Lecture on Interest Rates

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Lecture on Interest Rates


Josef Teichmann

ETH Zurich

Zurich, December 2011

Josef Teichmann Lecture on Interest Rates


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Mathematical Finance

Examples and Remarks

Interest Rate Models

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Goals

Basic concepts of stochastic modeling in interest rate theory,in particular the notion of numeraire.

No arbitrage as concept and through examples. Several basic implementations related to no arbitrage in R.

Basic concepts of interest rate theory like yield, forward ratecurve, short rate.

Some basic trading arguments in interest rate theory.

Spot measure, forward measures, swap measures and Blacksformula.

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References

As a standard reference on interest rate theory I recommend[Brigo and Mercurio(2006)].

In german language I recommend[Albrecher et al.(2009)Albrecher, Binder, and Mayer], whichcontains also a very readable introduction to interest rate theory

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Modeling of financial markets

We are describing models for financial products related to interestrates, so called interest rate models. We are facing severaldifficulties, some of the specific for interest rates, some of themtrue for all models in mathematical finance:

stochastic nature: traded prices, e.g. prices of interest raterelated products, are not deterministic!

information is increasing: every day additional information on

markets appears and this stream of information should enterinto our models.

stylized facts of markets should be reflected in the model:stylized facts of time series, trading opportunities (portfoliobuilding), etc.

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Mathematical Finance 1

A financial market can be modeled by

a filtered (discrete) probability space (,F, Q), together with price processes, namely K risky assets

(S1n , . . . , SKn )0nNand one risk-less asset S

0 (numeraire),

i.e. S0n >0 almost surely (no default risk for at least oneasset),

all price processes being adapted to the filtration.

This structure reflects stochasticity of prices and the stream ofincoming information.

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A portfolio is a predictable process = (0n, . . . , Kn)0nN, where

in represents the number of risky assets one holds at time n. The

value of the portfolio Vn() is

Vn() =Ki=0

inSin.

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Mathematical Finance 2

Self-financing portfoliosare characterized through the condition

Vn+1() Vn() =Ki=0

in+1(Sin+1 Sin),

for 0 n N 1, i.e. changes in value come from changes inprices, no additional input of capital is required and noconsumption appears.

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Self-financing portfolios can be characterized in discounted terms.

Vn() = (S0n )1Vn()Sin = (S0n )1Sin()

Vn() =

K

i=0in

Sin

for 0 n N, and recover

Vn() =

V0() + (

S) =

V0() +

nj=1

Ki=1

ij(

Sij

Sij1)

for self-financing predictable trading strategies and 0 n N.In words: discounted wealth of a self-financing portfolio is thecumulative sum of discounted gains and losses. Notice that weapply a generalized notion ofdiscountinghere, prices Si divided

by the numeraire S0

only these relative prices can be compared.8 / 5 3


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Fundamental Theorem of Asset Pricing

A minimal condition for modeling financial markets is the

No-arbitrage condition: there are no self-financing tradingstrategies (arbitrage strategies) with

V0() = 0, VN() 0

such that Q(VN() = 0)>0 holds (NFLVR).

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In the sequel we generate two sets random numbers (normalizedlog-returns) and introduce two examples of markets with constantbank account and two assets. The first market allows for arbitrage,then second one not. In both cases we run the same portfolio:

> Delta = 250

> Z = rnorm(Delta, 0, 1/sqrt(Delta))

> Z1 = rnorm(Delta, 0, 1/sqrt(Delta))

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L I R


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Incorrect modelling with arbitrage

> S = 10000

> rho = 0 > sigmaX = 0.25

> sigmaY = 0.1

> X = exp(sigmaX * cumsum(Z))> Y = exp(sigmaY * cumsum(sqrt(1 - rho^2) * Z + rho * Z1))

> returnsX = c(diff(X, lag = 1, differences = 1), 0)

> returnsY = c(diff(Y, lag = 1, differences = 1), 0)> returnsP = ((sigmaY * Y)/(sigmaX * X) * returnsX - returnsY) *

+ S

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L t I t t R t


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Plot of the two Asset prices

1.0

1.

