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A supermartingale approach to interest rates

Cezar Chirila

February 3, 2010

Cezar Chirila () A supermartingale approach to interest rates February 3, 2010 1 / 23

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Motivation

We are interested in developing an interest rate model, and for that weneed to model the prices of zero bonds for all maturities T .The main properties that the model focuses on is the absence of arbitrage,the completion of markets and the non-negative interest rate assumptions.


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Notations

Denote by B (t ,T ) the price of a zero bond with maturity T at time

t ≤ T .We consider the prices of the zero bonds as stochastic processesdefined on a filtered probability space (Ω,F , (F t )t ≥0,P).


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Arbitrage

An arbitrage is a self financing strategy θ such that the initial cost isnegative (V 0(θ) ≤ 0) and there exists some t > 0 such that V t (θ) ≥ 0

P − a.s and P(V t (θ) > 0) > 0 (By V t (θ) we have denoted the value attime t of the portfolio θ). Note that the notion of an arbitrage refers tothe class of probabilities equivalent to P.


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Numeraire

A numeraire N = (N t )t ≥0 is any strict positive value process of someself-financing strategy. Given a numeraire N we call a probability measureQN ≡ P an equivalent martingale measure for the numeraire N if allprimary security price processes expressed in numeraire units are

QN -martingales, that is, the processB (t ,T )

N t

t ≤T

is a QN -martingale.


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Theorem

Theorem : The condition for the absence of an arbitrage: if there exists anumeraire pair (N ,QN ), then there is no arbitrage in the set of admissiblestrategies.

Because we are interested in a model with no arbitrage opportunities, withrespect with the above theorem we need to impose the conditions on thezero bond prices:

B (t ,T ) = N t · E QN 1

N T |F t


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Particular case: Short rate models

Particular cases of this construction are the short rate models, which weget by considering

N t = e t

0 r s ds

Then B (t ,T ) = E (e − T t r s ds |F t ) and in case we consider r s as a Markov

process, B (t ,T ) = E (e − T t r s ds |σ(r t )).


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Non-negative interest rate

Another property that we want to introduce in our model is the fact thatfor two maturity dates, T < S , we should have B (t ,T ) ≥ B (t ,S ),equivalent with a non-negative interest rate assumption. That gives us the

relation:

N t E QN

1

N T |F t

≥ N t E QS

1

N S |F t


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Non-negative interest rate

By replacing t with T , we obtain:

1

N T ≥ E QN 1

N S |F T

To conclude, for our model to consider only non-negative interest rates, wemust have that

1N

t

is a QN -supermartingale.


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The general model

The most general case of constructing a model is by starting with anumeraire pair, and obtaining arbitrage-free prices of zero bonds from the

ecuationB (t ,T ) = N t · E QN

1

N T |F t

.


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The Rational Log Normal model

One way in which we can get the numeraire with the above properties is toconsider

1

N t = M t · g (t ) + f (t )

whereg ,f

are deterministic decreasing functions, andM

t is a positivemartingale. We consider a Brownian motion W , the filtration F W

generated by it, and the martingale M t as being:

M t = e t

0 σ(s )dW s −12

t 0 σ(s )2dt

.


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Zero bond prices

A zero bond price is given by:

B (t ,T ) = N t · E QN 1

N T |F t =

=1

M t g (t ) + f (t )E (M T g (T ) + f (T )|F t ) =

M t g (T ) + f (T )

M t g (t ) + f (t ).


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Caplet on Libor

We can use this model to get closed form formulas for options on theinterest rate. For example, let us consider caplet on Libor L from T 1 to

T 2, which pays ∆ max(L− K , 0) on the maturity date T 2, where∆ = T 2 − T 1 on the corresponding day convention, e.g. act /360.


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Caplet on Libor

On time T 1, the caplet has the discounted value:

∆(L− K )+

1 + ∆L= 1 −

1 + ∆K

1 + ∆L

+

= (1 + ∆K ) 1

1 + ∆K − B (T 1,T 2)

+

.Therefore, we can consider a caplet as a put on a zero bond. So we willfocus on giving a close price formula for the puts and calls on the zero

bonds.


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Pricing calls

Giving the arbitrage-free context, the price for a call on the bondB (T 1,T 2), payed at T 1, will be given by:

N 0 · E QN

(B (T 1,T 2) − K )+

N T 1|F 0


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Pricing calls

V 0((B (T 1,T 2) − K )+) =1

f (0)+g (0)· (M

T 1g (T

1) + f (T

1)) · E Q

N M T 1g (T 2)+f (T 2)

M T 1g (T 1)+f (T 1)− K +

=1

f (0)+g (0) · E QN (M T 1 · [g (T 2) − K · g (T 1)] + [f (T 2) − K · f (T 1)])+


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Notations

Let G = g (T 2) − K · g (T 1), F = f (T 2) − K · f (T 1) and R = 1f (0)+g (0) .

For the case where F G < 0, consider y is given by

e y T 1

0σ

(s )2

ds −

1

2 T 1

0σ

(s )2

ds G + F = 0, so y =ln −F

G + 1

2 T 1

0σ(s )2ds T 1

0 σ(s )2ds

We use d 1 =ln F −G − 1

2

T 10 σ(s )2ds

T 1

0 σ(s )2ds and d 2 =

ln F −G

+ 12

T 10 σ(s )2ds

T 1

0 σ(s )2ds .


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Pricing calls

We distinguish 4 cases depending on the sign of F and G .

F > 0,G > 0 V 0((B (T 1,T 2) − K )+) = R · (G + F )

F > 0,G < 0 V 0((B (T 1,T 2) − K )+) = R · (G · N (d 1) + F · N (d 2)F < 0,G > 0

F < 0,G < 0 V 0((B (T 1,T 2) − K )+) = 0


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Pricing calls

Translated back to the strike of our call, we get also a natural financialexplanation:

Strike Call price

K > max f (T 2)f (T 1) ,

g (T 2)g (T 1) R · (G + F )

max f (T 2)f (T 1) ,

g (T 2)g (T 1) > K > min f (T 2)

f (T 1) ,g (T 2)g (T 1) R (GN (d 1) + FN (d 2))

K < min f (T 2)f (T 1) ,

g (T 2)g (T 1) 0


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Pricing calls

The price of the bond B (T 1,T 2) is given byM T 1g (T 2)+f (T 2)

M T 1g (T 1)+f (T 1) and for fixed

T 1 and T 2 we get

max f (T 2)f (T 1)

,g (T 2)g (T 1)

> B (T 1,T 2) > minf (T 2)f (T 1)

,g (T 2)g (T 1)


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Pricing calls

The results say the price will be a constant on time if the strike is chosensuch that the call is always in the money

(K > max f (T 2)f (T 1) ,

g (T 2)g (T 1) > B (T 1,T 2)) or out of the money

(K < min f (T 2)f (T 1) ,

g (T 2)g (T 1) < B (T 1,T 2)).


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Pricing calls

Furthermore, our model implies the boundness of the bond prices, andfrom the relation

r (t ,T ) = − lnB (t ,T )T − t

we get that the interest rates are also bounded.


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Levy process

We will continue further with a generalized method of obtaining themartingale M t . We will consider a Levy process X , and set

M t =e σX t

Ee σX t

By considering X t = W t , a Brownian motion, we particularize for the case

of Rational Log Normal model, M t = e σW t −12σ

2t .


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