U.U.D.M. Project Report 2009:13
Examensarbete i matematik, 30 hpHandledare och examinator: Johan TyskJuni 2009
Short rates and bond prices in one-factor models
Muhammad Naveed Nazir
Department of MathematicsUppsala University
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Acknowledgement All prays to Almighty Allah who induced the man with intelligence, knowledge and wisdom. It is He who gave me ability, perseverance and determination to complete this thesis. Teachers are lighthouses spreading the light of knowledge and wisdom everywhere and guiding the new generation so that they can cruise safely towards their destination. They are performing the job, which Allah himself acknowledge as the noblest to all jobs; the job of teaching. They will get its reward not only from Allah but also in the form of immense respect that every student carries for them in the core of his heart. I offer my sincerest thanks and deepest gratitude to my research supervisor Prof. Johan Tysk for his inspiring and valuable guidance, encouraging attitude and enlightening discussions enabling me to pursue my work with dedication. I would like to say a big thanks to all the teachers who taught me during the entire program. They did not only teach me how to learn, they also taught me how to teach, and their excellence has always inspired me. I would like to take this opportunity to thank my colleagues, friends who were always there for evocative discussions, invaluable advice and support. Many thanks and my deepest gratitude to my parents who have kept me motivated through the extreme hard time and encourage me during the good times.
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Abstract Interest rate modeling has gained special attention during the last few decades which
has resulted in reliable models. These models are in common use for future evolution of interest rate; one way to accomplish this task is by describing the future evolution of the short rate. The goal of the thesis is to provide a detail analysis of bond pricing using one factor short rate model. The thesis explores and provides an insight into the practicality and the inter-variability between different models.
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Table of Contents Table of Contents ........................................................................................................................ 1
1 Introduction ......................................................................................................................... 2
2 Basics Definition .................................................................................................................. 3
2.1 Bond ............................................................................................................................. 3
2.2 Bond Characteristics .................................................................................................... 3
2.3 Bond Types .................................................................................................................. 4
2.4 Bond price .................................................................................................................... 6
2.5 Bond Yield ................................................................................................................... 6
2.6 Relation between Yield and Bond Price ...................................................................... 6
2.7 Bond Convexity and Duration ..................................................................................... 9
3 One Factor Short Rate Models ......................................................................................... 10
3.1 Short Rate Model ...................................................................................................... 10
3.2 Vasicek Model ........................................................................................................... 10
3.3 The Hull White Model (Extended Vasicek Model) ................................................. 16
3.4 Cox Ingersoll Ross Model ......................................................................................... 19
3.5 The Hull White Model (Extended CIR Model) ...................................................... 23
3.6 Dothan Model ........................................................................................................... 24
3.7 Black-Derman-Toy Model ........................................................................................ 26
4 Model’s Evaluation ............................................................................................................ 28
4.1 Vasicek Model ........................................................................................................... 28
4.2 Extended Vasicek Model ........................................................................................... 28
4.3 Cox Ingersoll Ross Model ......................................................................................... 29
4.4 Extended Cox Ingersoll Ross Model ......................................................................... 29
4.5 Black Derman Toy Model ......................................................................................... 29
4.6 Short Rate Model Versus Other Models ................................................................... 30
Conclusion: ............................................................................................................................ 31
References .................................................................................................................................. 32
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1 Introduction This thesis defines and analyzes a simple one-factor model of the term structure of interest rates. The issue of pricing interest rate derivatives has been addressed by the financial literature in a number of different ways. One of the oldest approaches is based on modeling the evaluation of the instantaneous short interest rate. This is still quite popular for pricing interest rate derivatives and for risk management purposes, and represents the most commonly used type of dynamical stochastic model for interest rates. Therefore, one of the aims of this study is to give information about these models and allow readers to compare them. In this thesis, we are discussing on-factor short rate models, Vasicek model (1977), Hull-White (extended Vasicek model) (1993), Cox Ingersoll Ross model (1985), Hull-White (extended CIR model) (1993), Dothan model (1978), Black-Derman-Toy model (1980). The thesis is organized as follows: Chapter 2 provides with the basic introduction to bonds. The Chapter to follow will discuss different short rate models for bond pricing. Finally, Chapter 4 summarises the results and discussions with concluding remarks.
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2 Basics Definition
2.1 Bond
A bond is a debt instrument which is issued for a period of more than one year for the purpose of capital rising by borrowing. A bond is a promise to repay the principal with interest (coupons) on a specific date (maturity). Some bonds do not pay interest but all bonds require a repayment principal.
2.2 Bond Characteristics
A bond depends on the following characteristics,
• Issuers Bonds are issued by public authorities, credit institutions, companies and supranational institutions in the primary markets. The most common process of issuing bonds is through underwriting. In underwriting, one or more security firms or banks, forming a syndicate, buy an entire issue of bonds from an issuer and re-sell them to investors.
There is little difference between bonds issued by companies and those issued by the national government. A federal government bonds have a low risk by default, where as corporate bonds are considered higher risk. International bonds are complicated by different currencies. These types of bonds are issued in foreign market to the issuer’s home market and the currency type is the currency of the foreign market. Examples of some international bonds are Euro bonds, Foreign bonds, Global bonds etc.
• Priority
Determined by the probability of when the issuer will pay back the money. The priority shows your place in the queue of the company for payments. If you have unsubordinated (senior) security then you will be the first in the queue to receive
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payment from bankruptcy of its assets. But if you have subordinated (junior) debt security then you will receive payment after the senior (unsubordinated) security holders have received their shares.
