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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: 4
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: 4
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: 5
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
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