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Solution of Derivative

Therefore we receive the high yield instrument and if

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Unformatted text preview: alculate at the previous nodes: delta B Put premium value of early exercise t=0 -0.6229 82.1175 7.37 0 node d -0.8870 110.7413 11.602 8.2322 node u -0.3939 52.3571 2.9372 0 We can see that the American and the European put option must have the same price, since it is never optimal to exercise the American put option early. The price of the put option is 7.37. c) The benefit of early exercise for a put option is to receive the strike price earlier on and start earning interest on it. The cost associated with early exercising a put is to stop earning income on the asset we give up. In this case, the strike is 120 Yen, and the Yen interest rate is not very favorable compared to the dollar interest rate. We would give up a high yield instrument and receive a low yield instrument when we early exercise the put option. This is not beneficial, and it is reflected by the non-optimality of early exercise of the put option. 70 Chapter 10 Binomial Option Pricing: 1 For the call option, the opposite is true: When exercising the call option, we receive a dollar and give up 120 Yen. Therefore, we receive the high-yield instrument, and if the exchange rate moves in our favor, we want to exercise the option before expiration. Question 10.18. a) We have to use the formulas of the textbook to calculate the stock tree and the prices of the options. Remember that, while it is possible to calculate a delta, the option price is just the value of B, because it does not cost anything to enter into a futures contract. In particular, this yields the following prices: For the European call and put, we have: premium = 122.9537 . The prices must be equal due to put-call-parity. b) We can calculate for the American call option: premium = 124.3347 and for the American put option: premium = 124.3347 . c) We have the following time 0 replicating portfolios: For the European call option: Buy 0.5371 futures contracts. Borrow 122.9537 For the European put option: Sell 0.4141 futures contracts. Borrow 122.9537 Question 10.20. a) The price of an American call option with a strike of 95 is $24.1650. The price of an American put option with a strike of 95 is $15.2593 b) Now, we have for the American 100-strike call option a premium of $15.2593 and for the European put option a premium of $24.165. Both option prices increase as we would have expected, and the relation we observed in question 10.19. continues to hold. 71 Chapter 11 Bionomial Option Pricing: II Question 11.2. By introducing a non-zero interest rate, we increase the cost of early exercise, because we pay the strike before expiration, and lose interest on it. We see that we only exercise the call with a strike of 70. The value of the European 70-strike call is $27.69, the value of immediate exercise is $30. The decisive condition, derived from put-call-parity, is now: C = Se −δ − Ke − r + P = 100 × 0.92311 − K × 0.92311 + P = 92.31164 − 0.92311K + P Therefore, we will exercise whenever 100 − K > 92.31164 − 0.92311K + P ⇔ P < 7.688 − 0.07688 K This condition is indeed fulfilled at a strike price of 70. Clearly, this boundary is attained earlier than the boundary of exercise 11.1., so we will stop early exercise at lower strike prices when the interest rates are high. Question 11.4. a) Early exercise occurs only at a strike price of 130. The value of the one period binomial European 130 strike put is $26.38, while the value of immediate exercise is 130 – 100 = 30. b) From put-call-parity, we observe the following: P = Ke − r − Se −δ + C = K × 0.9231164 − S + C = 92.31164 K − S + C Clearly, as long as K – 100 is larger than 92.31164K – 100 + C or C < 0.07688K, we will exercise the option early. Already at a strike of 120, 0.07688*120 = 9.2256 is smaller than the value of the European call option with a strike of 120 (with a price of $10.30), which means that we do not exercise prior to expiration. c) The value of a call falls when the strike price increases. From part b), we learned that the decisive criterion was that C < 0.07688K. Therefore, if this criterion is fulfilled for some threshold K(*), it is fulfilled for every K above K(*). 72 Chapter 11 Binomial Option Pricing: II Question 11.6. We now have from put-call-parity: P = Ke − r − Se −δ + C = K − S × 0.92311 + C = 0.92311K − 92.31164 + C We would exercise early if: K − 100 > K − 92.31164 + C C < −7.688 ⇔ , which can never be true. It is never optimal to early exercise, because the sole advantage of early exercise, receiving the interest on the strike earlier, has been removed. Question 11.8. For the following questions, we will report the first two of the 10 nodes. We have for the European call options of strike 70, 80, 90 and 100: Delta and B: K=70 0.98 -62.84 0.95 -58.76 K=80 0.94 -66.81 K=90 0.88 -68.35 0.88 -59.44 0.91 -55.96 0.79 -57.40 0.81 -53.66 K=100 0.77 -63.16 0.66 -50.19 0.68 -48.32 0.54 -39.16 Call option price and gamma, the required rate of re...
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