aschon
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I have a question regarding the replication of a cliquet option with the following payoff (local cap, global floor):
the sum in the brackets can be replicated using a series of forward start options/call spreads, but what about the global floor? is there a possibility to replicate this feature (even if its only an approxiamte solution)?
thanks in advance! |
In practice, this works, but how about in theory? |
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granchio
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no.
May I also suggest advice to tread very carefully with this stuff. |
"Deserve got nothing to do with it" - Clint |
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chiral3
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I don't want to put words in Granchio's mouth, but my guess is what he is saying is you can algebraically attack this thing all you want and it will not reasonably price the structure. Maybe I'm wrong. Here's a few thoughts:
1. The interesting thing about this structure is the cap, that's what drives the real pricing, not the floor
2. OK, so there's some replication lurking...
let r=r_i, sums over i to N and assume the floor is 0 so I don't keep typing. Also, I'll come from the put angle, although the same thong applies to calls
max(Sum(min(r,cap)),0)
= max(sum(cap-max(cap-r,0)),0)
=max(N*cap-Sum(max(cap-r,0)),0)
So you view the sum like ~= expectation this is just a strip or fwd starting puts and the whole thing is just a compound option. If you come from the call angle you'll get something like max(sum(r-max(r-cap,0)),0) which is kinda like a european call struck at a strip of fwd starting caplets.
3. So this is worthless, although in this day and age, with french banks, etc. this structure is fairly benign. If you are interested in pricing and replication a "heston bump and price" will work, where your bump is on the vol of vol or the corr (likely the former since corrs are pinned near -1 on most surfaces, like sx5e and spx).
4. if you want to replicate what the rest of the trading world sees you might want a fwd skew model or an estimate of its price. Bergomi has contributed a bunch to this but, for the structure you mention, isn't really hugely necessary. Agreed with Granchio, though, tread carefully....
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Solipsism (Listeni/ˈsɒlɨpsɪzəm/; from Latin solus, meaning "alone", and ipse, meaning "self") is the philosophical idea that only one's own mind is sure to exist. |
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aschon
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thx for your responses!
@chiral: the decompositon into compound options on a series of forward start puts is not really helpful if you ask me. it gives you some insights about the (volatility) risks involved in this thing, but the pricing is still not that easy.
i'll give it a try with the heston model. what about pricing this structure in a BS type of model where you incorporate the skew? i read somewhere that this gives you a somewhat good approximation, but im not really a fan of this idea.
i read the articles from Bergomi. why isnt his stuff/skew models necessary for this structure? could you elaborate a bit on this?
@granchio: will do. thx for the advice!
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In practice, this works, but how about in theory? |
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chiral3
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Just to be clear, the replication takes care of the global floor and cap, so I think it answered your original question. Compound options are easy to deal with, right?
You can blow the vol out in bs, price with skew, price with jumps, etc., to get closer but it is a bit more controlled if you come from the SV angle. I dug up a graph I made in 2007 and the pricing between BS and market/volvol and how they rank, worst to best, is like BS, replication, jump diffusion, heston, and finally market.
Bergomi is relevant, this is an accumulator. What I mean is that heston bump and price works fine for the structure you mention. There are some hairier structures, though, for which bump and price will just fail for. It's important to have an intuition why these models work or don't work for certain structures or at certain times. The bump and price works because you are subsidizing a mispricing of forward skew with artificially warping the current surface. You can see this with the bumped versus non-bumped surfaces created by the complex integral that defines the surface a la heston. For instance, the params {kappa, theta, eta, rho, V0} for the SPX last week, assuming a certain region, etc., are estimated to be {4.42, .078, .83, -.99, .029}. What gets you to a bunch of the trades that happened around that time is {4.42, .078, .81, -.99, .029}. OK, so the eta going from .83 to .81 isn't much but, historically, for the past 5 years, it is usually an inflation of maybe .25-.5. Anyway, when you plot these surfaces you can see the artificial skew adjustment, but this isn't real pricing of forward skew, like you'd ideally want with a strip of local caps. |
Solipsism (Listeni/ˈsɒlɨpsɪzəm/; from Latin solus, meaning "alone", and ipse, meaning "self") is the philosophical idea that only one's own mind is sure to exist. |
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granchio
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sorry I was in a bit of a hurry before.
By "no" I simply meant: you cannot replicate with vanillas - that is my normal understanding of replication.
Chiral has already said it all on the modelling.
I would only add a way of seeing why "BS" is the worst: basically whatever "BS" you do, you are trying to do local vol. Local vol will be the best BS. And you know how bad local vol is for this kind of things.
