Fin Dev Midterm - 2007

Description
derivatives
Categories
Published

View again

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
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

D242

Jul 23, 2017
Search
Similar documents

View more...
Tags

Related Search