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For a call option, we know that the vega is the derivative of the price wrt to the volatility. However the volatility, in that context, actually refers to the implied volatility of the specific call contract at this moment, which is inferred from the option price. So, knowing this, what is the purpose of the vega ? We are essentially saying "if the implied vol rises by 1%, the price rises by X", but the cause that will trigger a change in the implied vol is the price of the option itself.

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The question is backwards, you should ask why we need the price! – brian Jun 27 '14 at 2:33

Think of it this way, the change in option premium is roughly the sum of changes due to delta, gamma, theta, rho, and vega. Now for a long dated option, gamma and theta don't change dramatically day to day, and let's say you have delta hedged your option. Then your P&L is mostly just driven by changes in implied vol – i.e., vega. So vega matters a great deal when you are trying to manage risks of an option book – you can isolate different sources of risk (due to underlying, due to time, due to implied vol) and hedges them accordingly. Similarly, there are also strategies to trade (almost) purely on implied vol – if you carefully hedge out all the other risks, then your P&L will be completely driven by vega. This, of course, is because of the one-to-one mapping between implied vol and option pmreium, just like you said.

I thought about it more. If you are familiar with bonds, I think they provide a good analogy. As you may know, bond price and bond yields also form 1-to-1 mapping, with "DV01" or "Duration" measuring the sensitivity of bond price to yield changes. The value of "DV01" (or duration) is that they provide a consistent framework for measuring interest rate risks. Two bonds might be trading at very different prices (e.g., if one has much higher coupon than the other), but they may have similar duration, so the two have similar risks. Simply put, bond prices are not particularly meaningful, but bond yields are more comparable (that's why we plot the "yield" curve, not bond "price" curve...), and yield-based risk metrics are also more useful.

Options are the same. Prices are not particularly meaningful, while implied vols are more comparable across different options, allowing for easier comparison, hedging, and P&L attribution. As I mentioned above, if you can carefully manage other risks, it's possible to trade vega directly. In fact, we frequently look at stuff like spreads of two implied vols and trade on relative value in this way.

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but in that case, can we really say that the remaining risk is volatility risk and not just "other types of risks not really explained" ? For instance, I understand that the remaining pnl is driven by vega, but that's just a way of saying that the remaining pnl is due to a change in the option price... which doesn't say much – lezebulon Jun 26 '14 at 19:46
@lezebulon You're absolutely right that the attribution I described is an approximation and there are residuals, but the approximation is a VERY good one in practice. The point is – if you just say price changed, that doesn't tell you anything about how to hedge. But if you have two options, option A has a vega of 2, and option B has a vega of 4, then you know that to hedge B with A, you need two units of A (all else equal). The point is, you now have a much more "controlled" framework to hedge using different instruments. – haginile Jun 26 '14 at 20:10
but since option A and B are different contracts, there is no reason that a change in A's i.v will result in the same change of B's i.v right ? So I don't see how this hedge can work – lezebulon Jul 1 '14 at 19:35

As an option trader you may well prefer model and forecast volatility rather than the specific price dynamics of the stock and option contract. This is a straight-forward consequence of the Black-and-Scholes framework (*). If you own an option, knowing vega allows you compute the expected PnL on your position and manage risks with respect to your realized volatility forecasts.

(*) As a matter of fact, the real-world stock-price drift is not an input of the B&S formula.

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I don't really see how computing the vega with model volatility changes anything. If the calibration is done correctly the implied vol given by the model is the same as the i.v. from the market. The fact that you may have volatility forecast for the future doesn't really change the current value of the vega for a given contract, at the current time. – lezebulon Jun 26 '14 at 19:35
(1) your calibrated model gives you the current view of the market on the option price. If the markets were perfectly efficient then you would be right but markets are inefficient so you might have different expectations of the future so you might forecast changes in the implied vol or ultimately a different realized vol down to maturity (2) @haginile explains it very well: you use the current vega to assess the pnl impact of shifts in volatility. Therefore, you can hedge positions by looking at vegas. – pincopallino Jun 27 '14 at 5:14

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