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Strange


Total Posts: 1595
Joined: Jun 2004
 
Posted: 2019-10-27 15:38
So there is such a thing out there as UVM (uncertain volatility model), which offers an approach for conservative hedging. The gist of the model is that you switch between (or blend) simulation at higher and lower volatility depending of the gamma. I.e. if the hedger is long gamma, the model uses low volatility for next step, while if hedger is short gamma it uses high volatility. It has some theoretical problems, but all in all it's a useful tool.

Recently, I started experimenting with a UVM-like model to hedge products that have sensitivity to the forward skew AND are path dependent. I can't use a proper UVM, since there is path dependency involved. (for terminal payoffs or for combination of vanillas, the UVM model has a fairly reasonable implementation using backward propagation on a dual grid).

Here is what I am thinking, however (and that's where I need help). First, I would use mean volatility simulation to calibrate the gamma profile of the product (i.e. something that would be able to output expected gamma at point t based on path S[0, t]) and then I can use that calibration to actually run regular monte carlo using the high/low volatility approach. What would be wrong with that type of thought process?

"In Russia, every CDS ends in bullet payment"

nikol


Total Posts: 830
Joined: Jun 2005
 
Posted: 2019-10-27 20:30
You should get inconsistency between used value (mean vol based) and value predicted by MC (based on max/min scenarios) due to path dependence. However, if for every point in time future vol_min/vol_max scenarios are around mean vol at that point, then I see no issue.

Can't you use Bayesian approach by injecting vols as hypothesis, P(H_vol), and expectation (Value) given hypothesis P(Value|H_vol) to imply bounds for the vol out from resulting distribution P(H_vol|Value)? You could also do it at every t, so you will know needed scenarios.

This means, that for every observed Value (as Data) you can imply range of Vols (Posterior hypothesis).

If you use Longstaff-Shwarz method to value derivative, then you get interpolated E[V] at every time step, t, and hence gamma comes as a bonus.

PS. What I don't get intuitively is who pays for the seller and the buyer if they both follow UVM strategy and get positive PnL ? Trader of underlying?
Confused

nikol


Total Posts: 830
Joined: Jun 2005
 
Posted: 2019-10-27 20:46
Also,

briefly looking into this
https://wwwf.imperial.ac.uk/~ajacquie/index_files/UVM_EQF_140708.pdf

Look at page 3, after "we must have" formula for boundaries of Portfolio Value:

W- < EP[df*Ф-] < Пt < EP[df*Ф+] < W+

Can you convert it into probability space making it P(Пt) and, therefore, bounding the distribution of Пt?

TonyC
Nuclear Energy Trader

Total Posts: 1316
Joined: May 2004
 
Posted: 2019-10-28 04:10
I generally Monte-Carlo simulate across (sometimes modified) crr trees by generating a base 10 number, converting it to binary, and saying that a 1 is an up jump and a 0 is it down jump.

Why don't we grab lunch/dinner/drink sometime before Thursday evening so I can I understand your problem better.


flaneur/boulevardier/remittance man/energy trader

ronin


Total Posts: 508
Joined: May 2006
 
Posted: 2019-10-28 10:50
> The gist of the model is that you switch between (or blend) simulation at higher and lower volatility depending of the gamma.

God, is even that called machine learning now? We used to call that overvol in the good old days. We just bumped the vol by a fixed number - if you sell gamma, you charge positive overvol, if you buy gamma you charge negative.

> What would be wrong with that type of thought process?

The process would be fine in terms of gamma. But, by the sound of it, it would mess up your skew. You really want to charge an "overskew", which will sort out your overvols automatically.

Of course, it could be that this is really some over-engineered way to charge overskew. I don't know enough about it to be able to tell.

"There is a SIX am?" -- Arthur

Energetic
Forum Captain

Total Posts: 1505
Joined: Jun 2004
 
Posted: 2019-10-28 18:41
I don't think it has anything to do with ML.

I do know about one unsuccessful attempt to implement UVM in a bank environment. I don't see why UVM should work any better than LV/SLV MC simulations. (I know that you're flying solo so LV/SLV are not practical choices for you.)

