Fixed Income Securities and Interest Rate Modelling | Assignment

Fixed Income Securities and
Interest Rate Modelling 1
FIXED INCOME SECURITIES AND INTEREST RATE MODELLING
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Fixed Income Securities and
Interest Rate Modelling 2
Fixed Income Securities and Interest Rate Modelling
1.
a. Expected 6 month treasury rate
Po=97.4845, Pu = 0.5 Pd = 0.5
When r1=r1 ,u P1 , u=100 e−0.04 × 0.5=98.0199
The return then is 98.0199
97.4845 −1=0.00549
When r1=r1 ,d P1 ,d =100 e−0.01 ×0.5 =99.5012
The return here is 99.5012
97.4845 −1=0.0207
E [ r1 ]= ( 0.00549 × 0.5+0.0207 ×0.5 )
¿ 0.013095=1.31%
b. Market price of risk
λ= e−r0
E [ P1 ] −P0
P1 , u−P1 ,d
E [ P1 ]=99.5012 ×0.5+ 98.0199× 0.5
¿ 98.76055
λ= e−0.02× 0.5 98.76055−97.4845
98.0199−99.5012
¿−0.1980

Fixed Income Securities and
Interest Rate Modelling 3
c. Risk neutral probability for an upward movement in the interest rate
λ¿= e−ro E¿ [ P1 ] −Po
P1 ,u −P1 ,d
=0
E¿=P u¿ P1 ,u−P d¿ P1 , d∧P d¿+ P u¿=1∧0< Pu¿<1 ,
P d¿=1−P u¿
Hence Po=e−ro E¿ [ P1 ]
Solving, 98.76055=e−0.02 ×0.5 ( 98.0199+99.5012−99.5012 )
P u¿=98.76055 e−0.02 × 0.5−99.5012
98.0199−99.5012
¿ 0.1667
P d¿=1−0.1667=0.8333
d. Option
i. Price of option
r1=0.04 , thus0.04−0.02=0.02
max { 0.02 , 0 }=0.02
payoff of A=100× 0.02
¿ 2
ii. Replicating portfolio
Using replication
Payoff ¿ 2 with price=99.005
the 2 period has 98.0199∧99.5012 with price=97.4845