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qu¨¨
¨¨
¨¨
r1,1 ¨ d r2,1 ¨ r3,1 ¨
¨ q ¨¨
¨
¨¨
¨¨
¨
¨¨
r0,0 ¨ r1,0 ¨ r2,0 ¨ r3,0 ¨
¨
¨
¨
¨
r3,3 t=0 t=1 t=2 t=3 t=4 Let Zi ,j be the date i , state j price of a noncoupon paying security.
Will use riskneutral pricing to price every security so that:
Zi ,j = 1
[qu × Zi +1,j +1 + qd × Zi +1,j ]
1 + ri ,j (1) – where qu and qd are the riskneutral probabilities of an up and downmove
– so qd + qu = 1 and must have qd > 0 and qu > 0. There can be no arbitrage when we price using (3). Why?
6 Binomial Models for the Short Rate
If the security pays a “coupon”, Ci +1,j , at date i + 1 and state j then
Zi ,j = 1
[qu (Zi +1,j +1 + Ci +1,j +1 ) + qd (Zi +1,j + Ci +1,j )]
1 + ri ,j (2) – where Zi +1,. is now the excoupon price at date i + 1. If we use (3) or (2) to price securities in the lattice model then arbitrage is
not possible
 Regardless of what probabilities we use! Why is this? In fact it is very common to simply set qu = qd = 1/2
 and to calibrate other parameters to market prices. We will assume qu = qd = 1/2 in our examples.
7 Financial Engineering & Risk Management
The Cash Account and Pricing ZeroCoupon Bonds M. Haugh G. Iyengar Department of Industrial Engineering and Operations Research
Columbia University Binomial Models for the ShortRate
¨¨
¨
¨¨
¨
¨
¨
r2,2 ¨ r3,2 ¨
¨
¨
¨
qu¨¨
¨¨ ¨¨
r1,1 ¨ d r2,1 ¨ r3,1
q
¨
¨
¨
¨¨ ¨¨¨ ¨¨¨
¨¨
r0,0 ¨ r1,0
r2,0
r3,0 ¨
¨
¨
¨
¨
r3,3 t=0 t=1 t=2 t=3 t=4 We use riskneutral pricing to price every noncoupon paying security:
Zi ,j = 1
[qu × Zi +1,j +1 + qd × Zi +1,j ]
1 + ri ,j (3) – qu > 0 and qd > 0 are the riskneutral probabilities of an up and
downmove, respectively, of the shortrate.
There can be no arbitrage when we price using (3). Why?
2 The CashAccount
The cashaccount is a particular security that in each period earns interest at
the shortrate
 we use Bt to denote its value at time t and assume that B0 = 1. The cashaccount is not riskfree since Bt +s is not known at time t for any
s>1
 it is locally riskfree since Bt +1 is known at time t . Note that Bt satisﬁes Bt = (1 + r0,0 )(1 + r1 ) . . . (1 + rt −1 )
 so that Bt /Bt +1 = 1/(1 + rt ). Riskneutral pricing for a “noncoupon” paying security then takes the form:
Zt ,j =
=
= 1
[qu × Zt +1,j +1 + qd × Zt +1,j ]
1 + rt ,j
Zt +1
EQ
t
1 + rt ,j
Bt
EQ
Zt +1
t
Bt +1 (4)
3 RiskNeutral Pricing with the CashAccount
Therefore for a noncoupon paying security, (4) is equivalent to
Zt
Zt +1
= EQ
t
Bt
Bt +1 (5) We can iterate (5) to obtain
Zt +s
Zt
= EQ
t
Bt
Bt +s (6) for any noncoupon paying security and any s > 0. 4 RiskNeutral Pricing with the CashAccount
Riskneutral pricing for a “coupon” paying security takes the form:
Zt ,j 1
[qu (Zt +1,j +1 + Ct +1,j +1 ) + qd (Zt +1,j + Ct +1,j )]
1 + rt ,j
Zt +1 + Ct +1
(7)
= EQ
t
1 + rt ,j
= We can rewrite (7) as
Zt
Ct +1
Zt +1
= EQ
+
t
Bt
Bt +1
Bt +1 (8) More generally, we can iterate...
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This note was uploaded on 01/09/2014 for the course FIN 347 taught by Professor Martinhaugh during the Fall '13 term at Bingham University.
 Fall '13
 MartinHaugh

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