• Coupon Rate (Interest Rate)
Interest payment received by the bondholder is called “coupon”. Most bond issuer companies pay interest every sixth month, but it is also possible to pay monthly, quarterly or annually. The coupon is expressed as a percentage of its value. If the bond coupon rate is fixed (or stays fixed) and market rate rises then bond price are reduced or if market rate falls then bond price will rise.
• Redemption
The return of an investor's principal in a security, such as a stock, bond, or mutual fund. Or we can say as, both investor and issuers are exposed to interest rate risks because they agree either to receive or pay a set coupon rate over a specific period of time.
2.3 Bond Types
• Callable (Redeemable) bond gives rights to bond issuer but not obligation to redeem his issue of bonds before its maturity. Issuers have to pay bond holders premium. There are two types of this bond. o American Callable bonds can be called by the issuer any time after the call
protection. o European Callable bonds can be called by the issuer only on pre-specified dates.
• Convertible bonds give rights to bondholder but not obligation to convert their
bonds into different securities, like shares. This only applies on corporate bonds.
• Puttable bonds give rights to bondholder but not obligation to sell their bonds back to the issuer at a predefined date and price. Investors can ask for the repayment of the bond.
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• Asset-Backed Securities are bonds or currency notes secured by assets of the company. “It is defined as yield and redemption are guaranteed by assets of the issuer, exclusively earmarked for the satisfaction of the rights incorporated in the financial instruments themselves”, [Ref. 10].
• Corporate bonds are debt securities issued by the corporation. Like all bonds, their
values rise when market interest rate falls and fall when market interest rate rises. They involve risk of principal.
• Euro bonds are medium or long term interest bearing bonds created in international
capital markets. It is dominated in a currency other than the place where it is issued. It is also called global bond.
• Junk Bonds (high yield bonds) are issued from risky companies. So they offer
higher interest rates to compensate the risk.
• Municipal bonds are issued by various cities having some low interest rates and these bonds are tax free.
• Zero-Coupon bond (also called discount bond) is a bond that is redeemed at full
face value at the maturity date ‘T’. This is special type bond that does not pay tax. The price at time ´t´ of a bond with maturity date ‘T’ is denoted by p (t, T). We assume that
Market (frictionless) for T-bonds for every 0T > .
( ),p t t = 1 ∀ t.
For fixed t, the bond price ( ),p t T is different with respect to ‘T’ (time of maturity).
Calculated Zero coupon bond price = (1 )n
Mi+
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2.4 Bond price
A good sum is paid to buy a bond. Bond prices have an inverse relationship with interest rates. When interest rates rise, bond prices fall and when interest rates fall, bond prices rise. Here is the formula for calculating a bond’s price, i.e. basic Present Value (PV) formula:
Price = 2 ......(1 ) (1 ) (1 ) (1 )n n
C C C Mi i i i
+ + + ++ + + +
,
C = Coupon, n = no. of payment
i = interest rate, M = value at maturity
2.5 Bond Yield
The return received when bond is purchased and held to maturity. The yield is calculated by dividing the interest rate by the purchase price of the bond.
2.6 Relation between Yield and Bond Price
Yield is the return on an investment, expressed as annual percentage. For example 3% yield means that the averages 3% of investment is returned each year. There are several methods to calculate yield.
• Current Yield
The simplest way to calculate the current yield is
Current Yield = _ ($) *100Annual Interestprice
,
while current yield is easy to calculate, however it is not as accurate a measure as yield to maturity. So the modified current yield formula that takes into discount / premium account for bond is
Adjusted Current Yield = (100 Pr )*100. .
AnnualCoupon Market iceMarketprice Year to Maturity
−+
,
now the current yield for zero coupon bonds, which has only one coupon payment, is
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1
_ 1_ Pr
nFuture ValueYieldPurchase ice
= −
,
Where n = years left until maturity.
• Yield to Maturity
The current yields return the annual coupon payment, but this percentage does not take into account the time value of money. Yield to maturity (YTM) is the interest rate which is the present value of all cash flows. And cash flows are equal to the bond’s price. YTM require a complex calculation. It considers the following factors. Coupon rate: Higher bond’s coupon rates have higher yield. Price: Higher bond’s price, the lower yield. Remaining Year until maturity: By reinvesting the interest payment, can
earn more bonds.
The difference between the face value and the price is that if you keep a bond to maturity then you get the face value of bond. You might get less or more the price you paid. Yield to maturity can be calculated as,
Bond Price = ( )n
n
11-1+i 1cashflow * + Maturity*i (1+i)
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here, i = Interest rate. • Calculating yield for callable and puttable bond
The yield for callable bond is also called yield-to-call and the yield for puttable bond is also called yield-to-put. Yield-to-Call and Yield-to-put is a Yield to Maturity assuming that the bond will redeem at the Call or Put date. It can be calculated as
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1
100 1 1
100 100
M
i Mi
RPY Y=
= + + +
∑ .
• Yield to Worst
The lowest yield calculated is known as yield to worst. We can assume as if the issuer does not default. If market yields are higher than the coupon, the yield to worst would assume no prepayment. If market yields are below the coupon, the yield to worst would be assuming prepayment. We can say as, yield to worst that market yields are unchanged.
• Yield Curve and Maturity Date
A yield curve is the relationship between interest rates and time, determined by plotting the yields of all default-free coupon bonds against their maturities. The yield curve typically slopes upwards since longer maturities normally have higher yields, although it can be flat or even inverted.
