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# R19.P1.T3.FIN_PRODS_HULL_Ch26_ExoticOptions_Topic:Barrier-Options

##### Active Member
In reference to R19.P1.T3.FIN_PRODS_HULL_Ch26_ExoticOptions_Topic: UP-and-OUT Barrier-Options:-
Are UP-and-OUT Barrier-Options = PUT Options ? or can they be CALL Options too ..?

If they are CALL Options, why would one want to limit the Payoff and Profit by putting a cap on the upward movement of the price..? Is it to lower the premium of the option ?

Much gratitude for the insight on this.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @gargi.adhikari FYI, as part of our current wholesale revision of the R19 notes (that Deepa and I are working on), I am currently updating all of the underlying XLS, which will include an updated version of a working model for the barrier options. I will be excited to share that with you. I will share a version back here on this thread (of the barrier model) as soon as i have the first draft of the revised ...

First, the motivation for barrier options is to reduce their cost, as evidenced by relationships like c(uo) = c - c(ui). If the price of a European call, c, is X, then its price splits into two "pieces" because c(uo) + c(ui) = c. So, if you want to assume the risk of it getting knocked out, or if you want to assume the risk it won't get activated, you can pay significantly less to play for the truncated portion of the payoff curve. Otherwise, because the barriers only introduce the barrier, H, to an option that otherwise has the same properties, there can be no reason to want the barrier (because without the barrier, you get all of the same payoff plus the additional payoffs restricted by the barrier) except for one motivation that i can think of: for hedging (or even speculative) purposes, barrier options have uniquely different Greek properties. So i can imagine scenarios where the barrier option might be well-suited for customized gamma/vega exposure.

Re: Are UP-and-OUT Barrier-Options = PUT Options ? or can they be CALL Options too ..? The way that I think of barrier options is
• Start with a plain old vanilla call or put; strike (K) and current (S)
• Add a barrier, (H). If this barrier is above the current (S), that will be an "up-and..."; if this barrier is below current (S) that will be a "down-and..."
• Either it knocks-out or activates (knocks-in) if future (S) hits the barrier. But this can be confusing way to look at it, because the barrier merely determines if the option exists ("is alive"); the stock price determines whether you are on the alive side of the barrier, or on the knocked out side of the barrier.
• Ergo, re: Are UP-and-OUT Barrier-Options = PUT Options ? No, an up-and-out call is not a put option. A put pays off on price decline, but u-o call is still a call, so it won't pay on price decline. But it's a little worse, because it also won't pay if the price increases too much. I hope that helps!

##### Active Member
@David Harper CFA FRM Got it !!!!!! Thanks a ZILLION ! Every time is such an AHA moment - almost get goosebumps each time. You guys are undoubtedly the GURUS at this stuff !!!!!!!!!!!!!!!!!!!

##### Active Member
@David Harper CFA FRM Hi - I was revisiting this topic..and wanted to see how that payoff of { c(uo) + c(ui) } equals that of a regular call option c.....it isn't exactly the same is it..? it's truncated...? I've been trying to search for the payoff diagram ...
I've been trying to search all over for the payoff diagram...for { c(uo) + c(ui) } , I tried to construct the payoff diagram...doesn't look exactly like that of a plain vanilla Call Option but is kinda truncated...is this how the { c(uo) + c(ui) } is supposed to be..?= a Truncated Call Option..?- Wanted to make sure am getting this right...

Also wanted to mention that, I was not able to upload or attach a picture here for this post...says File too Large ... :-( like I do at other times....:-(

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @gargi.adhikari I don't think (am not aware of) that you can plot a payoff diagram of a barrier option. I could be wrong, but barrier options are path-dependent such that, for example, at future stock price of S(t) = X, you don't know if the option was knocked in or out because that's a function of the path! Ero, I am not confident that c(uo) + c(ui) = c, in terms of payoffs (I can imagine path-based exceptions.) The equality, c(uo) + c(ui) = c, given by Hull when H>K, is a relationship about the current prices of these options. IF you are interested, this is a very rough draft of my corresponding pricing model, see https://www.dropbox.com/s/fhd95vh9nbsscie/0728-barrier-option-v2.xlsx?dl=0

For example, here is the plot of the current (BSM) price of a vanilla call option (blue line), where X axis is stock price, and given other assumptions (σ=40%, Rf = 5%, T = 0.25, q = 5% for some odd reason I have that in there!). And then the red and blue lines parse this value into up-and-in call (red line) and up-and-out call (green line). Analytically, these equalities should hold in terms of PV prices. Although if you look at the XLS, the formulas are complex; I think i have in error somewhere in the cdi_cdo. I hope that explains why it holds for current price but not future payoff diagrams. Thanks!

##### Active Member
@David Harper CFA FRM Thanks so much for sharing this insight... I guess I misunderstood the relation : c(uo) + c(ui) = c thinking that this was the payoff...but now i see that the equality applies to the Current Pricing of the options ...

##### Active Member
@David Harper CFA FRM I guess I have another questions on this topic...

For BARRIER OPTIONS:-

"When the Frequency-of-Observation" Increases :-
1) the Value of the Knock-Out-Barrier-Option Decreases while
2) the Value of the KNOCK-IN-Barrier-Option INCREASES...?

I understand the Value-Decrease for the Knock-Out-Barrier-Options. But am having trouble understanding why would the Value Increase in case of the KNOCK-IN-Barrier-Options..

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @gargi.adhikari Yes this can seem strange, can't it!? Imagine that I am your counterparty and I will write you an one-year barrier option when the stock price is $100.00. The strike price is also$100.00, but we agree the barrier, H = $108.00. If this a knock-out (aka, up and out) barrier option, then it will cease to exist if the stock price breaches$108.00. You would prefer to test less frequently and I would prefer to test more frequently: the more frequently we observe whether S > H, then the more likely this option will be knocked-out. In extremis, we'd only test once at the end of the year; in this case, it might breach several times in the meantime, but still not get knocked out. Maybe we test monthly, but it could still breach on many days and survive. Daily observations, by comparison, decrease the value of this option to you (and the price I can charge you for it). Hence, for the knock-out barrier, an increase in frequency implies a decrease in value.

But keep all assumption the same, S = K = 100 and H = 108, but now imagine I sell you a knock-in (aka, up-and-in). Now which do you prefer? One observation at the end, monthly, or daily? You prefer to observe as often as possible! Because when we observe S > H, this option "comes into existence." In this case, the more frequently we observe, the more I get to charge you (higher price) because more observations imply greater value to you (as the option has a greater probability of coming into existence). I hope that clarifies!

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