I don't think anything is wrong with your thought process. You are trying to do something commonsensical quick-and-dirty and it might well be an improvement vs. not doing it.

I suspect you'd lose some (or much) of your edge though.

For every complex problem there is an answer that is clear, simple and wrong. - H. L. Mencken

Strange


Total Posts: 1595
Joined: Jun 2004
 
Posted: 2019-10-29 02:47
Thanks for the answers! It's surprising how excited people got over a pretty dry topic :)

@nikol
It will be inconsistent by definition since we are trying to match greeks which are vol-dependent. I.e. it's very probable that I will get some mismatched paths (i.e using low vol for short gamma points). However, it does not make the pricing bad, though, it essentially makes the calibration volatility yet another pricing input (so I have 3 pricing inputs instead of 2 now).

@TonyC
" Why don't we grab lunch/dinner/drink sometime before Thursday evening so I can I understand your problem better. "
Gonna be tricky, I have a bunch of stuff happening. When are you back from wherever you're going?

@ronin
" God, is even that called machine learning now? We used to call that overvol in the good old days. We just bumped the vol by a fixed number - if you sell gamma, you charge positive overvol, if you buy gamma you charge negative. "
UVM is an smart overhedging hack and has nothing to do with machine learning, overhedging is as old as the hills. The machine learning bit comes in trying to quasi-calibrate the model. I.e. the model will need to have some way of matching the path of S[0, t| i] to the calibrated set.

"Of course, it could be that this is really some over-engineered way to charge overskew. I don't know enough about it to be able to tell."
LOL, exactly - the whole point of using UVM is to hedge the skew in a conservative way :) The general idea is that it's much easier to come up with upper and lower bounds on volatility than to come up with good idea of joint spot/vol dynamics.

@Energetic
"I do know about one unsuccessful attempt to implement UVM in a bank environment. I don't see why UVM should work any better than LV/SLV MC simulations. (I know that you're flying solo so LV/SLV are not practical choices for you.)"
Well, as you rightfully point out, simplicity of implementation is one of the reasons. The other practical aspect is that this methodology should give me a quick and dirty way to evaluate various garbage the banks are trying to layoff.

"In Russia, every CDS ends in bullet payment"

ronin


Total Posts: 508
Joined: May 2006
 
Posted: 2019-10-29 12:23
> LOL, exactly - the whole point of using UVM is to hedge the skew in a conservative way :)

So I'm feeling particularly dumb today.

Why exactly does this work better than bumping the skew by a fixed number?

In fact, why does this even work at all? How does it not create strike arbitrage? Precisely at the points you presumably care most about?

"There is a SIX am?" -- Arthur

Strange


Total Posts: 1595
Joined: Jun 2004
 
Posted: 2019-10-29 19:02
@ronin
" In fact, why does this even work at all? How does it not create strike arbitrage? "

In a sense it's just an odd (and very conservative) case of stochastic volatility - evolution of spot depends not on some arbitrary set of inputs but rather on the risk of your position. I.e. at each time point t you are assessing the exposure and taking a step forward. Since you are pricing the whole arbitrary payoff (compound or otherwise) at each point, there is no reason it would result in strike arbitrage (e.g. call spread would never be worth more than the difference between strikes). Because you are providing extreme volatilities in local form (i.e. at each step would pick volatility for next step), calendar arbitrage is not possible either.

"Why exactly does this work better than bumping the skew by a fixed number?"
A lot of times (especially for complex payoffs) you don't even know what your exposure will be as the product evolves. That's where UVM comes in handy, as it gives an ability to overhedge the flips in convexity without trying to decompose the product. It's also an intuitive way of pricing the skew when there is no liquid listed market (my use case).

This said, UVM is very conservative. Avellaneda in his original paper shows any stochastic vol process that is bounded by these extreme volatilities will produce a cheaper price than under UVM itself.

