Vol Swaps

• Call delta > probability of call expiring ITM

  Absolute(Put delta) < probability of put ends up ITM

  
• Why does zero delta straddles have strike above spot?


  Ito-correction. Strike (%) of zero delta straddle = $\exp(r + \sigma^2/2)T$
• Long gamma position can sit on the bid and offer, short gamma position have to cross the bid-offer spread
• ATM option premium is $\approx 0.4*$$\text{vol} *$$\sqrt{T}$
• Profit of delta hedging is proportional to square of returns

  
• Why variance swap is a more correct measure for deviation?
Variance takes into account implied vol of ALL strike prices, while ATM implied will change with spot, even if vol surface doesn't change
• Why variance is a more correct measure for deviation vs volatility?

  Variance takes into account implied vol of ALL strike prices, while ATM implied will change with spot, even if vol surface doesn't change

  
• In terms of fair price:

  Vol Swap <= Gamma Swap <= Variance Swap

  Square root of variance strike is always above vol swap due to convexity. The fair price of gamma swap is between vol swaps and var swaps.
• Volatility Swap Payoff

  $(\sigma_{future vol} - \sigma_{swap ratevol}) *$$\text{vol notional/vega}$
• Variance Swap Payoff

  $(\sigma_{future vol}^2 - \sigma_{swap ratevol}^2) *$$\text{vega}/(2 * \sigma_{swapratevol})$
• Variance swaps are hedged with portfolio weighted 1/K^2

  K is the strike price of option
• Gamma swaps are hedged with portfolio 1/K

  K is the strike price of option
• Vega profile of vol swap is a flat line (as vega is equal to the vol notional)
• Var swaps are long skew and vol surface curvature.

  1/K^2 weighting means a larger amount of OTM puts are traded than OTM calls, so it is long skew.

  
• Volatility swaps are short vol of vol (volga) compared to variance swaps.

  
• In theory variance swap has zero delta, but in practice it has a small "shadow delta" due to spot-vol covariance.
• Variance swap vega decays LINEARLY with time
• Vanna can be thought of as the size of the skew position (in a similar way that vega is the size of a volatility position)

  Vanna = dVega/dSpot
• Options have their peak vega when they are approximately ATM
• When will the profit of delta hedging be NOT path dependent?

  Implied vol = realised vol
• Term structure is affected by:

  1. Call overwriting
  2. Structured Products
  3. Convertible Bonds
  4. Protection Buying

  
• Variance swaps require no delta-hedging
• Unlike P&L of a delta-hedged vanilla, the payoff at maturity for variance swaps (long) will always be positive when realized vol exceeds the strike
• For a delta-hedged position, the daily PNL is roughly driven by: the difference between realized and implied VARIANCE, multiplied by DOLLAR GAMMA)

  (assuming zero rate, constant vol, negligible higher order sensitivities)

  
• How you can still lose money on delta-hedged position despite being right about implied/realised spread

  
• Rules of thumb approximation: