PWIQF Barrier Options Formulas

plessas · August 9th, 2002, 9:43 pm

Greetings all,I have been fooling arround with the Barrier Option formulas given by Paul in his PWIQF book on pages 247-249 but I am a bit puzzled with the results:Lets try an example... I have an UP and IN CALL with:S=100, E=110, Sb=120, VOL=0.4, r=0.06, q=0.02 and T-t=2 and get a price of 25.606400Here comes the puzzling part... for a plain vanilla call option with the same characteristics (except the barrier Sb of course) I get a price of 20.828610So the vanilla call is cheaper than the barrier one?I have been doing the calculations for three days now and I am really giving up hope on finding out whats happening!Paul or anyone kind enough... please help me a bit here rgds,Dimitris

Paul · August 10th, 2002, 12:27 am

I wasn't aware of any error there. Have you checked the Collector's book? (An up and out plus an up and in should equal a vanilla...whcih they do in the formulae you mention. So if the up and in is wrong so is the up and out )P

Tanaka · August 10th, 2002, 9:04 am

QuoteLets try an example... I have an UP and IN CALL with:S=100, E=110, Sb=120, VOL=0.4, r=0.06, q=0.02 and T-t=2 and get a price of 25.606400 Hi plessas,there might be something wrong with your implementation, according to my own calculations the answer should be:20.820666 for the Up-In Call and 0.007945 for the Up-Out Call giving a total of20.828611 for the vanilla Call./Tanaka

sam · August 10th, 2002, 9:14 am

Using the same parameters, I make the up and in call to be worth 20.8 and the standard Euoprean call to be worth almost the same. This suggests that the up and out is worth zero, which I verified separately (in fact, its worth is very small). Maybe you miscalculated the up and in option value? Sam

sam · August 10th, 2002, 9:15 am

Ooops, looks like Tanaka got there first, I agree with his(/her) values!Sam

plessas · August 10th, 2002, 10:10 am

Thank you for your replies Paul, Tanaka and sam... I have checked my calculations with code I wrote in VB, with Excel and in paper (!) but using the formulas mentioned in Paul's book I still get the wrong answer!Keeping the notation Paul uses I have for the UP IN CALL:S=100, E=110, Sb=120, VOL=0.4, r=0.06, q=0.02, T-t=2 and get:a=0.91287, b=1.31453, d3=0.10196, d6=-0.74657, d8=-0.90038, d4=-0.46372, d5=-0.18088, d7=-0.33470 thus:UP IN CALL VALUE: S * exp( q * (T-t) ) * ( N(d3) + b * ( N(d6) - N(d8) ) ) - E * exp( -r * (T-t) ) * ( N(d4) + a * ( N(d5) - N(d7) ) ) which gives:UP IN CALL VALUE = 25.60640I cant check Collectors book cause I dont have it but I think there is something wrong with the formulas Paul.rgds,Dimitris

Yuka · August 11th, 2002, 7:36 am

Don't you have to discount S by the dividend yield? if you do so (i.e. with S*exp(-q*(T-t)) instead of S*exp(q*(T-t))) then you get an answer consistent with sam & tanaka (for both the up & in and up & out calls) & consistent with the vanilla call.Using Paul's formula as it stands, I get the same answers as plessas - I've attached my workings so y'all can let me know where I've gone wrong

Paul · August 11th, 2002, 9:17 am

Good man! You are correct. All of the Se^{q(T-t)} should be Se^{-q(T-t)}.P

plessas · August 11th, 2002, 9:39 am

YES! Finally... I was starting to calculate barrier option prices in my sleep!Do you need my address to send me a corrected signed by the author copy of the book? rgds,Dimitris

plessas · August 11th, 2002, 9:52 am

Ah and Yuka, if you ever come to Greece mail me... dinner is on me rgds,Dimitris

J · October 24th, 2002, 1:34 am

Hey,I got confused about how to hedge down-out European Call. What should the replicated portfolio be? Why do you do so?

Collector · October 24th, 2002, 3:09 am

J are you speaking about theory or practice? Practice: Why hedge? Hedging is for Gardners!Theory: let's say you want to static hedge a down-out-call with strike X and barrier H, to hedge this simplybuy H/X put's and strike H^2/Xif asset never hit barrier the put will expire worth less and you are fine. If asset hit barrier the value of the puts = to value of the calls you need to buy (if you are short the down-and-in option). so just sell put and use money to buy the calls, You are fine.This is based on put-call symmetry, first described by the Genius David Bates. To use put-call symmetry for hedigng barrier options was first described by Peter Carr. Search for their home pages and downloade plenty of papers on static and semi-static hedging of barriers.Static hedging has two puroses: A) you can theoreticly static hedge barriers using plain vanilla, the method in general only holds for symetric smile. B) the static hedge method also gives you a valuation fomula even if you not nessesary use static hedging.Peter Carr has plent of papers on this topic on his hompage. I also have a few papers on static hedging of complex barriers on my web cite (+one on this cite), using symmetry between plain vanilla barriers. For European options you can use in-out barrier parity to price/hedge out option from in-option. Also fo American options there are multiple static/semistatic hedge methods, but things are naturally getting more complicated

J · October 24th, 2002, 5:47 am

Hi Epsen,Thank your reply very much!!!J

karsty · March 5th, 2003, 3:06 pm

Hi ... query about the barrier option formulas in "PW introduces quantitative finance" ... are there some problems with the up & out puts and the up & in puts? e.g. S=85, E=100, Sb=90, t=0.4, sig=0.7, r=0, q=0, up & out put with E>Sb. I get: a=0.944, b=1.0588, d1= -0.1457, d2= -0.5885, d5= 0.0923, d7= 0.3302, d8= -0.1125,and my formula is E *exp(-rt) * (1-N(d2)-a*(N(d7)-N(d5)) ) - S * exp(-qt) * (1 - N(d1) - b*N(d8) ),giving a price of 56.9848! But the vanilla put price is only 24.7642... Also, shouldn't adding the up&out put and the up&in put give a vanilla put? In the case of E<Sb this seems unlikely as the S-portion of both formulas are the same.I am fairly dizzy with all these d's floating around ... but I'm fairly sure I've got my formulas right. Thanks

Anton · March 5th, 2003, 3:52 pm

karsty, Indeed, knock in + knock out = vanilla. moreover, the vanilla is always more expensive than the barrier equivalent, so there is definetely something wrong with the prices you report. you can also try to cross-chack your results with an online calculator, have a look at http://www.mathfinance.de/optioncalculator.htmlAnton