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lovenatalya
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Volatility Swap Hedging

April 6th, 2018, 10:44 pm

What are the hedging methods for volatility swap (rather than variance swap)? What are the possibilities of setting up a static, semi-static or dynamic hedging? 
 
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VolMaster
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Re: Volatility Swap Hedging

April 7th, 2018, 7:40 am

Vol Swap is more complicated to replicate and hedge due to the convexity adjustment which is required compared to variance swap. Ideally you need to buy/sell the entire surface (hence, trading the different delta with inverse weights to their moneyness), but that would expose you to convexity, which you will not be compensated with the vol swap. In practice the market makers either put the risk in their risk pool or replicate with vanillas and bear the convexity cost. When it comes to hedging the delta between the fixing it's rather simple calculation.
  
 
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lovenatalya
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Re: Volatility Swap Hedging

April 7th, 2018, 7:06 pm

@VolMaster:

Thank you for your reply. I have many questions some of which I suppose are due to unfamiliarity with the jargons.
Ideally you need to buy/sell the entire surface (hence, trading the different delta with inverse weights to their moneyness)
Is this to hedge the corresponding variance swap? Could you please elaborate on what you mean by "buy/sell the entire surface" -- by "surface" I suppose you mean the implied volatility surface, right? --- and what you mean by "trading the different delta with inverse weights to their moneyness", and why the former implies the latter? 
but that would expose you to convexity, which you will not be compensated with the vol swap. 
I suppose this confirms my speculation that your previous description applies to the corresponding variance swap.
In practice the market makers either put the risk in their risk pool or replicate with vanillas and bear the convexity cost. 
Are you saying people essentially leave the convexity risk unhedged, putting in the risk pool notwithstanding?

When it comes to hedging the delta between the fixing it's rather simple calculation.
What do you mean by "fixing" and "hedging the delta"? By the latter, do you mean hedging the delta of a vanilla option?

-----------------

Another question in addition to the questions above: 
Do you think Peter Carr and Roger Lee's paper Robust Replication of Volatility Derivatives offer any practical hedging approaches?
 
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VolMaster
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Re: Volatility Swap Hedging

April 8th, 2018, 4:29 am

@lovenatalya

Re your questions:

1. By surface I mean the volatility surface. The replication of a Var Swap, which provides a flat exposure to the implied volatility is done by trading the different strikes weighted to their vega, meaning that if you want to get flat vega exposure you need to trade the lower delta (hence lower vega) with higher weight to equate to the ATM strike. This actually what causes the convexity exposure, as a sharp move of the underlying vol/spot will affect your 2nd order derivatives to vega (volga/vanna)

2. Yes, the above is approximation to Var Swap ,  rather than Vol Swap. The only way to remedy this difference is by doing convexity adjustment to your weights, but this is not ideal.

3.  Market Makers sometimes just leave convexity unhedged,  and bear the risk.

4. Vol Swap (and Var Swap) are fixing based products, which fix on a daily/weekly basis. The risk of daily volatility comes from the underlying move between the fixings (which means that your products is essentially a strip of overnight ATM vanilla options). the way to hedge it is by trading the delta between the fixings, the same as trading the delta of vanilla option 
 
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lovenatalya
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Re: Volatility Swap Hedging

April 8th, 2018, 5:55 am

@Volmaster:

Thank you again for your detailed response. Before I read it through carefully, let me pose another question: 
Do you think Peter Carr and Roger Lee's paper Robust Replication of Volatility Derivatives offer any practical hedging approaches?
 
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VolMaster
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Re: Volatility Swap Hedging

April 8th, 2018, 11:32 am

Peter Carr (alongside Bruno Dupire) is the godfather of Volatility Derivatives, and as I know him and his work, he offers a practical hedging approach, yet it assumes no transaction cost for the hedging, which is far from being practical in some assets
 
frolloos
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Re: Volatility Swap Hedging

April 8th, 2018, 11:59 am

What are the hedging methods for volatility swap (rather than variance swap)? What are the possibilities of setting up a static, semi-static or dynamic hedging? 
Carr & Lee's approach is (almost) model free and from that perspective useful. But it's only an approximation to the volswap strike (correlation immune to first order). I don't think it's possible to have a full model free expression for the volswap strike. It's easy to see for example that the volswap strike under local volatility would be different than volswap under stochastic volatility.
The PDE approach is more flexible and has my preference. Google for instance Broadie & Jain volatility derivatives, or Javaheri, Haug, Wilmott Garch and volatility swaps to see how the PDE method works for vol derivatives. Broadie and Jain applied it to Heston model, the other paper to GARCH, but the PDE approach can be used for any model (eg local vol, LSV etc).
The PDE approach should also give you the correct hedge ratios, i.e. the number of variance swaps to use to hedge the volswap MtM.
Another approach would be to specify a model for the variance strike curve. Then you could have options on variance, and if you have those you could "easily" replicate the volswap using a strip of variance options.
Hope that helps.
 
