If an implied volatility of an out of the money call option goes to infinity,what happens to the delta of the said call option?


closed as off-topic by Ric, Quantopik, olaker Jun 15 '15 at 15:01

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    $\begingroup$ Well, first you should have a look at the black scholes formula and think about the implications of an infinite implied volatility $\endgroup$ – muffin1974 Jun 15 '15 at 6:22
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    $\begingroup$ This is really basic ... let one parameter go to infinity ... $\endgroup$ – Ric Jun 15 '15 at 9:27

The Black-Scholes delta: $$\partial_SC=N\left(\dfrac{\ln\left(\frac{S_0}{K}\right) +(r - q + \frac{1}{2}\sigma^2)(T - t)}{\sigma\sqrt{T - t}}\right)$$ As you can see this delta would go to$1$ if $\sigma\to\infty$ (and $t<T$).

  • $\begingroup$ Is there any economic interpretation behind what it means if the implied volatility goes to $\infty$? $\endgroup$ – muffin1974 Jun 15 '15 at 13:16
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    $\begingroup$ Remember a call option will never be more expensive than the stock. As the implied volatility goes to infinity, the price of the option approach to the stock. Since, the value of the option can't go over the stock, it'll act like the underlying stock. Therefore, the delta is 1. $\endgroup$ – SmallChess Jun 15 '15 at 15:26
  • $\begingroup$ If the volatility goes to infinity, the stock price will also grow to infinity and the the option will end in the money. That is, a whole unit of stock is needed for hedging. $\endgroup$ – Gordon Jun 15 '15 at 17:32
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    $\begingroup$ @Gordon I don't think that is correct. The current stock price is finite, but its future volatility is not. There exist distributions with finite expected value but infinite variance (e.g. cauchy), so you cannot infer that you expect infinite stock price from infinite variance. $\endgroup$ – emcor Jun 15 '15 at 17:35
  • $\begingroup$ I do agree with @emcor as the reasoning will be obviously wrong for put. The value is the discounted strike when vol is infinite so the infinite value of the stock would be incompatible with this. This was actually a good question it doesn't deserve to be closed $\endgroup$ – Jiem Jul 31 '18 at 8:39

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