Term Structure Models

Victor Lapshin, PhD, research fellow at Higher School of Economics,

Moscow, Russia

What is “Yield”?

• Term structure of interest rates.• [Zero-Coupon] [Spot] Yield Curve.• Discount function.

• We apply the terms “Yield” and “Interest Rate” interchangeably to either a loan interest rate or a deposit yield.

Different kinds of “Yield”

• Simple.• Compounded:– Annually.– Semiannually.– Quarterly.– Monthly.… etc.

• Continuous.

Expressed in terms of:• Per annum.• Per period.• “Pure” percent.

Day count convention:• 30/360.• 30/365.• Actual/365.• Actual/Actual.

etc…

Examples of Yields

• $100 at 01.01.2000 under “5% yield” will become at 01.01.2001:– $105.148 if it is 5% per annum compounded semiannually

under Act/360.– $121.551 if it is just 5% compounded quarterly.– $105.142 if it is 5% per annum continuously compounded

under Act/365.– $105.014 if it is 5% per annum under Act/365.– $105.000 if it is simply 5% gross.

Wealth Factor & Discount Factor

• Wealth Factor: a unit initial sum becomes w(t) in time t.

• Discount Factor: d(t) becomes 1 in time t.

• Both offer a sound basis for comparison.

)(

1)(

twtd

Subjectivity of d and w

• Wealth and discount factors depend on:– Whether we lend or borrow money.– Contract details.– Counterparty’s and our own credit risk.– Embedded options.

…etc.

• We need a way to objectively price instruments promising future cash flows.

Present Value

• Given a description of a financial instrument determine its “present value”.

• The problem admits different solutions.• We need some basic assumptions.

Our Basic Assumptions

• All promised cash flows will happen (no risk).• Only promised cash flows matter.• The present value is unique.• The PV is additive:

• PV(A and B) = PV(A) + PV(B).

• No trading restrictions and absolute liquidity:• PV(-A) = -PV(A).

• No taxation.

THE Present Value

• Under the above assumptions the present value of an instrument promising N+1 cash flows at times is

• d(t) is the discount function: the PV of a unit cash flow in the time t.

• Note: this is not exactly the case. Find a counterexample and a “mild technical condition” to ensure the above solution.

N

kkk tdF

1

)(

NkFk ,...,0 , Niti ,...,0 ,

The Discount Function

• Suppose thatPresent Value = Opportunity Cost.

• There is an “ideal market” where one can invest any amount (possibly, negative) of money for any term (equiv. bonds of all maturities and face values are traded).

• Then d(t) is the current price of a bond with time t to maturity and unit face value.

The Problem

• Find the discount function.• Available data:– Bond prices.– Bond descriptions.

• Easy?• Do we really observe PV?• How do the observed values correspond to the

PV?• What really are the observed values?

Formalizations• What is a “bond price”?– Closing price.• What if it wasn’t traded?• What if we desire an intraday estimation?

– Last trade price.• What if no trades?• What if trades are unreliable (very small volume)?• Timing issues.

– Bid & Ask quotes• Is “Price” a single number?• What if only one quote is present?• What if they are unreliable?

Dynamic & Static Models

• Timing issues:– Bond A was priced at 12:00.– Bond B was priced at 13:00.– Were they both priced with respect to the same

“true” discount function?

• Does the discount function change over time? How? More assumptions.

• We assume the static model.

14

The Problem

• Suppose we really observe the PV.• Zero-coupon bonds:

• Coupon-bearing bonds:

• The system is underdetermined.• Discount function values are desired in intermediate

points.

( )k k kP N d t

n

i ikik tdFP0 , )(

FdP )( ii tdd

More Assumptions

• We have too little data available => we need more a priori assumptions.

• Different assumptions lead to different problem formulations and to different results.– On the exact parametric form of the discount

function (parametric methods).– On some extreme properties of the discount

function (spline methods).