• Normal Yield Curve
A normal yield curve is a graph, illustrating the structure of interest rates over a period of time, which indicates that rates rise as maturities lengthen, i.e., short-term rates are lower than long-term rates. It is also known as “positive and upward-sloping” yield.
• Flat Yield Curve
A yield curve in which there is little difference between short term and long term rates for bonds of the same credit quality. Or a yield curve showing same yield for short maturity and long maturity bonds called Flat Yield Curve.
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• Inverted Yield Curve A rare situation, where long term interest rates have lower yields and short term interest rates have a negative yield curve (interest rates decrease as maturities lengthen).
A negative yield curve is a graph, depicting the structure of interest rates over time, which indicates that as maturities lengthen interest rates fall, short-term rates exceed the long-term rates.
• Spot Rate curve
It is the graphical depiction of the relationship between the spot rates and maturity.
2.7 Bond Convexity and Duration
As bond yields go higher, the price goes lower. This relationship between price and yield has a convex structure in nature. The term used to describe this relationship is also known as convexity.
Bond convexity is the sensitivity of Bond duration to changes in interest rates. Bond duration is the sensitivity of Bond prices to change in interest rates. In calculus terms, duration is the first derivative of the price/yield equation and convexity is the second derivative.
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3 One Factor Short Rate Models
3.1 Short Rate Model
The short rate describes the future evolution of the interest rate by describing the future evolution of the short rate. The short rate ( tr ) is the interest rate at which an entity can borrow money for a short period of time from time t. The current short rate does not specify the entire yield curve. However no-arbitrage arguments show that, under some fairly relaxed technical conditions, if tr as stochastic process under a risk-neutral measure (Q) then the price at time (t) of a zero-coupon bond maturing at time T is given by
( ) ( ), expT
s ttP t T r ds = Ε − ∫ F , (3.1.1)
where F is the natural filtration for the process, thus specifying a model for the short rate specifies future bond prices. This means that instantaneous forward rates are also specified by the usual formula.
( ) ( )( ), ln ,f t T P t TT∂
= −∂
. (3.1.2)
Here we consider models for the short rate with only one source of randomness, so called one-factor models.
3.2 Vasicek Model
The Vasicek model was introduced by Oldrich Vasicek in 1977 [8]. This model can be used for interest rate derivative valuation and also adapted for credit market, although it uses the wrong principle for credit market and implying negative probabilities [11]. Vasicek Model is an Ornstein-Uhlenbeck stochastic process.
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This model specifies that the instantaneous interest rate follows the stochastic differential equation.
( )t t tdr a b r dt dWσ= − + .
• Here tW is a Wiener process, modelling the random market risk factor. • σ (Standard deviation parameter), determines the volatility of the interest rate. • ( )ta b r− , is the drift factor that describes the expected change in the interest rate
at that particular time. • b: “Long term mean value”. The long run equilibrium value towards which the
interest rate goes back. • a: “Speed of reversion”. It gives the adjustment of speed and has to be positive in
order to maintain stability around for the long-term value.
• For example, when tr is below b, then the drift term ( )ta b r− becomes positive for positive a, generating a tendency for the interest rate to move upwards (toward equilibrium).
• Note that { }tX X= with t tX r b= − is an Ornstein-Uhlenbeck (O-U) process satisfying the SDE
t t tdX aX dt dWσ= − + .
Hence X is a Gaussian process, which implies Q ( tr < 0) > 0 for every [ ]0,t T∈ an unreasonable behaviour in practice. Pricing of bond option would be easy because nice properties of the O-U process.
Vasicek model was the first economics model to capture the value of mean reversion. Mean reversion is an important characteristic of the interest rate that is responsible to set it apart from other financial prices. Hence interest rates neither rise nor decrease
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indefinitely unlike stock prices. It moves within a limited range, showing a tendency to revert to a long run value. It gives an explicit formula for the (zero coupon) yield curve and gives explicit formulae for derivatives such as bond option. It can be used to create an interest rate tree.
• Calculating Bond Price Implied by the Short Rate
The Vasicek Model assumes that the short rate tr , under risk-neutral measure is given by
( )t t tdr a b r dt dWσ= − + . (3.2.1) Here a, b & σ are positive constant. We obtain after integrating
0 0 0
t tat au aut ur e r abe du e dWσ− = + + ∫ ∫ ,
( )0 0 1
tat at auue r b e e dWσ− = + − + ∫ ,
( )0
t a u tt ue dWµ σ −= + ∫ ,
where tµ is a deterministic function and [ ]t tE r µ= . The expected value [ ]tE r can be determined without explicitly solving the SDE. Let’s take integral for equation (3.2.1).
( )( )0 0
t
t u ur r a b r du dWσ= + − +∫ ,
Then
[ ] [ ]( )0 0:
t
t t uE r r a b E r d uµ = + −∫ , (3.2.2)
( )t td a bdt
µ µ= − .
This is a linear ordinary differential equation (ODE). Now we use the
integrating factor ate . That is
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[ ] ( )0 1at att tE r e r b eµ − = = + − , (3.2.3)
Further we can define it,
[ ] ( )22
0
2 2
0
22
: ,
,
1 .2
tatt t u
tat au
at
Var r E e dW
e E e du
ea
σ σ
σ
σ
−
−
−
= =
=
−=
∫
∫
Since normal random variables can become negative with positive probability, this is a weakness of Vasicek model. Now using the risk-neutral valuation framework, the price of a zero-coupon bond with maturity T at time t is
( ) ( ), exp |T
u ttB t T E r du = − ∫ F ,
Now consider that
( ) uX u r b= − . (3.2.4)
( )X u is the solution of the Ornstein-Uhlenbeck equation, so
( ) ( ) tdX t aX t dWσ= − + ,
with ( ) 00X r b= − . Applying Itô’s lemma formula, then the X (u) is given by
( ) ( )0(0)
uau assX u e X e dWσ−= + ∫ . (3.2.5)
Here we can see that X (u) is a Gaussian process with continuous sample paths.
