volitility smirk implies fat left tail?
- Thread starter cotton
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Hi cotton, to me the easiest way to grasp this is to understand: BSM makes an assumption that asset prices are lognormal; i.e., LN(s) is normal. If we infer implied volatility from a BSM (implied vol is model dependent, but in FRM we don't go beyond BSM) and if the asset price behaves exactly and unrealistically per the BSM assumption of lognormal prices, then the implied volatility plot would be a flat line; i.e., constant volatility regardless of S/X.
On the left tail, I prefer to focus on puts not calls (although on the left, for each OTM put, there is a corresponding ITM call). A lognormal assumption would predict a flat line for implied volatility; but if there is a smirk, the market is assigning a higher price to the OTM put, which generates the higher implied volatility. The higher price is implying that the probability of "ending up" in the left tail (i.e., low stock price, such that the put can be exercised) is greater than the probability predicted by the BSM lognormal assumption, which is the fat tail.
In short: if BSM constant volatility were true --> implied vol would be a flat line. But higher price on left defines the higher implied vol, which is way of saying the probability of low asset price is greater (heavier tail). thanks,
On the left tail, I prefer to focus on puts not calls (although on the left, for each OTM put, there is a corresponding ITM call). A lognormal assumption would predict a flat line for implied volatility; but if there is a smirk, the market is assigning a higher price to the OTM put, which generates the higher implied volatility. The higher price is implying that the probability of "ending up" in the left tail (i.e., low stock price, such that the put can be exercised) is greater than the probability predicted by the BSM lognormal assumption, which is the fat tail.
In short: if BSM constant volatility were true --> implied vol would be a flat line. But higher price on left defines the higher implied vol, which is way of saying the probability of low asset price is greater (heavier tail). thanks,
that makes a lot of sense. a follow-up question:
for OTM put, it works. how about IMC call on the left tail.
IMC call, which generates the higher implied volatility. The higher price is implying that the probability of "ending up" in the right tail (i.e., high stock price, such that the call can be exercised) is greater than the probability predicted by the BSM lognormal assumption, which is the fat right tail. Then this would contradict the fat left tail, inferred from OTM put, right?
for OTM put, it works. how about IMC call on the left tail.
IMC call, which generates the higher implied volatility. The higher price is implying that the probability of "ending up" in the right tail (i.e., high stock price, such that the call can be exercised) is greater than the probability predicted by the BSM lognormal assumption, which is the fat right tail. Then this would contradict the fat left tail, inferred from OTM put, right?
i would interpret the situation as:
OTM put implies high volatility which means probability of going either way of the current stock price is higher. But ITM put has a lower bound so that the volatility is lower because there is only high probability of going up but not so much probability of going further down(bounded down by zero) so they are priced lower. The higher implied volatility means higher price which suggests that for OTM put the probability of stock price ending in left tail is higher due to further distance from the lower bound zero. As the distance widens from lower bound the chances of reaching below current price increases as compared to when the put is ITM when the distance narrows and chances diminishes.
Similarly compare the ITM call, here the ITM call is already is in the money so its price is higher as implied by higher stock price. High implied volatility implies that there is very large chance of stock ending up than the current price because there is no lower bound but an infinite bound upto which stock price can increase so there is relatively much larger chances of stock price ending above the current price there are also more chances of price ending up lower due to wider distance from lower bound zero which implies a higher overall implied volatility thus making the option ITM expensive. The expensive ITM call already has gains so that it reflects higher price sue to already reflected profits.
thanks
OTM put implies high volatility which means probability of going either way of the current stock price is higher. But ITM put has a lower bound so that the volatility is lower because there is only high probability of going up but not so much probability of going further down(bounded down by zero) so they are priced lower. The higher implied volatility means higher price which suggests that for OTM put the probability of stock price ending in left tail is higher due to further distance from the lower bound zero. As the distance widens from lower bound the chances of reaching below current price increases as compared to when the put is ITM when the distance narrows and chances diminishes.
Similarly compare the ITM call, here the ITM call is already is in the money so its price is higher as implied by higher stock price. High implied volatility implies that there is very large chance of stock ending up than the current price because there is no lower bound but an infinite bound upto which stock price can increase so there is relatively much larger chances of stock price ending above the current price there are also more chances of price ending up lower due to wider distance from lower bound zero which implies a higher overall implied volatility thus making the option ITM expensive. The expensive ITM call already has gains so that it reflects higher price sue to already reflected profits.
thanks
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