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- Options Path-dependency: Can Volatility be captured by delta-hedging?
Options Path-dependency: Can Volatility be captured by delta-hedging?
Suppose a market-maker buys and delta-hedges a vanilla option.
Assume that whatever volatility is realised is constant and option is delta-hedged over infinitesimally small dt.
Then the market-maker will profit IFF RV > IV at which the option was purchased.
HOWEVER:
1. Magnitude P&L depends not only on RV-IV, but where that volatility is realised in relation to the option strike.
If underlying trades near strike, especially close to expiry (high gamma) --> absolute value of P&L will be larger.
2. If volatility is not constant, where and when the volatility is realised is crucial.
RV-IV will count more when underlying close to strike, especially close to expiry (large gamma).
For non-constant vol --> possible to buy and delta-hedge an option at an IV < subsequent RV and lose from delta-hedging!
Example: Performance delta-hedging 1y SX5E option
Index initially @ 3500 with ATM IV = 28.5%.
First 7 months --> index ranged between 3500-3800 --> realising ~ 20% vol.
After that period --> index fell rapidly to 2500 --> realising ~ 50% vol on the way
Over whole year --> RV = 36%.
Case 1: Long 2500-strike option purchased at 33.4% due to put skew.
Initially, the option made a loss due to low RV, but it was kept small by low gamma exposure far from strike.
As index sold off, gamma increased, volatility picked up and option ended up making a large profit.
Case 2: Long 4000-strike option purchased at 26%.
Initially, the option has a higher gamma and lost more due to lower RV.
As index sold off and volatility occured, delta-hedging failed to capitalise on it because very low gamma far from strike.
--> loss despite the fact the option was purchased at an IV 10% below subsequent RV.
When delta-hedging daily an option:
- P&L is a daily accrual of RV - IV scaled by time step
- magnitude of contribution of this daily accrual is weighted by current 
which is unpredictable since path-dependent.
The non-constancy of exposure to (RV2 - IV2) comes from the changing over lifetime of option.
To summarise:
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Fig 99: P&L of delta-hedged option depends on the spread between RV and IV --> positive correlation.
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This dependence is relatively strong if the underlying trades close to the strike.
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The sensitivity to RV tails off rapidly as the underlying moves away from the strike.
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This sensitivity is accounted for by the dollar gamma.
If there was such an instrument where the dollar gamma remained constant both over time and for changes in the underlying, then the exposure to volatility would be constant.