Thus 0
( ) tX u du∫ is also Gaussian process. ( )( ) ( )0 auE X u X e−= , then
( ) ( ) ( )0
01
t atXE X u du e
a− = − ∫ , (3.2.6)
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now the variance is
0 0 0( ) ( ) , ( )
t t tVar X u du Cov X u du X s ds = ∫ ∫ ∫ ,
0 0 0 0( ) ( ) ( ) ( )
t t t tE X u du E X u du X s ds E X s ds = − − ∫ ∫ ∫ ∫ ,
after solving it, we get,
( )2
23 2 3 4
2at atat e e
aσ − −= − + − . (3.2.7)
From equation (3.2.4), we get
( )( )0 0
t t
uE r du E X u b du − = − + ∫ ∫ . (3.2.8)
Now combining equation (3.2.8) with (3.2.6), we get
( )( ) ( )0
1t a T tt
ur bE r du e b T t
a− −− − = − − − − ∫ , (3.2.9)
And
( )T T
ut tVar r du Var X u du − = − ∫ ∫ ,
( ) ( ) ( )( )2
23 2 3 4
2a T t a T ta T t e e
aσ − − − −= − − + − . (3.2.10)
Now the zero coupon bond is,
( ) ( )( )( )
, , exp | ,
exp .
T
t u tt
T
u tt
B t T r E r du r
E r r du
= − = −
∫
∫
Here ur is a function of tr , so combining equation (3.2.9) and (3.2.10), and getting the bond price
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( ) ( ) ( )1, , exp2
T T
t u t u tt tB t T r E r r du Var r r du = − + − ∫ ∫ ,
( )( ) ( ) ( ) ( ) ( )( )2
23exp 1 2 3 4 ,
4a T t a T t a T ttr b e b T t a T t e e
a aσ− − − − − − − = − − − − + − − + −
( ) ( )
( )( )
( )
( ) ( )
2 2
2 2
22
2
1 1 12 2
exp ,1 2
4
a T t a T t a T t
t
a T t a T t
e e er b T t T ta a a a a
e ea a
σ σ
σ
− − − − − −
− − − −
− − −− + − − − + − =
− + −
( ) ( ) ( ) ( ) ( ) ( )2 2 2
22 2exp , , , , ,
2 2 4tA t T r b At T b T t A t T T t A t Ta a a
σ σ σ = − + − − − + − −
we get
( ) ( )( )exp , ,tA t T r D t T= − + , (3.2.11) where
( )( )
( ) ( ) ( ) ( )222
2
1, and
,, , .
2 4
a T teA t Ta
A t TD t T b A t T T t
a aσσ
− −−=
= − − − −
So by the result of equation (3.2.7), tr represent the Itô integral. The equation (3.2.11) is called an exponential affine bond price. The simple Gaussian structure of tr (short rate process) leads to a closed form solution of the bond price. Conclusions: The model is arbitrage free, in that sense, no bond or option price
produced by the model will allow or permit arbitrage.
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As a one-factor model, it does not capture the mode complex term structure shifts that occur.
The main disadvantage is, (under the theory) the value of interest rate can be negative.
The Hull-White Model is the further extension over Vasicek Model.
3.3 The Hull White Model (Extended Vasicek Model)
In Vasicek model, we found the poor fitting of the initial term structure of interest rate. So, for the need of exact fit, The Hull-White model introduced by “Johan C.Hull and Alan White” in 1990 [7]. It is a model of future interest rate. It belongs to the class of no-arbitrage models that are able to fit today’s term structure of interest rate. The Hull-White model is varying the time parameter in Vasicek model. It allows the deviation of analytical formulae that is strength of its normal distribution. But weakness is that, it also allows the negative interest rate with positive probability.
This model is short rate model. So, the stochastic differential equation for the extended Vasicek Model is written as
( ) ( )( ) ( )t t tdr a t b t r dt t dWσ= − + . (3.3.1)
Here, tW is a Brownian motion in the martingale measure. We will consider this
model when ( )a a t= but “b” and “σ ” are constants.
