Countec


Total Posts: 3 
Joined: Oct 2017 


Hi All
i am unable to figure out the formula how to compute the breakeven realized vol level if i am long a variance swap, short a volswap with equal notional but two different strikes.
meaning at what realized level would the net PnL be zero (there are two answers due to the var's convexity.)
sure the excel solver can very easily generate the answer but i'd like to know the mathematical formula behind.
any ideas? thanks 




kloc


Total Posts: 15 
Joined: May 2017 


Just write down vol swap and var swap payoffs as:
VolSwap = N*(RealVol  K_VOL)
and
VarSwap = N/(2*K_VAR)*(RealVol^2  K_VAR^2)
where RealVol is realized vol, K_VOL and K_VAR are vol and var swap fair strikes, and N is the CCY notional amount for your swaps. Factor 1/(2*K_VAR) makes them have identical vega notional (I'm assuming that's what you really mean when you say "equal notional").
Your total portfolio is now
Port = VarSwapVolSwap
All you need to do now is to solve the quadratic equation for RealVol...
kloc 



Countec


Total Posts: 3 
Joined: Oct 2017 


Many thanks kloc for this answer.
quadratic equation was the keyword i was missing out! 




Countec


Total Posts: 3 
Joined: Oct 2017 


i am having a bit of trouble to come up with the solution
i decomposed the convex part and the linear part (PnL of realized leg for both swap and PnL for fixed leg of swap) to determine a and c for below equation. assuming b=0
meaning
a= varunits x 100 x R_vol^2  vega notional x 100 x R_vol c= varunits x 100 x K_var^2 + vega notional x 100 x K_vol
varunits=veganotional/(2*K_Var)
would appreciate a hint
thanks




kloc


Total Posts: 15 
Joined: May 2017 


Well, the portfolio is:
Port = N*(1/(2*K_VAR)*(RealVol^2  K_VAR^2)(RealVol  K_VOL))
So you should solve for roots RealVol1 and RealVol2 of
Port = N*/(2*K_VAR)*(RealVol^2  2*K_VAR*RealVol + K_VAR*(2*K_VOL  K_VAR))
i.e. you can ignore the overall factor and just set a=1, b=2K_VAR and K_VAR*(2*K_VOL  K_VAR) in the quadratic equation.
kloc 




frolloos


Total Posts: 73 
Joined: Dec 2007 


I don't know how you plan to calculate k_vol (k_var is much easier to calculate). But a good and quite simple approximation for the volswap strike k_vol can be found here:
http://onlinelibrary.wiley.com/doi/10.1002/wilm.10566/abstract
The whole pdf can actually be downloaded for free 



logos01


Total Posts: 5 
Joined: Jul 2014 

 

frolloos


Total Posts: 73 
Joined: Dec 2007 


@logos01:
yes, Carr & Lee's paper gives a 1st order correlation immune approximation to the volswap strike. But as shown in my paper there is a much easier/straighforward approximation which also happens to match Carr&Lee's approximation for 6mth tenor. 



logos01


Total Posts: 5 
Joined: Jul 2014 


@frolloos Very interesting paper! I am impressed. Initially I thought there was a mistake in your Table 3 with the CarrLee prices. In reality you give the ATM vol in this table (only table 1 contains carrlee prices). I tried table 3 out of curiosity and for T=1 obtain
0.900000 CarrLee 887.126790 29.784674 27.796325 0.500000 CarrLee 886.957929 29.781839 28.092351 0.000000 CarrLee 886.907918 29.780999 28.223476 0.500000 CarrLee 886.933754 29.781433 28.095967 0.900000 CarrLee 886.996106 29.782480 27.836056
the first number is correlation, last is vol swap price (two others are var swap prices in variance/vol). This is close to the theoretical value and close to your own method. Still your method is much simpler (no replication at all). It is a bit amazing that a local approximation leads to so good results. Does this mean you can hedge with a single (theoretical) option (of course you will need to interpolate in practice). This is quite surprising compared to the straddle like replication of Carr and Lee.





frolloos


Total Posts: 73 
Joined: Dec 2007 


Thanks for your enthusiasm logos01  don't hesitate to spread the word how easy the method actually is! :)
Maybe the tables are not very clear: per maturity the first row is our approximation, the second row the ATM vol approximation (which is not recommended)
Thanks for running Carr & Lee for T=1, practically identical values to the D2=0 approximation, so that's good.
Regarding hedging: yes I do think it means you can hedge with a single option, but you'd need to rebalance frequently in order to always have the option with zero vanna and volga. 



 

Strange


Total Posts: 1580 
Joined: Jun 2004 


@frolloos, I was trying to replicate the analytical approximation you've referenced (maybe in a different thread) and I get very different results vs a replication model (variance swap  a quadratic payoff replicated with options on realized variance). 
“My dear, here we must run as fast as we can, just to stay in place. And if you wish to go anywhere you must run twice as fast as that.” 


frolloos


Total Posts: 73 
Joined: Dec 2007 


@Strange: I was able to match the result to those of Carr and Lee for 6mth maturity, and I think logos01 matched for other naturities as well. I am not sure how you did the crosschecking / comparison?
I can send you the matlab scripts I used if you want. Just pm me.
The advantage of the analytical aprrox is that it is very easy to implement compared to Carr and Lee but just as accurate. I've recently generalized it to forward startng volswap strike.
Anyway, let me know if you'd like more info.
Some historical background: it was my co author who had the interesting and correct trading intuition that the volswap strike is the d2=0 implied vol. I asked him if there is a proof his intuition is correct. he said there was none and that it can't be proved. all i did was to provide the proof and subsequently tried to generalize it to forward starting volswaps 




Strange


Total Posts: 1580 
Joined: Jun 2004 


@frolloos
> I am not sure how you did the crosschecking / comparison Well, the general thought is that vol swap = var swap  convexity adjustment, so I took the var/vol as quoted in the broker market and backed out the vol swap prices. My expectation was that the approximation would be somewhere in the ball park.
> The advantage of the analytical aprrox is that it is very easy to implement compared to Carr and Lee but just as accurate. I've recently generalized it to forward startng volswap strike.
That's exactly why I was interested  I don't really need to price them perfectly (I am on the buyside), but I do want to match the market somewhat.
> volswap strike is the d2=0 implied vol Oh, I think I see. That approximation breaks down if the vol is strongly correlated to the underlying, right? An equity derivatives trader would always feel that strikeless volatility product should trade at a premium to the fixed strike vol and display some delta.
PS. I recall there was a closed vol swap approximation for SABR with beta=1, but i don't have the skills to derive it.
PPS. you don't have a PM in your profile :) 
“My dear, here we must run as fast as we can, just to stay in place. And if you wish to go anywhere you must run twice as fast as that.” 


frolloos


Total Posts: 73 
Joined: Dec 2007 


When I was eqd pm I remember that for SX5E the varstrike does not match the theoretical strip of options, especially for longer tenors. This dislocation was due to structured products rehedging. Hence strip vs varstrike was one of the trades that could work. Is SX5E the index you are lookig at? Because if it is then there could be a mismatch between the quoted strikes and the analytical approximation/value for volswaps as well.
For SPX the strip is much more in line with the IDB varstrike quotes. So for SPX the volswap approximation also should be more in line with the quoted IDB volswap strikes as well.
The analyticalapprox should be robust across all correlations but yes if high correlations and long time to maturity accuracy decreases slightly. However for short tenors the approx should be very very close to true theoretical value if the smile is generated by a stochastic volatility process. but if the market prices volswaps using local volatility then the difference could be higher.
You can mail me at frolloos at yahoo dot com. 



