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d2 = d1 âˆ’ Ïƒ T . 1
+ âˆ’Î´ + Ïƒ 2 T ,
2 As T â†’ âˆž, we have that
eâˆ’Î´T N (d1 ) â‰¤ eâˆ’Î´T â†’ 0,
d2 â†’ âˆ’âˆž â‡’ N (d2 ) â†’ 0.
Hence,
VC (0, T ) â†’ 0, as T â†’ âˆž. 4 b. In this case, the price of the call option reads as
VC (0, T, r) = Seâˆ’Î´T N (d1 ) âˆ’ Keâˆ’rT N (d2 ),
with
1
S (0)
d1 = âˆš
ln
K
ÏƒT
âˆš
d2 = d1 âˆ’ Ïƒ T . 1
+ r âˆ’ Î´ + Ïƒ2 T ,
2 As T â†’ âˆž, we have that
eâˆ’Î´T N (d1 ) â‰¤ eâˆ’Î´T â†’ 0,
d2 â†’ âˆ’âˆž â‡’ N (d2 ) â†’ 0.
Since the function N is bounded between 0 and 1, we see that as T â†’ âˆž, VC (0, T, r) â†’
0.
c. Until the call option is exercised, the owner of the option can earn interest on the
strike price which he/she can invest at the riskfree rate. However, in forfeiting the
physical ownership of the asset, he/she also forfeits the possible dividend payments.
In part b., the interest rate was reintroduced, but it was still much smaller than the
dividend yield. In both cases, the longer the life of the option, the more dividend
payments are forfeited and the value of the option itself becomes negligible. It would
be interesting to look at the above problem if we assume Î´ << r and let T â†’ âˆž. 5 2. Consider the following BlackDermanToy interestrate tree modeling the future evolution
of annual eï¬€ective interest rates. The periodlength is one year. 0.06 0.055 0.05 0.05 rd rdd (i) (5 points) Calculate the interest rate denoted in the tree by rdd .
(ii) (5 points) Assume that the volatility of the eï¬€ective interest rates for the second year
equals Ïƒ1 = 0.1. Calculate rd in the above tree.
(iii) (10 points) Consider a twoyear interest rate cap with the notional amount of $1,000 and
an annual eï¬€ective cap rate of KR = 0.05. Calculate the price of this interest rate cap.
Solution:
(i) According to the BlackDermanToy model, we have
0.06
0.05
=
0.05
rdd â‡’ rdd = 0.052
= 0.0417.
0.06 (ii) Again, in the BDT tree, we have
ru = rd e2Ïƒ1 â‡’ rd = 0.055eâˆ’2Â·0.1 = 0.045. 6 (iii) This is a two year cap, so there is only one node at which payment can ta...
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This note was uploaded on 08/04/2013 for the course M 339W taught by Professor Cudina during the Fall '12 term at University of Texas at Austin.
 Fall '12
 CUDINA
 Math

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