Let 0
t
t uK a du= ∫ and applying Itô’s lemma integration by parts to tKte r ,
( ) 't t tK K Kt t t td e r e K r dt e dr= + , (3.3.2)
therefore
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( ) ( )
( )
,
,
,
,
t t t
t t t t
t t
t
K K Kt t t t t t t t
K K K Kt t t t t t t
K Kt t t
Kt t t
d e r e b r dt e a b r dt dW
e b r dt e a dt e b r dt e dW
e a dt e dW
e a dt dW
σ
σσ
σ
= + − +
= + − +
= +
= +
and integrative,
( )0 0 0
0 0 0
,
,
t u u
t u u
t tK K Kt u u u
t tK K Kt u u u
e r r e a du e dW
r e r e a du e dW
σ
σ−
− = +
= + +
∫ ∫
∫ ∫
( ) ( )0 0 0
t u t utt tK K K KK
t u u ur e r e a du e dWσ− + − +−= + +∫ ∫ . (3.3.3)
To generalize the result for any u t≤ , so we have
( ) ( ) ( )0 0 0
t u t u t ut tK K K K K K
t u u ur e r e a du e dWσ− − − − − −= + +∫ ∫ , (3.3.4)
usually ( )b t and ( )tσ are supposed to be constant so the equation (3.3.4) will be
( ) ( )0 0 0
t tb t u b t ubt
t u u ur e r e a du e dWσ− − − −−= + +∫ ∫ , (3.3.5)
( ) ( ) ( ) t tb t s b t u b t u
t s u u us sr e r e a du e dWσ− − − − − −= + +∫ ∫ , (3.3.6)
For bond price
In the term structure, a discount bond with S-time, T-maturity value has distribution as
( ) ( ) ( ) ( )( ), , exp ,P S T A S T B S T r S= − , (3.3.7) where
( ) ( )( )1 exp,
b T SB S T
b− − −
= ,
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( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( )22
3
log 0, exp exp exp 2 10,, exp , ,
0, 4P S bT aS aSP T
A S T B S TP S dS a
σ ∂ − − − − = − −
( ) ( ) ( ) ( ) ( )1 2, ,V t P t S d KP t T d= Φ − Φ . To fit this model to the initial term structure of interest rate, we have to calculate the zero coupon bond prices. We have shown that the price of a zero coupon bond at present with T-maturity is
( ) 00,T
sr dsQP T E e− ∫=
, (3.3.8)
let consider ( )0
t
sy t r ds= ∫ , so above equation will be
( ) ( )( )0, y TQP T E e−= . (3.3.9)
By using equation(3.3.5), ( )y t can be calculated,
( ) ( ) ( )00 0 0 0 0
yt t s t sb s u b s ubt
u u ut e r ds e a duds e dW dsσ− − − −−= + +∫ ∫ ∫ ∫ ∫ ,
( ) ( ) ( ) ( )0
0 0
1 y
btt t t tb s u b s u
u u uu u
r et e a duds e dW ds
bσ
−− − − −−
= − + +∫ ∫ ∫ ∫ ,
and
( ) ( ) ( ) ( )0
0 0
1 y
btt tbs bu bs bu
u u
r et e e a t u du e e t u dW
bσ
−− −
−= − + − + −∫ ∫ . (3.3.10)
From the short rate dynamics of Hull-White, it is easily seen that the interest rates
are normal. So ( )0
t
sy t r ds= ∫ is a sum of normal distributed random variables.
Therefore, it is also normal distribution. So, ( )y t has mean with
( ) ( ) ( )0
0
1
bttbs bu
u
r em t e e a t u du
b
−−
−= − + −∫ ,
and variance
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( )2 2
2 22 3 3
12 2 4 4
btbs t t ev t e
b b b bσ−
= − − + −
.
From the expression of the Laplace transform of Gaussian, the equation (3.3.9) can be written as
( )( ) ( )( )1exp2
QE y T Va r y T = − +
, (3.3.11)
we also know that
( ) ( )1exp2
m T v T = − +
,
after substituting values in the equation, then we get
( ) ( )0
0
2 22 2
2 3 3
1
exp1 12 2 2 4 4
btTbs bu
u
bTbs
r ee e a T u du
bT T ee
b b b bσ
−−
−
− − + − = + − − + −
∫, (3.3.12)
taking the logarithms and differentiating twice time with respect to ‘T’. Using the
equality of the instantaneous forward rate, ( ) ( )log 0,0,
P Tf T
T∂
= −∂
, we can
get ‘a’ as
( ) ( ) ( ) ( )2
20,0, 1
2bTf T
a T bf T eT b
σ −∂= + + −
∂. (3.3.13)
3.4 Cox Ingersoll Ross Model
The Cox Ingersoll Ross model was introduced in 1985 by ‘John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross’ as an extension of the Vasicek Model [6]. The CIR bond pricing model assumes that the risk natural process for the Short interest rate is given by
( ) tdr a b r dt rdWσ= − + .
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• ( )a b r− is the same as in the Vasicek Model. The standard deviation
factor trσ , avoid the possibility of negative or null rates if the condition 22 ab σ is met.
The CIR model specifies that the interest rates follow the stochastic differential equation,
( )t t t tdr a b r dt r dWσ= − + , (3.4.1) taking integral
( )t t
t u s s su ur r a b r ds r dWσ− = − +∫ ∫ , (3.4.2)
Applying Itô’s lemma ( ) 2t tf x x= and t tx r= , so equation will be
( )2 2 212 . 2 .2
t t
t u s s s s su ur r r a b r ds r dW r dsσ σ = + − + + ∫ ∫ ,
2 2 2 . .t t
u s s s s su ur r abds ar ds r dW r dsσ σ = + − + + ∫ ∫ ,
2 2 23 2 2 2 . . .
t t
u s s s s su ur r abds ar ds r dW r dsσ σ= + − + +∫ ∫ ,
( )2 2 2 3 2 2 2 . .t t
u s s s s su ur r ab ar r ds r dWσ σ= + − + +∫ ∫ ,
and
( )2 2 2 32 2 2 .t t t
t u s s s su u ur r ab r ds a r ds r dWσ σ= + + − +∫ ∫ ∫ , (3.4.3)
If 0µ = then
( )0 0 0.
t t
t s s sr r a b r ds r dWσ= + − +∫ ∫ . (3.4.4)
Thus, the unconditional mean is
( ) ( )( )0 0
tQ Qt sE r r a bt E r ds= + − ∫ ,
21
solving the equation ( ) ( )( )0 0
tt r a bt s dsΦ = + − Φ∫ , which can be transformed
into the ODE (ordinary differential equation) ( ) ( )'t a t abΦ + Φ = . So, the unconditional mean is
( ) ( )0Q at
tE r b r b e−= + − , (3.4.5) can also write as,
{ } ( ) ( )( )1a t u a t uQt u uE r r e b e− − − − = + −F .
