Solution of Derivative

80 chapter 11 binomial option pricing ii in particular

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Unformatted text preview: 0.000 1.819 0.909 3.169 0.000 0.000 0.000 1.804 79 0.000 0.000 0.000 Part 3 Options 7.713 6.711 11.194 5.414 10.263 15.742 3.582 9.006 15.146 21.342 0.000 7.110 14.420 21.373 27.745 14.114 21.762 28.506 34.444 29.575 35.580 40.857 42.080 46.757 52.196 American Put 0.000 0.000 0.000 0.000 0.000 0.458 4.353 7.979 0.909 3.201 5.477 10.548 16.468 0.000 0.000 0.000 3.582 9.131 15.651 22.509 0.000 0.000 1.804 6.870 11.637 0.000 0.000 1.835 0.000 0.000 0.000 7.110 14.668 22.254 29.575 14.114 22.254 29.575 36.159 29.575 36.159 42.080 42.080 47.406 52.196 Question 11.18. We chose the stock of IBM from June 1st 1997 to May 31st 2002 (http://finance.yahoo.com). Please note that we must calculate the continuously compounded returns before we can calculate the weekly standard deviation. (c.f. table 11.1 of the textbook). We obtain the annual standard deviation by multiplying our weekly estimate by the square root of 52. 80 Chapter 11 Binomial Option Pricing: II In particular, this yields: period 97/6 – 02/05 weekly annual 0.050 0.360 97/6 – 98/5 98/6 – 99/5 99/6 – 00/5 00/6 – 01/5 01/6 – 02/5 0.040 0.049 0.052 0.063 0.040 0.287 0.355 0.376 0.457 0.287 97/6 – 97/11 97/12 – 98/5 98/6 – 98/11 98/12 – 99/5 99/6 – 99/11 99/12 – 00/5 00/6 – 00/11 00/12 – 01/5 01/6 – 01/11 01/12 – 02/5 0.039 0.041 0.043 0.056 0.055 0.050 0.060 0.067 0.037 0.041 0.279 0.298 0.311 0.401 0.395 0.364 0.432 0.486 0.268 0.297 We can observe a time trend in the volatility estimate: Volatility is rising throughout the nineties, up to a record level of over 45 percent in 2000/2001. Only the last year of data shows a reversal in the volatility. There does not seem to be an additional pattern to be detected when we conduct the semiannual volatility estimates. 81 Part 3 Options Question 11.20. We will use the methodology introduced by Hull, which is described in the main textbook. We can calculate: u = 1.2005 d = 0.8670 p = 0.4594 n = 4.0000 S = 50.0000 F = 46.0792 t = 1.0000 h = 0.2500 American Call 95.7188 50.7188 79.7304 35.6215 69.1230 66.4127 23.1772 24.1230 59.3195 57.5771 14.3195 13.4682 50.0000 47.9597 8.4551 7.2380 49.9170 4.9170 43.3480 41.5791 3.7876 2.2141 34.6340 36.0474 0.9970 0.0000 30.0263 0.0000 26.0315 0.0000 82 K= dividend = r= sigma = time to div 45.00 4.0000 0.0800 0.3255 0.2500 Chapter 11 Binomial Option Pricing: II European Call 95.7188 50.7188 79.7304 35.6215 66.4127 69.1230 23.1772 24.1230 59.3195 57.5771 14.2721 13.4682 50.0000 47.9597 49.9170 8.4338 7.2380 4.9170 43.3480 41.5791 3.7876 2.2141 34.6340 36.0474 0.9970 0.0000 30.0263 0.0000 26.0315 0.0000 83 Chapter 12 The Black-Scholes Formula Question 12.2 N 8 9 10 11 12 Call 3.464 3.361 3.454 3.348 3.446 Put 1.718 1.642 1.711 1.629 1.705 The observed values are slowly converging towards the Black-Scholes values of the example. Please note that the binomial solution oscillates as it approaches the Black-Scholes value. Question 12.4 a) T 1 2 5 10 50 100 500 Call Price 18.6705 18.1410 15.1037 10.1571 0.2938 0.0034 0.0000 The benefit to holding the call option is that we do not have to pay the strike price and that we continue to earn interest on the strike. On the other hand, the owner of the call option foregoes the dividend payments he could receive if he owned the stock. As the interest rate is zero and the dividend yield is positive, the cost of holding the call outweighs the benefits. b) T 1 2 5 10 50 100 500 Call Price 18.7281 18.2284 15.2313 10.2878 0.3045 0.0036 0.0000 84 Chapter 12 The Black-Scholes Formula Although the call option is worth marginally more when we introduce the interest rate of 0.001, it is still not enough to outweigh the cost of not receiving the huge dividend yield. Question 12.6 a) Using the Black-Scholes formula, we find a call-price of $ 16.33. b) We determine the one year forward price to be: F0,T(S) = S * exp(r*T) = $ 100 * exp(0.06*1) = $ 106.1837 c) As the textbook suggests, we need to set the dividend yield equal to the risk-free rate when using the Black-Scholes formula. Thus: C(106.1837, 105, 0.4, 0.06, 1, 0.06) = $ 16.33 This exercise shows the general result that a European futures option has the same value as the European stock option provided the futures contract has the same expiration as the stock option. Question 12.8 a) We have to be careful here: Now we have to take into account the dividend yield when calculating the 9-month forward price: F0,T(S) = S * exp((r-delta)*T) = $ 100 * exp((0.08-0.03)*0.75) = $ 103.8212. b) Having found the correct forward price, we can use equation (12.7) to price the call option on the futures contract: C(103.8212, 95, 0.3, 0.08, 0.75, 0.08) = $ 14.3863 c) The price we found in part b) and the prices of the previous question are identical. 12.7a, 12.7b and 12.8b are all based on the same Black-Scholes formula, only the way in which we input the variables differs. Question 12.10 Time decay is measured by the greek letter theta. We will show in the following that the statement of the ex...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.

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