATIONAL FINANCE
Numerical Methods in Finance (Implementing Market Models)
©Finbarr Murphy 2007
Lecture Objectives Interest Rate Modelling Traditional Interest Rate Models Vasicek 77
MSc
COMPUTATIONAL FINANCE
CIR 95
©Finbarr Murphy 2007
Agenda Page
Interest Rate Modelling
1
2 2
Traditional Term Structure Models
10
MSc
COMPUTATIONAL FINANCE
3
3
©Finbarr Murphy 2007
Interest Rate Modelling Definition: Interest Rate The percentage of an amount of money that's paid
for its use over a specified time period Very few market instruments do not use interest rates
in their effective valuation
MSc
COMPUTATIONAL FINANCE
Definition: The Term Structure of Interest Rates The term structure of interest rates is a graphic
representation of how interest rates vary with maturity; it shows the relationship between the yield from a financial instrument and its maturity. 4
©Finbarr Murphy 2007
Interest Rate Modelling Yields across different outstanding maturities can be
plotted to create a yield curve The term structure of interest rates is better known
as the yield curve.
MSc
COMPUTATIONAL FINANCE
Yield curves come in different shapes but they always
refer to instruments of a homogeneous (the same) nature. As interest rates change - or as expectations of future
interest rates change - investors will typically switch between maturities to try to achieve capital gains (or at least to avoid capital losses). 5
©Finbarr Murphy 2007
Interest Rate Modelling In this lecture, we are primarily concerned with how
one might model an interest rate/term structure In the same way that the modelling of stock price
MSc
COMPUTATIONAL FINANCE
behaviour gave us models to calculate option prices, the ability to model the term structure of interest rates allows us to model any number of fixed income derivatives There are two basic approaches: See slide:
6
©Finbarr Murphy 2007
Interest Rate Modelling For the moment, we will look at modelling interest
rates using traditional term structure models We begin with a look at a pure discount bond
MSc
COMPUTATIONAL FINANCE
This is denoted by s ˆ P( t , s ) = Et exp − ∫ r (τ ) dτ t
Where r(τ) is the short rate path between t and s And Êt is the risk neutral expectation at time = t
7
©Finbarr Murphy 2007
Interest Rate Modelling Consider a constant rate of 6% (the green line) The PV discount bond, in this case, can be calculated
MSc
COMPUTATIONAL FINANCE
as exp(-r(s-t))
8
©Finbarr Murphy 2007
Interest Rate Modelling But the actual value will be the expected short path To see this, consider the integral of the blue line,
MSc
COMPUTATIONAL FINANCE
discretised into 10 Δt steps
Δt
9
©Finbarr Murphy 2007
Interest Rate Modelling Early term structure models supposed that the entire
term structure was driven by one source of uncertainty So called, 1-factor models This single source of uncertainty is the short rate
MSc
COMPUTATIONAL FINANCE
We start with the Vasicek (1977) model
10
©Finbarr Murphy 2007
Agenda Page
Interest Rate Modelling
1
2 2
Traditional Term Structure Models
9
MSc
COMPUTATIONAL FINANCE
3
11
©Finbarr Murphy 2007
Traditional Term Structure Models The Vasicek (1977) model The SDE representing the uncertainty in the short
rate is give by the equation
dr = α ( r − r ) dt + σdz
Where
MSc
COMPUTATIONAL FINANCE
r(t) is the short rate at time t dz is a Weiner increment r is the long term mean rate of r The volatility of the short rate is assumed to be a constant σ α is the rate at which r reverts to r
12
©Finbarr Murphy 2007
Traditional Term Structure Models The short rate is assumed to follow the Ornstein-
Uhlenbeck process r, α and σ can be calculated using regression
techniques Under Vasicek, the discount rate is given by
MSc
COMPUTATIONAL FINANCE
P(t , s ) = A(t , s )e
− rB ( t , s )
The equivalent continuously compounded yield is
given by
ln A(t , s ) B(t , s ) R(t , s ) = − + r s −t s −t 13
©Finbarr Murphy 2007
Traditional Term Structure Models Where
(
1 B(t , s ) = 1 − e −α ( s −t ) α
)
And 2 R∞ σ ln A( t , s ) = 1 − e −α ( s −t ) − ( s − t ) R∞ − 3 1 − e −α ( s −t ) α 4α
MSc
COMPUTATIONAL FINANCE
(
and
)
(
1σ R∞ = lim R( t ,τ ) = r − τ →∞ 2 α2
2
14
)
2
©Finbarr Murphy 2007
Traditional Term Structure Models These functions can be easily coded in MatLab Try it!
The spot rate volatility structure is determined by
two parameters σ and α
MSc
COMPUTATIONAL FINANCE
σ −α ( s −t ) ( ) ( ) σ R t, s = 1− e α(s − t)
15
©Finbarr Murphy 2007
Traditional Term Structure Models
MSc
COMPUTATIONAL FINANCE
Here is the Vasicek code for the discount bond
16
©Finbarr Murphy 2007
Traditional Term Structure Models Cox-Ingersoll-Ross The Vasicek model has two major short falls Interest rates can be negative in the model The negative correlation between interest rates and interest
rate sensitivity is ignored
MSc
COMPUTATIONAL FINANCE
CIR (1995) introduce a model where the volatility of
the short rate increases with the square root of the level of the rate This precludes negative rates Displays more variability in high rate environments
and visa versa 17
©Finbarr Murphy 2007
Traditional Term Structure Models The CIR SDE is given by
dr = α ( r − r ) dt + σ r dz
As with Vasicek 77, the discount rates and
MSc
COMPUTATIONAL FINANCE
continuously compounded yields are given by
P(t , s ) = A(t , s )e
− rB ( t , s )
ln A(t , s ) B(t , s ) R(t , s ) = − + r s −t s −t 18
©Finbarr Murphy 2007
Traditional Term Structure Models With
MSc
COMPUTATIONAL FINANCE
A( t , s ) = φ2 B( t , s ) = φ2
( (
φ3
φ1 ( s −t ) e − 1 + φ1 φ1 ( s −t ) e −1 φ1 ( s −t ) e − 1 + φ1
φ1e
φ2 ( s −t )
)
)
And where
φ1 = α + 2σ 2
2αr φ3 = 2 σ
2
α + φ1 φ2 = 2 19
©Finbarr Murphy 2007
Traditional Term Structure Models
MSc
COMPUTATIONAL FINANCE
Here is the CIR Code
20
©Finbarr Murphy 2007
Traditional Term Structure Models The CIR volatility term structure of spot rates is given
by
MSc
COMPUTATIONAL FINANCE
σ r σ R (t , s ) = B (t , s ) s −t
21
©Finbarr Murphy 2007
Traditional Term Structure Models Here is a graph of 100 random movement in the short
MSc
COMPUTATIONAL FINANCE
rate. Note the convergence to the mean
22
©Finbarr Murphy 2007
Recommended Texts Required/Recommended Clewlow, L. and Strickland, C. (1996) Implementing derivative
models, 1st ed., John Wiley and Sons Ltd. — Chapter 6, 7
Additional/Useful Hull, J. (2005) Options, futures and other derivatives, 6th ed.,
Prentice Hall
MSc
COMPUTATIONAL FINANCE
— Chapters 4
23
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