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Skew and Convexity Trades

Since VS are priced off the entire skew curve --> significant exposure to skew and convexity.


Variance swaps are long skew and convexity.  

In theory exposure to skew is neutral, but typical shapes of skew surfaces mean that increasing the skew increases the convexity, and really it is this which affects the VS price. 


Straddles vs. variance swaps

One possibility for owning convexity is to trade variance swaps against delta-hedged straddles.

Variance swaps --> long volatility and long convexity

Straddles --> long volatility


VS - Straddle --> leaves a long convexity exposure


This hedge is only effective providing that the asset remains close to the straddle strike. 

--> straddle's exposure to volatility (gamma/vega) is reduced as underlying trends away from strike. 

To maintain as pure as possible an exposure to convexity --> straddle will need to be re-struck as asset moves. 



Variance swaps plus naked short puts

One way of trading realised skew is to combine a put with a variance swap.


An investor wants to express a long vol position via variance swaps. 

The P&L at maturity is simply given by:  P&L = NVega × (σ2 −K2) / 2K


Volatility is directional --> it helps to explain commonly observed shape of index skew curve: 

- relatively steep put skews --> expectation of increased volatility on the downside

-  flatter call skews --> likelihood of more benign volatility on the upside


To a first approximation this type of skew curve can look somewhat like the payoff from a long put. 

--> it should be possible to partially offset the observed negative correlation of volatility and market direction by trading short naked puts against long variance swap positions.


Consider an investor who wishes to be long volatility, believing that RV will be above that encapsulated by the skew: 

- he would be willing to be long volatility at a higher than prevailing variance strike if the market sells off

- in return for being long volatility at a lower strike if the market rallies 


This exposure can be approximated by combining a long VS with a short put.


Suppose we sell a put of strike SP for absolute premium P for NPut notional of underlying.

The combined P&L of the short put and long variance swap at time T is:


Vs put

where the adjusted strike is now dependent on the underlying’s price at maturity ST and is written as:



We can set the notional of the put such that on the upside our BE is equal to the lowest RV if market has rallied. 


Shorting puts against a long VS position neutralises some of the long downside puts embedded within a VS.

Combinations of short options can be used to remove exposure to RV if asset moves above/below a certain level.


This observation  leads to the concept of third generation volatility products such a conditional VS.

Since conditional VS are priced from only part of the volatility skew --> more direct skew exposures.

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