"In Russia, every CDS ends in bullet payment"

ronin


Total Posts: 508
Joined: May 2006
 
Posted: 2019-10-29 21:16
OK, ok. I'll read up on it when I have five minutes. Which, on present form, will be some time in 2027. If all goes well...

But in all seriousness: if you don't know which way your exposures lie, should you really be trading it?

I can see how that sort of a dumb black box might be attractive at the middle office level ("yay! we don't need expensive risk managers!"), but I can't work out why you would use it on a trading desk.

"There is a SIX am?" -- Arthur

deeds


Total Posts: 457
Joined: Dec 2008
 
Posted: 2019-10-30 16:36

@ronin - would suggest there is likely arbitrage...maybe not in the way discussed here
- saw Bruno Dupire last night talking about available arbitrage in parameter space...an old idea he was recapping, maybe with new examples (probably to fill a slot in the very pleasant monthly bloomberg quant series in NY)
- at a high level
- since market participants use the same models, reparameterization from date to date leaves convexity on the table
- very roughly, analytical procedure is to draw convex hull around skew region to allow hyperplane separation of profit and loss regions and identify a point that always ends up in profit (am trying to get the slides)
- demonstrated in several products and models, including SABR, Heston...in addition, one of the illustrations was that both sticky strike and sticky delta have this vulnerability
- very pleasant...will share reference when/if i find, if any interest

ronin


Total Posts: 508
Joined: May 2006
 
Posted: 2019-10-30 22:41
@deeds,

Definitely - would love to see that.

The problem with these regions that always end up in profit is that you never get the opportunity to trade them...!

But I'd be interested in having a look.

"There is a SIX am?" -- Arthur

Strange


Total Posts: 1595
Joined: Jun 2004
 
Posted: 2019-10-31 14:11
@ronin
"But in all seriousness: if you don't know which way your exposures lie, should you really be trading it?"
It's less about exposures being unknown and more about exposures changing with time/underlying which is hard to deal with. For example, take a simple cliquet - you have the local caps (long vol) and a global floor (short vol) but depending on your path, level of vol and slope of the skew one can dominate the other. The usual approach is to overhedge some of the parameters of the structure itself (e.g. moving the cap or the floor against you). UVM seem to offer a nice alternative to the overhedge approach.

PS. after tinkering with it, I decided against using it, it's just too unstable.

"In Russia, every CDS ends in bullet payment"

ronin


Total Posts: 508
Joined: May 2006
 
Posted: 2019-10-31 19:35
I'm wondering if that is such a great example.

The buyer is long low strike vols, short high strike vols. The strikes reset so he doesn't know exactly what strikes he is long and short at any point, but he knows he is long low strikes and short high strikes.

Ergo, he bumps the skew down. It marks down low strikes that he is long, at least relatively to the higher strikes that he is short. The position is marked conservatively.

I'm racking my brain trying to think of a payout where you would genuinely have no idea where you are long and where you are short. Some sort of cliquet-resetting range accrual? Nope - that's a simple smile bump. Lookbacks? Nope, that's just like the cliquet. Passport options? Nope - the details are hazy, but the overal exposure is clear.

And we already deep in the territory where you wouldn't be using simple conditional vol scenarios anyway.

Just don't see it.

"There is a SIX am?" -- Arthur

Strange


Total Posts: 1595
Joined: Jun 2004
 
Posted: 2019-10-31 23:17
The local cap is being reset periodically, thus creating pockets of long convexity after each reset. On the other hand, the global floor is with respect to the value of the portfolio. The uncertainty of the exposure comes from that path dependency. Take, for example, a very plausible scenario where you are resetting into your local cap but the whole portfolio is already at the global floor. It's not clear what convexity profile this position presents, it's going to depend on many things (e.g. time to maturity).

There is a bunch of other structures (like some fairly common autocallbles) where the value of the barrier exposure changes with time and spot, making them very hard to track. When managing a book of these, usually people overhedge the barriers one way, check the value of the overhedges on regular basis and change the direction of the overhedge when the value flips.

FWIW, none of these structures are managed using UVM. I am simply illustrating that in plenty of cases you do not know which way the exposures lie and up doing various tricks to deal with that problem.

