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Hull: Options, Futures, and Other Derivatives, Ninth Edition Chapter 14: Wiener Processes and Ito’s Lemma Multiple Choice Test Bank: Questions with Answers 1. A variable x starts at 10 and follows the generalized Wiener process dx = a dt + b dz where time is measured in years. If a = 2 and b =3 what is the expected value after 3 years? A. 12 B. 14 C. 16 D. 18 Answer: C The drift is 2 per year and so the expected increase over three years is 2×3 = 6 and the expected value at the end of 3 years is
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  Hull: Options, Futures, and Other Derivatives, Ninth EditionChapter 14: Wiener Processes and Ito’s e!!a ultiple Choice #est $an%: &uestions 'ith (ns'ers 1. A variable  x  starts at 10 and follows the generalized Wiener process dx  = a dt   + b dz   where time is measured in years. If a = 2 and b  = what is the e!pected valueafter  years A.12#.1$%.1&'.1(Answer) % *he drift is 2 per year and so the e!pected increase over three years is 2+ = & andthe e!pected value at the end of  years is 10,& = 1&. 2. A variable  x  starts at 10 and follows the generalized Wiener process dx  = a dt   + b dz  where time is measured in years. If a =  and b  =$ what is the standard deviation of the value in $ years A.$#.(%.12'.1&Answer) # *he variance per year is $ 2 or 1&. *he variance over four years is 1&+$ = &$. *he standard deviation is  64 8 = . 3. A variable  x  starts at 10 and follows the generalized Wiener process dx  = a dt   + b dz   If a =  and b  =$ what is the standard deviation of the value in three months A.1#.2%.'.$Answer) #  *he variance per year is $ 2 or 1&. *he variance over three months is 1&+0.2- = $.  *he standard deviation is  4 2 = . 4.  *he variance of a Wiener process in time t  is A. tB. t squared  C.  the suare root of t D.   t to the power of 4 Answer) A *he variance of a Wiener process is 1 per unit time or t  in time t .5.  *he process followed by a variable  X   is dX = mX dt+sX dz  What is the coe/cient of dz   in the process for the suare of  X  .  A.  sX   B.  sX  2C. 2  sX  2  D. msX  Answer) %rom Itos lemma the coe/cient of dz   is  X    f   sX    ∂∂  where  f =  X  2 . #ecause  X   X    f    2 =∂∂  the coe/cient of dz   is 2  sX  2 . 6.  *he process followed by a variable  X   is dX = mX dt+sX dz  What is the coe/cient of dt in the process for the suare of  X  . A. 2 mX  2 +  s 2  X  2B. 2 mX  2 C. mX  2 ,2  s 2  X  2 D. mX  2 +s 2  X  2 Answer) Arom Itos lemma the coe/cient of dt   is 2222 21  X   f   X  s X   f  mX  ∂∂+∂∂ where  f =  X  2 . #ecause  X   X    f    2 =∂∂  and 2 22 =∂∂  X   f   the coe/cient of dt   is 2 mX  2 +  s 2  X  2 7. Which of the following is true when the stoc3 price follows geometric #rownian motionA.*he future stoc3 price has a normal distribution#.*he future stoc3 price has a lognormal distribution%.*he future stoc3 price has geometric distribution'.*he future stoc3 price has a truncated normal distributionAnswer) #  Itos lemma show that the log of the stoc3 price follows a generalized Wiener process. *his means that the log of the stoc3 price is normally distributed so that the stoc3 price is lognormally distributed. 8. If a stoc3 price follows a 4ar3ov process which of the following could be trueA.Whenever the stoc3 price has gone up for four successive days it has a 506 chance of going up on the 7fth day.#.Whenever the stoc3 price has gone up for four successive days there is almost certain to be a correction on the 7fth day.%.*he way the stoc3 price moves on a day is una8ected by how it moved on the previous four days.'.#ad years for stoc3 price returns are usually followed by good years.Answer) %A 4ar3ov process is a particular type of stochastic process where only the current value of a variable is relevant for predicting the future. 9toc3 prices are usually assumed to follow 4ar3ov processes. *his corresponds to a wea3 form mar3et e/ciency assumption. 9. A variable  x  starts at zero and follows the generalized Wiener process dx  = a dt   + b dz  where time is measured in years. 'uring the 7rst two years a = and b =$. 'uring the following three years a =& and b =. What is the e!pected value of the variable at the end of - yearsA.1&#.20%.2$'.0Answer) %'uring the 7rst two years the drift per year is  and so the total drift is +2 or &. 'uring the ne!t three years the drift per year is & and the total drift is &+ = 1(.  *he total drift over the 7ve years is &,1( =2$. :iven that the variable starts at zero its e!pected value at the end of the 7ve years is therefore 2$. 10. A variable  x  starts at zero and follows the generalized Wiener process dx  = a dt   + b dz  where time is measured in years. 'uring the 7rst two years a = and b =$. 'uring the following three years a =& and b =. What the standard deviation of the value of the variable at the end of - yearsA.&.2#.&.5%.5.2'.5.5  Answer) ' *he variance per year for the 7rst two years is $ 2  or 1&. *he variance per year for the ne!t three years is  2  or ;. *he total variance of the change over 7ve years is 2+1&,+;= -;. *he standard deviation of the value of the variable at the end of the 7ve years is therefore 59 7.7 = 11. If a variable  x  follows the process dx  = b dz   where dz   is a Wiener process which of the following is the process followed by  y  = e!p<! . A.dy = by dz  B.dy = 0.5b 2  y dt+by dz C.dy = (  y+0.5b 2  y   dt+by dz  D.dy = 0.5b 2  y dt+b dz  Answer) #Itos lemma shows that the process followed by  y  is dy = 0.5b 2 e!p(  x  dt +b e!p(  x) dz. 9ubstituting  y = e!p(  x    we get the answer in #. 12. If the ris3>free rate is  r   and price of a nondividend paying stoc3 grows at rate m   with volatility  s  at what rate does a forward price of the stoc3 grow for a forward contract maturing at a future time   .  A.m B. m ?  s 2 @2 C. m ? r D. r  ?  s 2 @2Answer) % *his is the application of Itos lemma in 9ection 1$.&. 13. When a stoc3 price !   follows geometric #rownian motion with mean return m  and volatility  s  what is the process follows by  X    where  X   = ln !. A. dX = m dt + s dz  #. dX = m#r) dt + s dz  %. dX = m   #s 2  ) dt + s dz 
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