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Quantitative Finance

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  • $\begingroup$ Delta hedging is inherently statistical - even in absence of jumps, there are paths where standard deviation is exactly matching the implied volatility yet you make or lose money. example - you buy an ATM option but the underlying realizes smaller moves in a single direction but then realizes large moves when the option is OTM and your gamma is much lower. $\endgroup$ Commented Mar 28, 2017 at 0:25
  • $\begingroup$ @StudentInFinance, The answer of Daneel Olivaw concerning the fact that jumps are not hedgeable is perfectly correct. However, note that if you used real market data (and notably real IVs) to your theoretical replication error: $.5\Gamma(S_t,\sigma)S_t^2( (dS_t/S_t)^2 - \sigma^2 dt)$ also adds the mark-to-market error (out-of-model risk) due to the fact that IV changes as well i.e. $ \nu (\sigma_{t+dt}-\sigma)$ where $\nu$ is the BS vega and $\sigma_{t+dt}$ the new implied volatility for your option ($\sigma$ = the IV you've used to hedge). $\endgroup$ Commented Mar 28, 2017 at 8:08
  • $\begingroup$ Thus, rather than simple BS hedging, investment banks incorporate some rules to capture the spot/IV dynamics (so called sticky rules, or stickiness assumptions). $\endgroup$ Commented Mar 28, 2017 at 8:10