Exercise 10.3

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Exercise 10.3

Postby rer » Wed Nov 28, 2012 7:52 pm

Hi Mark,

I'd like to query your solution to Ex. 10.3, which asks us to compare the number of price evaluations required to statically replicate a discrete barrier option using either weak static replication or the auxiliary variable replication technique (by the way, my book is 4th printing of 2nd Edition).

I can't understand your answer of approximately n^2 * N, where n is the number of barrier times we look at and N is the number of options required to kill the value of the portfolio below the barrier. I would have thought it was approximately n * N^2, using the following argument:

At one look-up time, the 1st short put struck at B kills value at (x_k-1, T_N-1). We need to calculate the value of this put at x_k-1. x_k-2, ... x_1, 0, which is a total of N times. The 2nd short put struck at x_k-1 that kills value at (x_k-2, T_N-1) must be evaluated at x_k-2, x_k-3, ..., x_1, 0, which is a total of N-1 times. Carry on down the partitions until we have the last put struck at x_1 that is only evaluated at 0, so a total of 1 time. Therefore the number of price evaluations for one look-up time is N + N-1 + N-2 + .... + 1 = N * (N + 1) / 2 which is the of the order of N^2. We have to do this for all the other n-1 look up times as well, so the total number of price evaluations is of the order of n * N^2.

I just can't see how it could be N * n^2, although you do mention in the text on page 251 that the auxiliary pricing method is linear in time, whereas the other method is quadratic, so perhaps I am confused...? Please help!

Also, just thought I should ask whether the notional amounts of the short puts used to kill the value below the barrier (found about half-way down page 248) are correct, as I would have though the first short put struck at B should be shorted by a notional P(x_k-j-1, T_N-1)/(x_k-j-1 - x_k-j) rather than simply P(x_k-j-1, T_N-1) to ensure the value (which has already been killed at x_k-j) is properly killed at x_k-j-1. If this needs to be corrected, there is also another instance a few lines up for the notional amount of the short put struck at B.

Thanks for all your help!

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Re: Exercise 10.3

Postby mj » Fri Nov 30, 2012 12:56 am

i think we have the issue that each option with expiry $T_j$ has to be evaluated at at time $T_i$ for all $i < j.$ This is where I am getting the $n^2$ from. It is a very long time since I thought about this stuff.
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