Similarly, we can write equation (3.4.3) as
( ) ( ) ( ) ( )2 2 2 2 20 0 0
2 2t tQ Q Q
t s sE r r ab E r ds a E r dsσ= + + −∫ ∫ , (3.4.6)
substituting the value of ( )Q
tE r into (3.4.6) and applying second derivative, then we can get variance
( ) ( ) ( )2
01 12
bt bt btt
bVar r e r e ea
σ − − − = − + − , (3.4.7)
or
{ }( ) ( )( )( ) ( )( )222 2 1
2
a t u a t u a t uu
t u
r e e b eVar r
a a
σ σ− − − − − −− − = +F .
In [6], they aimed to have relationship between term structure of interest rates and yields on risk free securities that differ only in their time to maturity. The explanation of the term structure gives extra information to predict the effect on the yield curve if we change the underlying variables. The instantaneous short rate dynamics corresponds to a continuous time first-order autoregressive process where the randomly moving interest rate is elastically pulled toward a central location or long term value, b, which leads to mean reversion property.
If 2 2abσ > then tr can be zero.
If 2 2abσ ≤ then the upward drift is sufficiently large to make the original inaccessible.
22
So from (3.4.7) equation we get that there is no explicit form for the solution to the CIR model. It is known that the model has unique positive solution. If the interest rate reaches zero then it can subsequently become positive. More generally, when the rate is low (close to zero), then the standard deviation also becomes close to zero.
For calculating the increment of bond price in the volatility of CIR model, we use the
formulation under the risk-neutral measure ‘Q’ is ( )t t t tdr a b r dt r dWσ= − + . We assuming here, a pure discount bond price is a function of current interest rate ‘r’, time ‘t’ and maturity ‘T’. Under free-arbitrage condition ‘B(t,T,r)’ satisfies the following PDE for CIR model [6].
( ) σ∂ ∂ ∂+ − + − =
∂ ∂ ∂
=
22
2
1 0,2
( , , ) 1.
B B Ba b r r rB
t r r
B T T r (3.4.8)
The CIR model assumes that the short rate tr satisfies
( ) .t t t tdr a b r dt r dWσ= − + ; 0 0r > . Here ‘a’ and ‘b’ are positive constants. The main advantage of the CIR model over the Vasicek model is that the short rate is guaranteed to remain non-negative [5]. Unlike the Vasicek model, the CIR model is not Gaussian and is therefore considerably more difficult [5] to analyze. But zero coupon bond price can still be computed analytically at time t with T-maturity is [6]
( ) ( ) ( ),, , tB t T rP t T A t T e−= , (3.4.9) where
( )( )
( ) ( )
( ) ( )( )( )
2
2
2 2
2 2, log ,1 2
2 1, , 2 .
1 2
h a T
hT
hT
hT
ab heA t Th a e h
eB t T h a
h a e h
σ
σ
+ =
+ − +
−= = +
+ − +
23
We have
( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( )
,
22 ,
2
, , , ,
, , , .
t
t
B t T r
B t T r
P t T A t T B t T er
P t T A t T B t T er
−
−
∂= −
∂∂
=∂
Here ( )2
2 , 0P t Tr
∂>
∂, since ( ), 0A t T > . By taking the second derivative with
respect to tr , we get that the bond price is convex as a function of interest rate ‘r’ for CIR model. We can calculate the bond price by placing equation (3.4.8) in equation (3.4.9).
3.5 The Hull White Model (Extended CIR Model)
Extension of the CIR model is given by
( ) ( ) ( ) .t t t tdr a t b t r dt t r dWσ= − + . (3.5.1)
Here ( ) ( ) ( ), ,a t b t tσ are non-random functions of t. [Ref. 7] The CIR model is extended by Hull and White by introducing time varying parameter, a. It has a time varying mean. The market term structure of rates at the current time is equivalent to solving a system with an infinite number of equations, one for each possible maturity. There is no analytical expression for ‘a(t)’ but observed yield curve is available in literature. And there is no guarantee that a numeric approximation of ‘a(t)’ would keep the rate ‘r’ positive. Hence extension is not analytically tractable [16].
24
However, in [15] a special of (3.5.1) is solved explicitly. That is done in the case when
( )( )
δσ
=2 ,a t
t positive constant. That must be greater than ½ to ensure that the origin is
inaccessible.
3.6 Dothan Model
Dothan introduced in 1978 a constant price of risk, which is equivalent to directly assuming a risk-natural dynamics of type [4]
t t t tdr ar dt r dWσ= + . (3.6.1)
Here ´ ,́́ ´a σ are non-negative constants. For detail see [Ref. 4]. In [Ref. 4], tr presented as
t t tdr r dWσ= . (3.6.2)
In Dothan model, the interest rate has a log-normal distribution which overcomes the drawback of negative values of the interest rate in Vasicek model.
