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Fin Dev Midterm - 2007

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  Mid-term Financial Derivatives Date: 27  th  July 2007    Total = 40  marks  Time = 1 hour 30 minutes   Note: This is an open book / open notes / open formula sheet exam – students are allowed to bring in the textbook (Hull), notes, the formula sheet and the normal distribution table. But no exchange of notes / books /formula sheets / tables between the students are allowed during the exam 1.   What happens to the following: a.   A look back call as we increase the frequency with which we observe the asset price in calculating the minimum  b.   A down and out call as we increase the frequency with which we observe the asset price in determining whether the barrier has been crossed [3 + 3 = 6 marks] 2.   Arithmetic average rate options were assumed to be newly issued, and there was no historical average to deal with. Show that no generality was lost in doing so. [5 marks] 3.   Assume Binomial Option Pricing Model. Show that a contingent claim that pays $1 when the stock prices reaches (S u i  d n-I ) and $0 otherwise can be replicated by a portfolio of calls. [5 marks]  4.   Show that “At – the – Money” options have the maximum time value [5 marks] 5.   At what stock price is the theta of a European Call the smallest? [5 marks] 6.   Let {W(t); t ≥ 0} be a Brownian motion. Verify whether 2)( 2 t W   is a martingale or not. [5 marks] 7.   Consider a 9-month call option with an interest rate of 8% and n = 3 (i.e. there are nodes at the 3 month time point, 6 months time point and the final one at the 9 month time point). The current volatility is 10% but the volatility changes  between each time point at the rate of 7%. Let the current spot price be $160 and the strike price is $176. The option is of the American type. What is the premium that is to be paid for this option? Also, find the premium to be paid if the option was a put option with all the other information remaining the same. [5 + 4 = 9 marks]  Solutions Solution to question 1: a.   As we increase the frequency, we observe a more extreme minimum, which increases the value of a look back call.  b.   As we increase the frequency with which the asset price is observed, the asset  price becomes more likely to hit the barrier and the value of a down and out call goes down. Solution to question 2: Let the historical average from m prices be A as of time zero. The terminal payoff for a call is then         −+−−++=    −+++  ∑∑ == 0,11max0,1max 00 nmmA X nmS  X nmS mA niinii           −+−+++−++++= ∑ = 0,1111max11 0 nmmA X nnmnS nmn nii  So it becomes 11 +++ nmn  options with strike price     −+−+++ 111 nmmA X nnm . Solution to question 3: Consider the butterfly spread with strike prices X L , X M  and X H  such that 1111 −−+− −−+−− <<=<< ini H iniini M ini Lini d Su X d Su d Su X d Su X d Su  with 2 X M  – X H  – X L  = 0. This portfolio pays off  Lini  X d Su  − −  dollars when the stock  price reaches ini d Su  − . Furthermore, its payoff is zero if the stock price finishes at other  prices.   Solution to question 4: We have to prove that the strike price X that maximizes the option’s time value is the current stock price S. Note that time value is defined as V = C – max(S – X, 0).  Now, ( ) S  X if   X C  X V and S  X if   X C  X V T  x N e  X C   rT  >>+∂∂=∂∂><∂∂=∂∂⇒−=∂∂  − 010 σ    Thus, the time value is maximized at S. Solution to question 5: It is as t r   XeS  )2( 2 σ   + =  To derive it, note that '''''' '''''' )(2)()( )(2)()(  x xrSN t  x xSN  x N   xt  x N rXe t  x xSN  x N  S  rt  −+−=−−+−=∂Θ∂  − σ  σ  σ    The last equality takes advantage of the fact that '''' )()(  x xSN  xt  x N  Xe  rt  =− − σ    And the Black Scholes formula for European call. With N ’’ (x) = -x N ’ (x) and x ’  = 1/(S σ√ t), it is not hard to see that the particle derivative is 0 if and only iff -   σ 2  + (x σ / √ t) – 2r = 0. From here the result follows easily.  Solution to question 6: 22)(0),(| 2)(2)(0),(| 2)(0),(| 2)( 22222  st  sW   suuW   sW t W   E  suuW   sW  E  suuW  t W  E  −+=≤≤−+ ≤≤=≤≤  Thus it is not a martingale.


Jul 23, 2017
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