Splines I

• Suppose we have to draw a curve connecting N points.

• Find function such that

• We need some additional assumptions to identify

)(f

ii yxf )(

)(f

Splines II

• Extreme property: minimal potential energy.

• Solution:

min)('' 2 dxxfk

],[

,)(

1

23

ii

iiii

xxx

dxcxbxaxf

continuous is )(' f

Word Usage

• If we postulate spline nature of the discount function, it is a parametric method (we assume a specific functional form).

Yield Curve

• Assume continuous compounding:

• t is the time: [t] = year.• r(t) is the continuously compounded yearly

interest rate: [r(t)] = 1 / year.

ttretd )()(

A Dumb Example: Bootstrapping

• Assume we observe the PV.• Assume that the interest rate is constant

between maturities of the bonds in the dataset:

– Find r1 from (unique if F > 0).

– Find r2 from

– Find r3 etc.

],[ ,)( 1 kkk TTtrtr

1 1,11

Tt

i

tri

i ieFP

21 21 1,2,22

TtT

i

tri

Tt

i

tri

i ii i eFeFP

Bootstrapping

Pro• Very simple. Very fast.• Exactly replicates observed

prices.

Contra• Discontinuous interest rates.• Exactly replicates observed

prices.• May fail (result in negative

interest rates).• No economic sense in rates

for large t’s.

A Simple Example: Nelson-Siegel

• Assume that prices are observed with errors.• Assume that the spot forward rate has the

form

• Spot forward rate

t

et

tf

210)(

t dxxfetd 0 )()(

Nelson-Siegel Model Estimation

• Now

• Use Nonlinear Least Squares to obtain parameter values:

),,,()( 210 ii dtd

N

ki

n

i kik dFP1

,,,

2

2100 ,210

min),,,(

Nelson-Siegel

Pro• Relatively Simple. Relatively

fast.• Sensible yield curve shapes.• Stable.• Ready for extensions.

Contra• Negative spot forward rates

are possible.• Arguable economic intuition

behind the parametric form.• Inflexible yield curve shape.• Fixed inter-temporal yield

correlations (due to fixed parametric form).

• Incapable of accommodating complicated term structures of interest rates.

An Example of a Spline Method

• We choose to model the spot forward rate f(t).

• To ensure that f(t)>0 we let f(t) = g2(t) for some unknown g( ).∙

• We require that the solution g( )∙ satisfy the maximal smoothness condition:

max

0 )(

2 min)('T

gdxxg

Spline Example II

• We suppose that the given bond prices are the PVs observed with independent errors.

• We suppose that the standard deviation of the observation error is equal to half the bid-ask spread:

• wherekk

n

ikkik wtdFP

0, )(

2kk

k

BidAskw

)1,0(~ Nk

Spline Example III

• So we have the following problem:• Find function g with such that

• Multiple criteria optimization problem.

],0[' max2 TLg

max

0

2

2

10

2

0,2

min)('

min)(exp1

T

N

k

t

x

n

ikik

k

dxxg

dxxgFPw

k

Spline Example IV

• Assign weight α to the smoothness criterion.

• Tikhonov regularization.• Variational calculus / Optimal control

problem.

min)('

)(exp1

max

0

2

2

10

2

0,2

T

N

k

t

x

n

ikik

k

dxxg

dxxgFPw

k

Spline Example V

• The solution:

for

• The coefficients have to be determined via nonlinear optimization.

0,

0,cossin

0,expexp

)(

211

211

211

ii

iiiii

iiiii

CttC

ttCttC

ttCttC

tf

)0(')0(' ),0()0( ],,[ 1 iiiiii tftftftfttt

niC ii ,...,1 ,,2,1

Sinusoidal-Exponential Splines

Pro• Non-negative spot forward

rates.• Flexible smoothness /

precision interplay.• Capable of replicating

various yield curve shapes.

Contra• Sophisticated formulae:

difficult to understand and do algebra & calculus.