0 0 0
t t
t s s sr r ar ds r dWσ= + +∫ ∫ , (3.6.3)
2 20 20 0 0
1 1 1 1ln ln2
t t t
t s s s ss s s
r r ar ds r dW r dsr r r
σ σ−= + + +∫ ∫ ∫ , (3.6.4)
20 0 0 0
1ln2
t t t
sr ads dW dsσ σ= + + −∫ ∫ ∫ ,
20
1ln2tr at W tσ σ= + + − , (3.6.5)
20
1exp2t tr r at W tσ σ = + −
, (3.6.6)
and
( ) ( ) ( )21exp2t u t ur r a t u W W t uσ σ = − + − − −
,
25
( ) ( )21exp2t u t ur r a t u W Wσ σ = − − + −
. (3.6.7)
Moreover, the conditional mean and the variance of tr are:
( ) ( )( )21exp2
t uW WQt u u uE r r a t u E eσσ − = − −
F F ,
( ) ( )( )
( ) ( )
20
2 2
1 exp ,2
1 1 exp exp ,2 2
t uWQu
u
r a t u E e
r a t u t u
σσ
σ σ
− = − − = − − −
F
( )( )expt u uE r r a t u = − F , (3.6.8)
( ) ( )( )2
212 1t u
a t u W WQt u u uVar r E r e e
σ σ − − −
= −
F F ,
( ) ( ) ( )( ) ( ) ( )22
1222 2
0 0 2 1t s t st uWa t u t u WQ Q
ur e e E e e E eσσσ σ− −
−− − = − +
F F ,
( ) ( )( )222 1a t u t ut u uVar r r e eσ− − = − F . (3.6.9)
The zero coupon bond price ( ) ( ), , tP t T F T t r= − is given [17] for all p R∈ by
( ) ( )2 2 2 28
2 2 10 0
2 1 4, exp ,4
tp
urp
z z du dzF r e eu u u z
σ τ σ ττ θσ σ
− ∞ ∞ −+
+ = −
∫ ∫ , (3.6.10)
we have,
26
( ) ( )
( )( ) ( )( )
( )
2
2
2
2
0 0
, ,
exp |
exp |
, ,
s t
st
t
T
s tt
T B B p s tt tt
B p sr u
F T t r P t T
r ds
r e ds
e P e ds du B dy
σ σ
τ σ στ
− − −
∞ ∞ −−
−∞
− =
= Ε − = Ε −
= ∈ ∈
∫
∫
∫ ∫ ∫
F
F (3.6.11)
0 0, , ,u y τ> ∈ℜ > where
2y
z eσ
= , ( ) ( ) ( ) ( )2
222
3 00
2cosh, sinh sin , , .
tt vvev t e e d v ttt
πξ ξ πξθ ξ ξ
π
∞ − −= >∫ And
pK denotes the modified Bessel function of the second kind of order p [4]. The above formula is a rather complex equation since it depends on three integrals. A triple numerical integration is needed to reduce the ‘explicit formula’. Under arbitrage
free conditions, the value of an interest rate ( ), ,B t T r satisfies the following PDE for Dothan model.
2
2 22
1 02
B B Bar r rBt r r
σ∂ ∂ ∂+ + − =
∂ ∂ ∂, (3.6.12)
Where
( ) ( ), , 1 , B , ,0 1B T T r t T= = . By solving the PDE (3.6.12), we can get bond price.
3.7 Black-Derman-Toy Model
The BDT model was introduced by Fischer Black, Emanuel Derman, and Bill Toy [3]. It was first developed for in-house use by Goldman Sachs in the 1980s and was eventually published in the Financial Analysts Journal in 1990. The BDT model is a one-factor model; a single stochastic factor (the short rate) determines the future evolutions of all interest rate. In the model the short interest rate is log-normally distributed. The model does not have a closed-form solution for bond price.
27
Calibrating the BDT model is a term structure volatilities would mean that the short rate volatility needs to decay with time. The future short-rate volatility would have to take on extremely low values. This would also affect the future term structure of volatilities. In fact, for a Gaussian short-rate model, it can be shown that the future term structure of volatilities is directly proportional to the future short-rate volatilities. Several authors have shown that the implied continuous time limit for the BDT model is given by the following SDE [18],
( ) ( )( ) ( )ln lnt t t
td r a t r dt t dW
tσ
σσ
′= + +
, (3.7.1)
If tσ is a decreasing function then tσ ′ becomes smaller than zero. In this case the BDT model satisfies the mean reversion property. If tσ is an increasing function then tσ ′ becomes greater than zero. In this case, the BDT model will grow and it has no mean reversion effect. If tσ is a constant function then 0tσ ′ = . In this case, the BDT model becomes a specific model,
ln ( ) ,t td r a t dt dWσ= + (3.7.2) By taking logarithm, we obtain the exact solution of the instantaneous short rate.
( )0 0 0ln ln
t t
t sr r a s ds dWσ= + +∫ ∫ , (3.7.3)
( )
( )
0 0
0 0
ln ln ,
exp ,
t
t t
t
t t
r r a s ds W
r r a s ds W
σ
σ
= + +
= +
∫
∫
( ) ( )expt
t u t uur r a s ds W Wσ = + − ∫ . (3.7.4)
28
4 Model’s Evaluation
4.1 Vasicek Model
Advantages: • It is analytically tractable. • It has an explicit solution since the distribution of tr is normal.
Disadvantages: • Vasicek model is a one-factor model, so it can not capture the more complex
structure shift that occurs. • It has a fundamental drawback that the short term interest rate ‘r’ can become
negative. • Since the short rate is normally distributed for every ‘t’, there is a positive
probability that ‘r’ is negative and it is not acceptable at the economic point-of-view.