1

1.2

1.3

X

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Plot of the value process: an arbitrage

0

20

40

60

cumsum

(returnsP)

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Correct arbitrage-free modelling

> Delta = 250> S = 10000

> rho = 0 > sigmaX = 0.25> sigmaY = 0.1

> time = seq(1/Delta, 1, by = 1/Delta)

> X1 = exp(-sigmaX^2 * 0.5 * time + sigmaX * cumsum(Z))

> Y1 = exp(-sigmaY^2 * 0.5 * time + sigmaY * cumsum(sqrt(1 - rho^2) *+ Z + rho * Z1))

> returnsX1 = c(diff(X1, lag = 1, differences = 1), 0)

> returnsY1 = c(diff(Y1, lag = 1, differences = 1), 0)> returnsP1 = ((sigmaY * Y1)/(sigmaX * X1) * returnsX1 - returnsY1) *

+ S

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Plot of two asset prices

1.0

1.1

1.2

X1

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Plot of the value process: no arbitrage

10

8

6

4

2

0

cumsum

(returnsP1)

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Theorem

Given a financial market, then the following assertions areequivalent:

1. (NFLVR) holds.

2. There exists an equivalent measure P Q such that thediscounted price processes are P-martingales, i.e.

EP( 1

S0NSiN|Fn) =

1

S0nSin

for0 n N.

Main message: Discounted (relative to the numeraire) pricesbehave like martingales with respect to one martingale measure.

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What is a martingale?

Formally a martingale is a stochastic process such that todays best

prediction of a future value of the process is todays value, i.e.

E[Mn|Fm] =Mmform n, where E[Mn|Fm] calculates the best prediction withknowledge up to time m of the future value Mn.

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Random walks and Brownian motions are well-known examples ofmartingales. Martingales are particularly suited to describe(discounted) price movements on financial markets, since theprediction of future returns is 0. This is not the most generalapproach, but already contains the most important features. Twoimplementations in R are provided here, which produce thefollowing graphs.

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Pricing rules

(NFLVR) also leads to arbitrage-free pricing rules. Let X be thepayoff of a claim Xpaying at time N, then an adapted stochasticprocess (X) is called pricing rule for X if

N(X) =X.

(S0, . . . , SN, (X)) is free of arbitrage.

This is obviously equivalent to the existence of one equivalentmartingale measure P such that

EPX

S0N|Fn

=n(X)

S0n

holds true for 0

n

N.

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The previous framework for stochastic models of financial marketsis not bound to a discrete setting even though one can perfectlywell motivate the theory there. We shall see two examples and

several remarks in the sequel the one-step binomial model.

the Black-Merton-Scholes model.

Hedging.

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One step binomial model

We model one asset in a zero-interest rate environment just beforethe next tick. We assume two states of the world: up, down. Theriskless asset is given by S0 = 1. The risky asset is modeled by

S10 =S0, S

11 =S0 u>S0 orS

11 =S0 d

where the events at time one appear with probability qand 1 q(physical measure). The martingale measure is apparently giventhrough u

p+ (1

p)d= 1, i.e. p= 1d

ud.

Pricing a European call option at time one in this setting leads tofair price

E[(S11 K)+] =p (S0u K)++ (1 p) (S0dK)+.

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E l d R k


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Black-Merton-Scholes model 1

We model one asset with respect to some numeraire by anexponential Brownian motion. If the numeraire is a bank accountwith constant rate we usually speak of the Black-Merton-Scholesmodel, if the numeraire some other traded asset, for instance a

zero-coupon bond, we speak of Blacks model. Let us assume thatS0 = 1, then

S1t =S0exp(Bt2t

2 )

with respect to the martingale measure P. In the physical measureQa drift term can be added in the exponent, i.e.

S1t =S0exp(Bt2t

2 + t).

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E am les and Rema ks


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Black-Merton-Scholes model 2

Our theory tells that the price of a European call option on S1 attime T is priced via

E[(S

1

T K)+] =S0(d1) K(d2)yielding the Black-Scholes formula, where is the cumulativedistribution function of the standard normal distribution and

d1,2 = log S0

K

2T

2T .

Notice that this price corresponds to the value of a portfoliomimicking the European option at time T.

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Hedging

Having calculated prices of derivatives we can ask if it is possible

to hedge as seller against the risks of the product. By the veryconstruction of prices we expect that we should be able to build at the price of the premium which we receive a portfolio whichhedges against some (all) risks. In the Black-Scholes model thishedging is perfect.

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A time series of yields

AAA yield curve of the euro area from ECB webpage.

Yield curves exist in all major economies and are calculatedfrom different products like deposit rates, swap rates, zerocoupon bonds, coupon bearing bonds.

Interest rates express the time value of money quantitatively.