This all aside, people use UVM for mostly-vanilla call spreads on illiquid names. These are a byproduct of the convertible bond issuance and are pretty long dated (3-6 years). Because of long maturity and some mismatch in expiration, it's hard to know if you are simply dealing with a skew position or is it going to be a pure convexity position at some point. So for a name that shows no visible spot/vol correlation, it's much easier to come up with max and min vol levels and back out the sk10 number to quote.

"In Russia, every CDS ends in bullet payment"

ronin


Total Posts: 508
Joined: May 2006
 
Posted: 2019-11-01 14:46
I think we are talking cross purposes here. The skew exposure in your cliqet example doesn't flip, so you don't need some on-the-fly adjustment. Same thing with autocallables.

And where the exposure does flip, like corridors, that just tells you that your primary exposure is smile rather than skew.

I am happy to concur with call spreads locally reducing to just gamma before expiry.

So is that basically it - vanilla call spreads? OK. I guess...

"There is a SIX am?" -- Arthur

Strange


Total Posts: 1595
Joined: Jun 2004
 
Posted: 2019-11-01 21:21
Darn, I was not expressing my thoughts clearly and we ended up in the weeds.

1.What is the general the idea behind UVM?

(a) in absence of magical hedges and MtM, your "actual" risk is gamma
(b) at inception sign of convexity exposure is uncertain due to path dependency
(c) so you overhedge long gamma by managing it at low vol
(d) and overhedge short gamma by managing it at high vol
(e) the cost of these overhedges is discounted to today and included in price

That thought process makes it sort of a reasonable universal model. However, it does make some implicit assumptions:

(a) you are in position to dictate pricing to you can manage your position conservatively
(b) you are living in the world where you do not really trust the existing vol market and can avoid marking to market

Of course, both assumptions are usually false, thus limiting the usefulness of the model.

2. Why/when is pricing and managing on UVM is better than making explicit vol and skew bumps?

(a) UVM is a conservative model for managing uncertain convexity exposure (see above)
(b) even in vanilla options, a combination of vanna and theta can quickly flip your convexity exposure
(d) you can imagine simple cases when pricing with both conservative vol and skew will end with unpleasant management path

Imagine that you are selling a vanilla 5 year 100/150 call spread (let's keep it simple):
(a) you think the fair vol is 75 (stock's been realizing anywhere between 50 and 100 vols)
(b) at inception, your exposure is long vega and short skew
(d) you price it with bumped-down vol (let's say 50) and conservative skew (let's say sk10=3.5, if you can get away with it)
(e) a volatile selloff ensues, you end up short vol and managing the position at bumped-down vol - you mark vols up and take a loss
(f) UVM would have forced you to manage the initial position at high vol since it's short gamma

Does it make sense now?

"In Russia, every CDS ends in bullet payment"

nikol


Total Posts: 830
Joined: Jun 2005
 
Posted: 2019-11-01 22:03
jeez.

Just one though/idea:

Can't you formulate this whole thing in terms of HJB-equation with switching vol up-down threshold used for hedging?

If I imagine this process correctly, it resembles me Stoikov-Avellaneda MM-formulation.. (Avellaneda again is not for nothing, I guess)

ronin


Total Posts: 508
Joined: May 2006
 
Posted: 2019-11-03 14:17
Yeah. This is beginning to sound a bit like that story about NASA spending millions on a space pen, and the Russians using a pencil.

Which, by the way, isn't true - https://www.scientificamerican.com/article/fact-or-fiction-nasa-spen/

But still.

Yes, you could hedge a call spread with some massively over-engineered contraption like this. Or, you could just book each leg with the appropriate vol bump.

Up to you, really.

"There is a SIX am?" -- Arthur

willis


Total Posts: 23
Joined: Feb 2005
 
Posted: 2019-11-04 02:12
How do you know the appropriate vol? Or more basic, what delta to run?

[edit] in response to ronin's response -

"Yes, you could hedge a call spread with some massively over-engineered contraption like this. Or, you could just book each leg with the appropriate vol bump."
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