# Using Daily or Annual Volatility to Price an Option

From Joshi's Quant Interview book:

The statistics department from our bank tells you that the stock price has followed a mean reversion process for the past 10 years, with annual volatility of 10% and daily volatility 20%. You want to sell a Euro option and hedge it. Which volatility do you use?

Now, I see from the answers, since we want to hedge it, we should use the higher daily volatility since hedging will require (at least) daily rebalancing so we are exposed to daily volatility and thus use the 20% to price the option. This makes sense to me.

Now, Joshi has 2 follow-up questions that I am less sure about.

(1) What would happen if we BOUGHT an option off the bank using the 10% volatility?

-- To me it seems, this would be 'good' for us as it would be cheaper and perhaps the bank may be hurting themselves by 'underselling' the option and not being able to hedge it appropriately. Is my understanding correct? Is there something else to add?

(2) What if we could statically hedge the option today, does that change which volatility we would use?

-- I'm not sure. I assume if we could statically hedge, we could use either since we don't need to rebalance daily and so are not exposed to daily volatility. Is this correct?

Always great if you can buy the option on a cheaper vol. The choice you're faced with after purchase is the frequency at which you hedge. The disparity between daily and annual vol indicates (as the question states) a level of mean reversion or negative autocorrelation. The spread you've got implies a daily autocorrelation of -0.998!

Buying the structure from the bank at 0.1, you'd want to be hedging frequently, locking in large daily moves under the expectation that tomorrow's move will have the opposite sign (thus reducing the variance between t_0 -> t_n).

If you were able to statically hedge the structure then what matters is the volatility at which your static hedge is priced. If your static hedge is 10% and you're selling at 20%, then great - you're able to lock in the spread (law of one price) between the two.

-- If your static hedge is more expensive than 20% vol, then I think you'd want to be dynamically hedging the delta disparity to minimise the theta decay between your long and short by monetising the model delta. --

^ Not the case:

Hedging residual delta from a model vs market price won't help here to minimise decay. Actually what it will do is smooth out the equity curve on a daily level from a marked perspective (each day has less volatile p/l), but introduce more uncertainty into the final payoff. If you didn't hedge then your final payoff is certain (the spread), but the path is more volatile. This is analogous to hedging an option at realised volatility or implied volatility. If hedging at IV then you smooth out the equity curve, but the final payoff is uncertain. When hedging at future realised vol, the final payoff is certain, but the path is more volatile. As per Wilmott: https://web.math.ku.dk/~rolf/Wilmott_WhichFreeLunch

• A couple quick questions: To be clear, am I correct in saying that if you buy the option from the bank at the lower 0.1 volatility, then by hedging your position, you will gain money/profit? Also, how does hedging delta minimize theta decay between long and short?
– jmac
Mar 15, 2023 at 4:27
• q1) if you long at 0.1, and short at 0.2, then you've made money. Your future p/l is locked in, there's just a m2m disparity between long and short because of where/who you're marking with. q2) I got this wrong (and will edit my answer), hedging residual delta from a model won't help here to minimise decay. Actually what it will do is smooth out the equity curve on a daily level (each day has less volatile p/l), but introduce more uncertainty into the final payoff. If you didn't hedge then your final payoff is certain (the spread), but the path is more volatile. This is confirmed by sims. Mar 15, 2023 at 14:10
• @Newquant could you briefly show how you got the implied autocorrelation? Ty! Mar 15, 2023 at 19:06