1 2 t 2 as t we have that et n d1 et 0 d2

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Unformatted text preview: d2 = d1 − σ T . 1 + −δ + σ 2 T , 2 As T → ∞, we have that e−δT N (d1 ) ≤ e−δT → 0, d2 → −∞ ⇒ N (d2 ) → 0. Hence, VC (0, T ) → 0, as T → ∞. 4 b. In this case, the price of the call option reads as VC (0, T, r) = Se−δT N (d1 ) − Ke−rT N (d2 ), with 1 S (0) d1 = √ ln K σT √ d2 = d1 − σ T . 1 + r − δ + σ2 T , 2 As T → ∞, we have that e−δT N (d1 ) ≤ e−δT → 0, d2 → −∞ ⇒ N (d2 ) → 0. Since the function N is bounded between 0 and 1, we see that as T → ∞, VC (0, T, r) → 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 risk-free 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 δ << r and let T → ∞. 5 2. Consider the following Black-Derman-Toy interest-rate tree modeling the future evolution of annual effective interest rates. The period-length 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 effective interest rates for the second year equals σ1 = 0.1. Calculate rd in the above tree. (iii) (10 points) Consider a two-year interest rate cap with the notional amount of $1,000 and an annual effective cap rate of KR = 0.05. Calculate the price of this interest rate cap. Solution: (i) According to the Black-Derman-Toy model, we have 0.06 0.05 = 0.05 rdd ⇒ rdd = 0.052 = 0.0417. 0.06 (ii) Again, in the BDT tree, we have ru = rd e2σ1 ⇒ rd = 0.055e−2·0.1 = 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.

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