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Interest Rate mechanics 1

Prices of zero-coupon bonds (ZCB) with maturity Tare denotedbyP(t, T). Interest rates are governed by a market of (defaultfree) zero-coupon bonds modeled by stochastic processes(P(t, T))0tT forT 0. We assume the normalizationP(T, T) = 1. T denotes the maturityof the bond, P(t, T) its price at a

time tbefore maturity T.

The yield

Y(t, T) = 1T t log P(t, T)

describes the compound interest rate p. a. for maturity T.

f is called the forward rate curveof the bond market

P t T = ex T f t s ds 30/53




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Interest Rate mechanics 2

The short rate process is given through Rt=f(t, t) for t 0defining thebank account process

(B(t))t0 := (exp( t

0

Rsds))t0.

No arbitrage is guaranteed if on the filtered probability space(,F, Q) with filtration (Ft)t0,

E(exp( Tt

Rsds)|Ft) =P(t, T)

holds true for 0 t Tfor some equivalent (martingale)measure P.

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Simple forward rates

Consider a bond market (P(t, T))tT with P(T, T) = 1 andP(t, T)>0. Let t T T. We define the simple forward ratethrough

F(t; T, T) := 1

T T

P(t, T)

P(t, T) 1

.

and the simple spot rate through

F(t, T) :=F(t; t, T).

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ApparentlyP(t, T)F(t; T, T) is the fair value at time t of acontract payingF(T, T) at time T.

Indeed, note that

P(t, T)F(t; T, T) =P(t, T) P(t, T)

T T ,

F(T, T

) = 1

T T 1P(T, T) 1.Fair value means that we can build a portfolio at time tand atprice P(t,T)P(t,T

)TT

yielding F(T, T) at time T:

Holding a ZCB with maturity Tat time thas value P(t, T),being additionally short in a ZCB with maturity T amountsall together to P(t, T) P(t, T).

at time Twe have to rebalance the portfolio by buying withthe maturing ZCB another bond with maturity T, precisely

an amount 1/P(T, T). 33/53




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Caps

In the sequel, we fix a number of future dates

T0


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Floors

At time Tithe holder of the floor receives

( F(Ti1, Ti))+.

Let t T0. We writeFll(t; Ti1, Ti), i= 1, . . . , n

for the time tprice of the ith floorlet, and

Fl(t) =n

i=1

Fll(t; Ti1, Ti)

for the time tprice of the floor.

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Swaps

Fix a rate Kand a nominal N. The cash flow of a payer swap atTi is

(F(Ti1, Ti) K)N.

The total value p(t) of the payer swap at time t T0 isp(t) =N

P(t, T0) P(t, Tn) K

ni=1

P(t, Ti)

.

The value of a receiver swap at t

T0 is

r(t) = p(t).The swap rate Rswap(t) is the fixed rate Kwhich givesp(t) = r(t) = 0. Hence

Rswap(t) =

P(t, T0)

P(t, Tn)

n= P(t, Ti) , t [0, T0]. 36/53 Lecture on Interest Rates



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Swaptions

A payer (receiver) swaption is an option to enter a payer (receiver)swap at T0. The payoff of a payer swaption at T0 is

N(Rswap(T0) K)+n

i=1

P(T0, Ti),

and of a receiver swaption

N(K Rswap(T0))+ ni=1

P(T0, Ti).

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Spot measure

From now on, let Pbe a martingale measure in the bond market(P(t, T))tT, i.e. for each T >0 the discounted bond priceprocess

P(t, T)B(t)

is a martingale. This leads to the following fundamental formula ofinterest rate theory

P(t, T) =E(exp( Tt

Rsds))|Ft)

for 0 t T.38/53




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Forward measures

For T >0 define the T-forward measure PT

such that for any

T >0 the discounted bond price process

P(t, T)

P(t, T), t [0, T]

is a PT

-martingale.

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Forward measures

For any T


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For any time derivative X FT

paid at T

we have that the fairvalue viamartingale pricingis given through

P(t, T)ET

[X|Ft].

The fair price of the ith caplet is therefore given by

Cpl(t; Ti1, Ti) =P(t, Ti)ETi[(F(Ti1, Ti) )+|Ft].

By the martingale property we obtain therefore

ETi[F(Ti1, Ti)|Ft] =F(t; Ti1, Ti),

what was proved by trading arguments before.

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Blacks formula

Let X N(, 2) and K R. Then we have

E[(eX K)+] =e+2

2 log K ( + 2)

K log K

,

E[(K eX)+] =K

log K

e+

2

2

log K ( + 2)

.

