Chapter 10

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Chapter 10

Postby MattT » Fri May 18, 2012 6:40 pm

Hi Mark,

I was hoping you could clarify two points for me in chapter 10 (all references to page numbers are for 2nd edition, 4th printing):

1) For the static replication of continuous barriers you write (page 246):
The difference in price between the replicating portfolio and the down and out put must be less than than the maximum (discounted) value of the replicating portfolio on the line (B,t)...As N goes to infinity this difference will go to zero...

You seem to be assuming some kind of continuity of the replicating portfolio here. My concern is, if the payoff of the replicating portfolio behind the barrier is large, then it is possible perhaps that the value of the portfolio at (B,t), where t lies in between one of the barrier times (t_j), could be quite large and potentially ruin the convergence of the replicating portfolio price to the down and out option price. Why does this not happen (or if it does, can you recommend some literature which examines the convergence in more detail)?

2) For the put-call symmetry section (reflection arguments) where interest rate is 0 you write (page254):
An interesting fact about this replicating portfolio is that the composition of the portfolio does not depend on \sigma....In fact, all that really matters is that when spot is on the barrier B, the smile should be symmetric - a call option of strike K/B must trade with the same implied vol as a put option of strike B/K

I think this is probably a typo? It seems to me that we want the smile to be "invariant under geometric reflection". So a call option of strike K/B must trade with the same implied vol as a put of strike B^3 /K (since then the log of these strikes are the same distance from the log of the barrier).

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Re: Chapter 10

Postby mj » Wed May 23, 2012 12:52 am

1) well ultimately value is given by expectations and so will be continuous -- any solution of the heat equation is smooth after all.

2) looks like it should be K and B^2/K
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Re: Chapter 10

Postby ragtimearmy » Tue Jul 16, 2013 7:47 pm

Dear Mark, I've been puzzling on the method in 10.2, i mean on "killing the difference portfolio's value at the points (B,tj) with tj counting downwards". Could you please advice me first of all on what is the "difference portfolio" and also to explain in other words the algorithm of the "kill" through the adding of the puts struck at B expiring at tj. I am not even shure if the proceeds from dissoling of the portfolo on touching the barrier for buying the puts.
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Re: Chapter 10

Postby mj » Tue Jul 23, 2013 3:55 am

we construct a sequence of portfolio's P_j

P_0 matches the final pay-off.

P_j matches the final pay-off and is zero on the barrier B at time t_k for the last j times.

To construct P_{j+1}, we value P_j at the last time we haven't killed the value at.
We remove the value at that time by taking puts struck at B expiring at the next time .
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