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lovenatalya
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Re: Volatility Swap Hedging

April 8th, 2018, 7:35 pm

What are the hedging methods for volatility swap (rather than variance swap)? What are the possibilities of setting up a static, semi-static or dynamic hedging? 
Carr & Lee's approach is (almost) model free and from that perspective useful. But it's only an approximation to the volswap strike (correlation immune to first order). I don't think it's possible to have a full model free expression for the volswap strike. It's easy to see for example that the volswap strike under local volatility would be different than volswap under stochastic volatility.
The PDE approach is more flexible and has my preference. Google for instance Broadie & Jain volatility derivatives, or Javaheri, Haug, Wilmott Garch and volatility swaps to see how the PDE method works for vol derivatives. Broadie and Jain applied it to Heston model, the other paper to GARCH, but the PDE approach can be used for any model (eg local vol, LSV etc).
The PDE approach should also give you the correct hedge ratios, i.e. the number of variance swaps to use to hedge the volswap MtM.
Another approach would be to specify a model for the variance strike curve. Then you could have options on variance, and if you have those you could "easily" replicate the volswap using a strip of variance options.
Hope that helps.
Thank you very much for introducing me to the PDE approach and the references thereof. I will look into it. I am more interested in the vol swap with the volatility [$]\sigma[$] described by the so called inverse gamma process 

[$]d\sigma = -\kappa(\sigma-\theta) dt+\eta \sigma dB_t[$]

where [$]\kappa[$], [$]\theta[$] and [$]\eta[$] are time dependent (or constant) functions, such as described in this paper. Have you seen a PDE treatment of vol swap based on this process? If not, do you think the PDE approach can be readily applied to this process?
  
Would you please explain what "variance strike curve" is? By that nomenclature do you mean the weight of vanilla options as a function of different strikes to hedge a variance swap? If so, is that not already known? If not, are there any references for models on the variance strike curve?

With the approaches you mentioned, does it actually help, with all the transaction cost taken into account, to hedge away the convexity VolMaster says is usually left unhedged?
 
frolloos
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Re: Volatility Swap Hedging

April 10th, 2018, 11:33 am

Please ignore my remark about variance strike curve - was written on the fly and I'm actually not sure whether what I wrote makes sense in applying it to volatility swaps. But in any case, variance (strike) curve models are akin to market models in rates: you directly model the dynamics of variance strikes. For example see Bergomi "Smile dynamics II" and "Smile dynamics III" papers.

I have not seen the paper you mentioned. But I don't see why the PDE method shouldn't work with it. What you need in the PDE method is the variable the accumulated variance [$] I_t = \int_0^t v_u du [$] where [$] v_u [$] is the instantaneous variance. The volatility derivative price is assumed to be [$] F = F(t, v_t, I_t) [$], and using [$] dI_t = v_t dt [$] together with Ito will give you a PDE just like equation (16) in Broadie & Jain's paper. Together with the appropriate boundary conditions I think it can be solved.

To be honest I don't quite understand what Volmaster means. I suspect he means that volswaps are hedged using variance swaps (makes sense), but the question is how often you rebalance the hedge. If discrete rebalancing (again that makes sense) then there will be convexity mismatch (volswap is linear, varswap is convex), and this convexity is left unhedged (i.e. you don't rebalance your hedge continuously). But I'll leave it to Volmaster to confirm whether what I think he means is right.
 
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lovenatalya
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Re: Volatility Swap Hedging

April 11th, 2018, 5:06 am

@frolloos:
I see. Thank you.

@VolMaster:
Would you care to kindly respond to frolloos' query regarding the unhedged convexity?
 
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VolMaster
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Re: Volatility Swap Hedging

April 12th, 2018, 4:39 am

Let's start by acknowledging the fact that there is a great difference between the academic/theoretical world of quant finance and the day-to-day practice. As someone who has been trading FX volatility and volatility products for nearly a decade I rarely saw vol/var options traded (neither vol knock-out/millage options, or other volatility based products). The common practice is to price Var Swap as a weighted Vol Surface (depends on the assets, there might be different model for more risky assets, such as high-vol EM currencies or small indices, that will be priced using SABR or local-vol/heston model). For Vol Swap (which is more commonly used in FX market) market makers usually price the strike price as an average between the Delta-Neutral and ATMF vol. 
Once the product needs to be hedged (either using Var Swap or via Vanilla options) there is a convexity issue (due to the linear nature of Vol Swap compared to the convex Var Swap). Market Makers usually tend to leave it unhedged, especially in recent years, where market has become much less volatile. 

Hope that answers your question
 
frolloos
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Re: Volatility Swap Hedging

April 14th, 2018, 8:49 am

Regarding my previous comment on variance strike curve models: so if you have a model for the variance strike curve, then indeed it is possible to derive the dynamics for instantaneous variance, consistent with the initial variance strike term structure (a bit like HJM approach). You can then proceed to price volatility derivatives using PDE or MC. As far as I know, the first paper on this market modellig approach is by Dupire, "Arbitrage pricing with stochastic volatility".
 
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lovenatalya
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Re: Volatility Swap Hedging

April 14th, 2018, 9:34 pm

@VolMaster and @frolloos:

Thank you both for your valuable insights. I will come back with more questions later after I have done some study. 

@frolloos:
Do you know more papers on the variance strike curve market modelling approach? 
 
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fomisha
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Re: Volatility Swap Hedging

May 22nd, 2018, 8:03 pm

What are the hedging methods for volatility swap (rather than variance swap)? What are the possibilities of setting up a static, semi-static or dynamic hedging? 
Check out this post on consistent valuation and risk management of vol derivatives and vanilla options:
https://www.linkedin.com/pulse/vola-dyn ... fomytskyi/
 
frolloos
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Re: Volatility Swap Hedging

May 23rd, 2018, 1:07 am

What are the hedging methods for volatility swap (rather than variance swap)? What are the possibilities of setting up a static, semi-static or dynamic hedging? 
Check out this post on consistent valuation and risk management of vol derivatives and vanilla options:
https://www.linkedin.com/pulse/vola-dyn ... fomytskyi/
That looks interesting. Is there a technical paper on your valuation and risk mgt method or is it proprietary?