M11_MCDO8122_01_ISM_C11 - Chapter 11 The Black-Scholes...

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Chapter 11 The Black-Scholes Formula ± Question 11.1 You can use the NORMSDIST function of Microsoft Excel to calculate the values for 1 () Nd and 2 . NORMSDIST( z ) returns the standard normal cumulative distribution evaluated at z . Here are the intermediate steps towards the solution: 1 d 0.3730 2 d 0.2230 1 0.6454 2 0.5882 1 0.3546 2 0.4118 Hence 008 025 41 0 6454 40 0 5882 3.399 Ce −. ×. . −× × . = and 40 0 4118 41 0 3546 1.607 Pe × . . = You could also use BSCall and BSPut to arrive at the answer. ± Question 11.2 N Call Put 8 3.464 1.718 9 3.361 1.642 10 3.454 1.711 11 3.348 1.629 12 3.446 1.705 50 3.3918 1.5997 The observed values are slowly converging towards the Black-Scholes values of the example. Note that the binomial solution oscillates as it approaches the Black-Scholes value.
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130 McDonald • Fundamentals of Derivatives Markets ± Question 11.3 1. T Call-Price 1 7.8966 2 15.8837 5 34.6653 10 56.2377 50 98.0959 100 99.9631 500 100.0000 As T approaches infinity, the call approaches the value of the underlying stock price, signifying that over very long time horizons the call option is not distinguishable from the stock. As T gets very large, the present value of the strike goes to zero; we essentially get the stock price for zero cost. If it isn’t exercised, the stock will be worth less than the present value of the strike (which is already very low). 2. With a constant dividend yield of 0.001 we get: T Call-Price 1 7.8542 2 15.7714 5 34.2942 10 55.3733 50 93.2296 100 90.4471 500 60.6531 The owner of the call option is not entitled to receive the dividends paid on the underlying stock during the life of the option. We see that, for short-term options, the small dividend yield does not play a large role. However, for the long-term options, the continuous lack of the dividend payment hurts the option holder significantly, and the option value is not approaching the value of the underlying; rather, it is approaching 0 . T Se δ ± Question 11.4 1. T Call Price 1 18.6705 2 18.1410 5 15.1037 10 10.1571 50 0.2938 100 0.0034 500 0.0000
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Chapter 11 The Black-Scholes Formula 131 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. 2. T Call Price 1 18.7281 2 18.2284 5 15.2313 10 10.2878 50 0.3045 100 0.0036 500 0.0000 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 much larger dividend yield. ± Question 11.5 1. Since everything is yen-denominated, the home interest rate is 15% r =. and the foreign interest rate is 35% . f r P (95, 90, 0.1, 0.015, 0.5, 0.035) = 1.0483 yen. 2. Since everything is now euro-denominated, the home interest rate is r and the foreign interest rate is 15%. f r C (1/95, 1/90, 0.1, 0.035, 0.5, 0.015) = 0.000122604 euros. 3. The relation is easiest to see when we look at terminal payoffs. Denote the exchange rate at time t as .
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This note was uploaded on 01/16/2011 for the course FIN 512 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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M11_MCDO8122_01_ISM_C11 - Chapter 11 The Black-Scholes...

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