m48-ch19 - Chapter 19 Monte Carlo Valuation Question 19.1....

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Chapter 19 Monte Carlo Valuation Question 19.1. The histogram should resemble the uniform density, the mean should be close to 0.5, and the standard deviation should be close to 1 / 12 = 0 . 2887. Question 19.2. The histogram should be similar to a standard normal density (“bell” shaped). Since a uniform distribution has a mean of 0.5 and a variance of 1/12, the mean of 12 i = 1 u i 6 is zero and the variance (& standard deviation) will be one since var à 12 X i = 1 u i ! = 12 var (u i ) = 1. Question 19.3. The mean of e x 1 should be close to e 1 / 2 = 1 . 6487 and the mean of e x 2 should be close to e . 7 + 1 . 5 = 9 . 025. Question 19.4. The standard deviation of the estimate will be s n / n where s n is the sample standard deviation of the n simulations. Since s n is close to 2.9, n = 84000 should give a standard error close to 0.01. Question 19.5. 1 /S 1 = exp ( . 08 + . 3 2 / 2 + . 3 z ) / 40 generates the simulations. The mean of which should be close to e . 035 + . 3 2 / 2 / 40 = . 02525. This should also be the forward price. Question 19.6. The simulations should be generated by S 1 = 100 exp ( . 06 . 4 2 / 2 + . 4 z ) where z is standard normal. The claim prices should be e . 06 S α where α is the relevant power and the S α is the average from the simulations. These values should be close to 100 α exp ³ 1 ) ³ . 06 + α 2 . 4 2 ´ . Using this, the three values should be close to 12461, 9 . 51, and . 000135 respectively. 249
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Part 5 Advanced Pricing Theory Question 19.7. The fve values should be close to 10366.56, 1.004, 96.95, 10 4 , and 1 , 261 , 120 respectively. Question 19.8. By log normality P (S t < 95 ) = P ³ 100 exp ³ . 1 . 2 2 / 2 ´ t + . 2 tz ´ < 95 ´ P Ã z< ln ( 95 / 100 ) ( . 1 . 2 2 / 2 ) t . 2 t ! with t = 1 / 365 this is N ( 4 . 9207 ) = 4 × 10 7 . This magnitude negative return should, on av- erage, occur once every 2.5 million days. With t = 1 / 252 (i.e. one trading day) this becomes N ( 4 . 0965 ) = 2 . 097 × 10 5 ; making such a drop is similarly unlikely.
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m48-ch19 - Chapter 19 Monte Carlo Valuation Question 19.1....

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