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Countec


Total Posts: 3
Joined: Oct 2017
 
Posted: 2017-10-16 20:26
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: 12
Joined: May 2017
 
Posted: 2017-10-16 21:06
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 = VarSwap-VolSwap

All you need to do now is to solve the quadratic equation for RealVol...

-kloc

Countec


Total Posts: 3
Joined: Oct 2017
 
Posted: 2017-10-16 22:21
Many thanks kloc for this answer.

quadratic equation was the keyword i was missing out!

Countec


Total Posts: 3
Joined: Oct 2017
 
Posted: 2017-10-16 23:57
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: 12
Joined: May 2017
 
Posted: 2017-10-17 05:18
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: 22
Joined: Dec 2007
 
Posted: 2017-10-21 16:26
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
 
Posted: 2017-12-18 10:50
k_vol can be approximated by a specific quantity a straddles at first order. More fancy approximations are due to Carr and Lee in Robust replication of volatility derivatives.
A good practical implementation is detailed in the book Applied Quantitative Finance for Equity Derivatives by Jherek Healy.

frolloos


Total Posts: 22
Joined: Dec 2007
 
Posted: 2017-12-22 04:36
@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
 
Posted: 2018-01-29 23:51
@frolloos
Very interesting paper! I am impressed. Initially I thought there was a mistake in your Table 3 with the Carr-Lee prices. In reality you give the ATM vol in this table (only table 1 contains carr-lee 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: 22
Joined: Dec 2007
 
Posted: 2018-01-30 08:45
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.
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