Assume it to be true that $dS = S\mu dt + \sigma(t)S dW$ where $\sigma$(t) is known.

Consider a call option with expiry $T$, currently $t = 0$.

For all $t \in [0,T]$, $\sigma(t) < \sigma_{impv}$ where $\sigma_{impv}$ is the implied volatility used to price the option.

How do we arbitrage?

My first thoughts were to go short gamma, since realized volatility is less than implied volatility.

Is there another way?

  • 4
    $\begingroup$ Short the option and long the replicating portfolio (self-financing delta hedge) would be the way to go if you can only trade that call option, that you know for sure that this dynamics will realise and that there are no frictions. But if the gamma of that option is close to zero (far OTM/ITM) you won't be able to lock in anything obviously. Also have a look at this quant.stackexchange.com/questions/33205/… $\endgroup$
    – Quantuple
    Mar 29 '17 at 19:11
  • 3
    $\begingroup$ I believe Paul Wilmott did some work on hedging with implied volatility or with the actual volatility; iirc hedging with the actual gives you a guaranteed final payoff but no guarantees on intermediate P&L while hedging on implied gives you better behaved P&L but the total profit is path dependent. Edit: here's a link: wwwf.imperial.ac.uk/~ajacquie/IC_AMDP/IC_AMDP_Docs/Literature/… $\endgroup$
    – Bram
    Mar 29 '17 at 19:52
  • 2
    $\begingroup$ Possible duplicate of Continuous delta hedge formula $\endgroup$ Jun 10 '17 at 10:19

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