Double Barrier Options (Double Knock-outs)

[quotes]

I need advice on pricing a double knock-out European option whose pay-off at the end is only allowed so long as the underlying has not breached an upper and a lower bound within the option life.

e.g.

A European Call with strike 20 dollars with both 25 dollar up-and-out and 15 dollar down-and-out constraint.

Also I have a guess that

25 up-out * 15 down-out + 25 up-in * 15 down-out = 15 down-out

Can anyone give some advice on pricing such an option? Any recommendations on books or paper is appreciated.

 

[John]

Just run enough MC simulations and average the discounted option payoffs. Make sure your stock dynamic processes, i.e. the drifts and volatilities are consistent with those observed on the option markets though. So calibration of the model parameters is the key. So calibrate your model (of your choice) with vanilla options then use it to price exotics.

 

[quotes]

do you mean use the implied volatility of a call to monte-carlo the double barrier?

 

[financeguy]

do you mean use the implied volatility of a call to monte-carlo the double barrier?

this is one approach
more popular among market making desks is PDE methodology for DKOs, where you assume some form of SV/LV blended process and work backwards from the payout vs the MC where you project the paths out forward through time

 

[Adrián Facchi]

You can always run enoughe Monte Carlo simulations (using local volatility model I assume) but this algorithm is maybe too much time-consuming because you need a small time-step if the barrier is continuous. I would rather compute the conditioned probability of the stock touching the barrier once known the initial and final value of a given time-step of the MC simulation.

Check Musiela & Rutkwowski pg 199 for a closed formula to price a single stock barrier option, from there you can deduce the probability of the underlying touching a down-and-out barrier and analogously any other barrier. I would assume constant volatility throughout the time step (the one given by the local volatility function).

I'm currently having trouble computing this conditioned probability for baskets. I need to take in account the correlation between assets and haven't been able to deduce anything useful, matters get even worse when the basket is of type BestOF/WorstOF. There is a post that I recently submitted to stackexchange.com but no one has given an indication on that regard.
http://quant.stackexchange.com/ques...he-correlation-in-barrier-options-on-a-basket

I hope you find useful this information

 

[quotes]

Thank you guys for your kindly advice.
Double Barrier seems to be infinitely harder than One Barrier.
The transition density is an infinite series because of infinite reflections between the upper and lower boundary.
It was not until late 1990's until mathematicians developed fast series solution to this problem
Marc Yor talked about it in Mathematical Methods for Financial Market a little bit.