• Requires nonlinear optimization to fit the model to data.

Timing Issues

• What if the price data correspond to different times?

• Has the discount function (the yield curve) changed since last price observation?

• How?• We need either:– A way to ensure that all prices correspond to the

same snapshot of the market (snapshot models).– A model of yield curve time dynamics (dynamic

models).

Snapshot Models

• Tend to be very unstable over time in illiquid environment:

Snapshot Models

• One bond just hasn’t happened to be traded this particular day.

Snapshot Models

Pro• Simple. Fast. Tractable.• Works with a snapshot

(there are applications where this is just what is required).

Contra• Unstable in time for illiquid

markets.• Meaningless results if no

long-term (or short-term) bonds are traded.

• Need to supply full set of data.

• Unfit for derivatives pricing: arbitrage opportunities.

Dynamic Models

• We need a way to describe the stochastic dynamics of the entire yield curve.– Simplify the problem.– Employ “higher maths”.

Simplified Dynamic Models

• Model only finite number of key parameters:– Instantaneous (spot) interest rate.– Spot and long rates.– Yields for several key terms.– Several general factors.…etc.

• The yield curve (all forward rates) may be derived from the stochastic dynamics of parameters (even if it’s a single spot rate).– Forward rate = “expected” future rate (have to

consider risk-neutrality issues).

Simple Models

Dynamic -> static• Low-dimensional dynamic

models imply non-realistic zero-coupon yield curves: negative or infinite.

Static -> dynamic• Consistency problems:

arbitrage opportunities.

Consistency Problems

• Very few static models may be embedded into a stochastic dynamic model in an arbitrage-free manner (Bjork, Christensen, 1999, Filipovic, 1999).

• Nelson-Siegel model allows arbitrage with every non-deterministic parameter dynamics.

0 1 2( ) t t

x x

t t t tt

xr x e e

Consistency Problems II

• Nearly all arbitrage-free dynamic models are primitive.

• All such models are affine (Bjork, Christensen, 2001, Filipovic, Teichmann, 2004).

01

,( ) ( ) ( )N

t i i ti

r x h x Y x

Simple Dynamic Models

• Vasicek (Vašíček, pronounced “Washeeczech”) model:– Ornstein-Uhlenbeck process: tractable Gaussian

distribution. Negative rates possible.• Cox-Ingersoll-Ross (CIR) model:

– Non-central χ2 distribution: semitractable, positive rates.

• Both models exhibit awful yield curve shapes.

ttt dWdtrkdr )(

tttt dWrdtrkdr )(

Simple Dynamic Models

Pro• Simple. Dynamic. • Sometimes tractable.• Simple simulation.• Relatively simple parameter

inference.

Contra• Incompatible with snapshot

models (unrealistic yield curve shapes).

• Unrealistic.• Inflexible.

Further Developments

• Models calibrated to the yield curve.– Hull, White (1990-1994): – Ho, Lee (1986); Black, Derman, Toy (1990), etc.

• General multidimensional affine models.• Whole yield curve models and market models.– Heath, Jarrow, Morton (1992).– Brace, Gatarek, Musiela (1997).– Filipovic (1999).

ttt dWdtr(t)kdr )(

Whole Yield Curve Models

• Heath-Jarrow-Morton (1992).– Consider a yield curve r(t) represented as a set of

individual forward rates: f(t, T), t – current (calendar) time, T – time of maturity.

– For each individual forward rate consider individual dynamics:

– Infinite number (indexed by T) of stochastic differential equations.

N

s

st

s dWTtdtTtTtdf1

),,(),,(),(

Whole Yield Curve Models II

• Musiela parametrisation (1994):

• No-arbitrage condition requires

• One infinite-dimensional SDE instead of many one-dimensional.

N

s

st

st dWdtdf

1

),(),()(

N

s

x ss dxxfx1

0)()()(')(

vlapshin@hse.ru