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A supermartingale approach to interest rates
Cezar Chirila
February 3, 2010
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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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P i i ll
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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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P i i ll
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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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P i i ll
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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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P i i ll
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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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L ss
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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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