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Blacks formula for caps and floors

Let t

T0. From our previous results we know that

Cpl(t; Ti1, Ti) =P(t, Ti)ETit [(F(Ti1, Ti) )+],

Fll(t; Ti1, Ti) =P(t, Ti)ETit [( F(Ti1, Ti))+],

and that F(t; Ti1, Ti) is an PTi-martingale.

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We assume that under PTi the forward rate F(t; Ti1, Ti) is anexponential Brownian motion

F(t; Ti1, Ti) =F(s; Ti1, Ti)

exp

1

2

ts

(u, Ti1)2du+

ts

(u, Ti1)dWTiu

fors

t

Ti1, with a function (u, Ti1).

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We define the volatility 2(t) as

2(t) := 1Ti1 t

Ti1

t

(s, Ti1)2ds.

The PTi-distribution of log F(Ti1, Ti) conditional onFt isN(, 2) with

= log F(t; Ti1, Ti) 2(t)2

(Ti1 t),2 =2(t)(Ti1 t).

In particular

+2

2 = log F(t; Ti1, Ti),

+ 2 = log F(t; Ti1, Ti) +2(t)

2 (Ti1 t).

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We have

Cpl(t; Ti1, Ti) =P(t, Ti)(F(t; Ti1, Ti)(d1(i; t)) (d2(i; t))),Fll(t; Ti1, Ti) =P(t, Ti)((

d2(i; t))

F(t; Ti1, Ti)(

d1(i; t)))

where

d1,2(i; t) = log

F(t;Ti1,Ti)

12 (t)2(Ti1 t)(t)Ti1 t

.

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Concrete calculation of caplet price

Consider the setting t= 0, T0 = 0.25y and T1 = 0.5y. Marketdata give us P(0, T1) = 0.9753,F(0, T0, T1) = 0.0503 and(u, T0) = 0.2 is constant, hence we can calculate

(t)2(T0

t) = 0.2

0.25,

and therefore by Blacks formula gives the caplet price for = 0.03

0.25 0.9753 (0.0503 ( log(0.0503) log(0.03) + 0.5 0.2 0.250.2

0.25

)

0.03 ( log(0.0503) log(0.03) 0.5 0.2 0.250.2 0.25 )),

where is the cumulative distribution function of a standardnormal random variable, which yields 0.004957.

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Exercises

Simulate a simple interest rate model:

We choose a simple interest rate model of Vasicek type, i.e.

Rt= exp(

0.2t)0.05 + 0.03

t

0

exp(

0.2(t

s))dBs.

First we simulate the bank account, i.e. we calculate thevalue B(t) for different trajectories of Brownian motion.Write an R-function called vasicek with input parameter t anddiscretization parameter n which provides the value ofB(t).Use the following iteration for this: B(0) = 1, R(0) = 0.05and

B(ti+ 1

n ) =B(t

i

n)

1+(R(t

i

n)0.2R(ti

n)

t

n+0.03

t

nN)

t

n

,

where N is a standard normal random variable. 48/53




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Bank account in the Vasicek model

> B 0 = 1

> X0 = 0.05

> b = 0 > beta = -0.2

> alpha = 0.03

> time = 1> n = 250

> m = 2 0

> x y for (j in (1:m)) {

+ B = vector(length = n + 1)+ X = vector(length = n + 1)

+ X[1]


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Bank account Scenarios with Vasicek-short-rate

0.9

0

0.9

5

1.0

0

1.0

5

1.1

0

B

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Second we take the simulation results and calculate the bondprice (or any other derivative price) by the law of largenumbers

P(0, t) =E(1/B(t)) 1mmi=1

1/B(t)(i).

Collect the result again in a function called vasicekZCB withinput parameters t, n and m. For large m we should obtainnice yield curves.

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Exam

No arbitrage theory: discounting, numeraire, martingale

measure for discounted prices, arbitrage. Notions of interest rate theory: yield, forward rate, short rate,

simple forward rate, zero coupon bond, cap, floor.

one calculation with Blacks formula in the forward measure.

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[Albrecher et al.(2009)Albrecher, Binder, and Mayer] HansjorgAlbrecher, Andreas Binder, and Philipp Mayer.

Einf uhrung in die Finanzmathematik.

Mathematik Kompakt. [Compact Mathematics]. Birkhauser

Verlag, Basel, 2009.

[Brigo and Mercurio(2006)] Damiano Brigo and Fabio Mercurio.

Interest rate modelstheory and practice.

Springer Finance. Springer-Verlag, Berlin, second edition, 2006.

With smile, inflation and credit.

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Lecture on Interest Rates

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