• Another drawback of Vasicek model is, it assumes ‘r=0’ and this assumption implies the conditional volatility of changes in the interest rate to be constant indefinite on the level of ‘r’.
4.2 Extended Vasicek Model
Advantages: • It reproduces the current zero-coupon yield curve. • It can be fitted very accurately to the initial term structure and its
implementation is relatively straight forward, based on a lattice structure.
Disadvantages: • It can generate negative short rate since tr is a normal random variable for each
‘t’. • It is sensitive to its parameters in the sense; a small change in parameter values
may result in a large change in bond price.
29
4.3 Cox Ingersoll Ross Model
Advantages: • It disallows negative interest rates. • The conditional volatility depends on the level of the short rate. • Explicit formulae for term structure, option price, etc. • It’s based on a coherent statistical model for behaviour of interest rates.
Disadvantages
• Statistical model is too simple. • Term structure is not flexible enough to fit observed term structure. • All bond price are 100% correlated (not empirically true) • This model is not analytically tractable.
4.4 Extended Cox Ingersoll Ross Model
Advantages: • It can be tractable although it has a more restricted set of term structure
compared to the extended Vasicek model, as a result of limitation imposed by
the tr term on the volatility parameter. • It always fits in term structure.
Disadvantages:
• ECIR model is unable to capture the dynamics of the whole yield curve. • Underlying statistical model of interest rate behaviour is possibly not verified. • All bond prices are 100% correlated (not empirically true). This is true for any
one-factor model of the short rate.
4.5 Black Derman Toy Model
Advantages: • It can be directly calibrated to an input term structure of pure discount bond
prices and cap volatilities.
30
• The interest rates always positive. Disadvantages [14]:
• Because of the one-factor model, substantial inability to handle conditions where the impact of a second factor could be relevance.
• Inability to specify the volatility of yields of different maturities independently of future volatility of the short rate.
• BDT model has a fundamental idea of short rate process, a mean reversion, does not hold under following circumstances. If the continuous time BDT risks neutral short rate process has the form,
( )ln lnln ln lnt tt t t t t
ud r u r dt dWt t
σ σ∂ ∂ = − − + ∂ ∂ ,
where tu is the median of the short rate distribution at time t. For
constant volatility (σ =const), ln t
tσ∂
∂= 0, the BDT model does not
display any mean reversion.
4.6 Short Rate Model Versus Other Models
Advantages: • The model is generally tractable, very amenable to numerical and Monte-Carlo
simulation methods. • Derivatives price can be computed quickly. This is very important for risk-
management purpose when many securities need to be priced frequently. • There are parsimonious to provide “sanity checks” on more sophisticated models
that can often be calibrated to “fit everything”. Models that are calibrated to “fit everything” can be unreliable due to the problems associated with over-fitting. Of course short rate models with deterministic time –varying parameters (e.g. Ho-Lee, Hull and White) are also susceptible to over-fitting.
31
Disadvantages:
• A model with three or more factors tends to suffer from the curse of dimensionality in that case Monte-Carlo simulation becomes the only practical pricing technique.
Conclusion:
There is no one model perfect for all type of problems. Some bond models work well in one situation but have low efficiency in other situations. So our study concludes that the Black-Derman-Toy model is better than the other models which we discussed above.
32
References 1. Klose, C.; Li C. Y. (2003). Implementation of the Black, Derman and Toy Model.
Seminar Financial Engineering, University of Vienna. 2. Black F., Derman E. and Toy W., (1990), p. 33-39. A one-factor model of interest
rates and its application to Treasury bond options, Financial Analysts Journal. 3. Dothan L. (1978), volume 6, p. 59-69. On the term structure of interest rates, Journal
of Financial Economics. 4. Cairns, A. (2004), Interest Rate Models: An Introduction, Princeton University Press. 5. Cox, John C., Jonathan E. Ingersoll, Jr., and Stephen A. Ross, (1985). Volume 53, no.
2, p. 385-407. A Theory of the Term Structure of Interest Rates, Econometric. 6. Hull J. and White A., (1990) volume 3, number 4, p. 573-592. Pricing Interest Rate
Derivative Securities, the Review of Financial Studies. 7. Vasicek, Oldrich, (1977). Volume 5, p. 177-188. An Equilibrium Characterization of
the Term Structure, Journal of Financial Economics. 8. Hull J. and White A., (1993). Volume 28, p. 235-254. One-Factor Interest Rate
Models and the Valuation of Interest Rate Derivative Securities, Journal of Financial and Quantitative Analysis.
9. http://www.group.intesasanpaolo.com/scriptIsir0/isInvestor/eng/glossario/eng_glossari
o.jsp --->Accessed on date 25th March, 2009.
10. Brigo and Mercurio (2006). Interest rate models -- theory and practice, Springer.
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11. http://personal.fidelity.com/products/fixedincome/price.shtml --->Accessed on date 25th March, 2009.
12. http://www.investopedia.com/university/advancedbond/advancedbond2.asp --->
Accessed on date 25th March, 2009.
13. Riccardo Rebonato. (2002) p. 268. Modern Pricing of Interest-Rate Derivatives: The Libor Market Model and Beyond, Princeton University Press.
14. Jamshidian, F. (1995) volume 2, p. 61-72. A simple class of square-root interest rate
models, Applied Mathematical Finance.
15. Björk, T., (1998). Arbitrage Theory in Continuous Time, Oxford University Press New York.
16. Pintoux, C., Privault, N., (2009). The Dothan pricing model revisited, Preprint.
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