Anything that chucks in some stoch vol will be better. We have played a bit with stochastic local vol and it seemed encouraging, but maybe it was just a fluke.
On why to tread carefully: over the years, there has been a lot of pain caused by these babies. Which is not a big surprise... if you have a lot of stoch vol exposure, IMHO is very easy to get burned... (think about expected std dev of your P&L in the future... how far are you from random?)
Some books, especially in europe, are still full, and they might gladly offload if you go in a bit naive.
I guess US books have less legacy, and that's probably why SPX ones are reasonably active in IDB market
PS: In the limit of the period = to one day, those are daily returns... like in varswaps...so the shorter the period, the more you want to compare to var derivs, and be sure to use a consistent model. Which reminds me that we might have discussed this issues in another thread
PPS: on the modelling: the old Avellaneda thing, which W*lmott popularised in ~2000 or so, was very cute.
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"Deserve got nothing to do with it" - Clint |
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aschon
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thx for the detailed answers!
@chiral: i might be missing something here, but to me this doesnt look like a "plain vanilla" compound option, since the underlying is a series of forward starting puts and not just a single option. but to be honest, im not too familiar with compound options yet, so i have to do some research on these first.
i found the older thread where you put up the graph. very helpful, thx!
@granchio: what do you mean by a "consistent model" here?
i like the idea of Avellanda to model the volatility as an uncertain quantity and i also looked up the W***tt paper, but as of right now im more interested in the replication/hedging of these cliquet structures.
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In practice, this works, but how about in theory? |
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granchio
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>what do you mean by a "consistent model" here?
that the same model is used for the different payoffs
>im more interested in the replication/hedging of these cliquet structures
what do you mean by that? I am sorry if it sounds stupid, but what do you want to achieve?
Short term reduction of mark-to-model P&L fluctuations?
Longer term reduction of realised P&L uncertainty?
The two are very different (the former being much easier), and depending on where you work, the emphasis will be on one or the other.
Also critical question: which instruments do you want to use to hedge/replicate?
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"Deserve got nothing to do with it" - Clint |
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aschon
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i was hoping that there might be a way to approximate the mentioned cliquet payoff with a bunch of plain vanilla options. i wasnt looking for an exact solution, just something that gets you as close as possible to the cliquet payoff. sorry, if i wasnt clear about that.
> "which instruments do you want to use to hedge/replicate?"
im not really sure yet. since the main risk lies in the forward vol/vega, variance or vol swaps might be an alternative. what do you think?
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In practice, this works, but how about in theory? |
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granchio
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>...what do you think?
alas it would take more time than I can dedicate to a forum discussion...
but I do recommend considering the 2 questions I asked you about what is the aim of your hedging...
as well as how would you compute meaningful hedging ratios. |
"Deserve got nothing to do with it" - Clint |
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chiral3
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| I agree with Granchio. I trade and hedge this structure and my goal is minimizing p&l volatility (economic and accounting). It helps that it has negative vega so netting is helpful. There is really no way to do this without taking some risks. Also depends on what ttm you're dealing with because there are some near term risks and some far term risks. |
Solipsism (Listeni/ˈsɒlɨpsɪzəm/; from Latin solus, meaning "alone", and ipse, meaning "self") is the philosophical idea that only one's own mind is sure to exist. |
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aschon
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@granchio: as of right now my focus is more on reducing mark to model p&l fluctuations, but im also interested to know more about the other aspect. But why is reducing your mark to model p&l fluctuations so much easier than minimizing the variance of the realized p&l? This is not clear to me yet.
@chiral: How do you hedge the vega risk (especially the risk comin from forward vols), ie what type of instruments do you use? Vol derivatives?
Thx for your help so far! |
In practice, this works, but how about in theory? |
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granchio
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<<
@granchio: as of right now my focus is more on reducing mark to model p&l fluctuations, but im also interested to know more about the other aspect. But why is reducing your mark to model p&l fluctuations so much easier than minimizing the variance of the realized p&l? This is not clear to me yet.
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because you control the model. Silly example: say you have a parametric volsurface with just N params. you compute the exposure to those N params, and cover them with N traded instruments. for small moves, you're hedged. then you rebalance.
Of course your realised P&L at expiry has got nothing to do with it. but short term it might pay your bonus.
(for big moves... well for big moves not even delta hedging of a vanilla in ideal blackscholes scenario works, so forget it.)
Of course don't take this as advice - though IMHO it is what most "replication" in the sell-side boils down to, effectively, alas.
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"Deserve got nothing to do with it" - Clint |
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