The BS model assumes a constant vol accross strike, K, and time,t. i.e Dvol/DK = 0 and Dvol/Dt = 0.
The market has a non-const vol so that it prices out and in the money options higher relative to the ATM, hence the vol smile or vol skew. If there is a crash then the ITM and OTM options have that extra bit of premium so the seller is protected in the case of a crash (like an insurance policy).
Try messing around with put-cal parity with arbitrary vol surfaces and see if you can arb!!!!!
brickie - Implied Volatility Skew attempts to account for different implied volatilities between options at the same strike price in different months, or between different strike prices in the same month. The higher the Implied Volatility (IV), the higher will be the price of the option, given that all other variables are the same.
This is particularly useful when constructing a multi-leg strategy, as normally you seek to sell expensive options (high IV) and buy cheap ones (lower IV).
For example, you may see that for a particular share/index the IV of the June 100 calls is 20 and the IV of the Sept 100 calls is 15. You suspect that the price of the underlying will be close to 100 at expiry of the June calls, so you could sell the June 100 calls and buy the Sept 100 calls. At June expiry, with the underlying at say, 99, the June calls will expire worthless but there will be some value left in the September calls. Given the IV skew, the premium that you have received from the sale of the June calls will be greater than the loss of premium you have had to pay for the September calls, because:-
a) the IV is higher for the near month calls, so there is more time value to decay (and keep, because you are short).
b) timevalue decays most quickly in the run up to expiry, so even if IV was the same, the time decay during June for the September series will be less than for the June series.
This is a calendar spread. As always, the devil is in the detail. Particularly with stocks, you need to have some idea as to WHY there is an IV Skew. Even if the near month IV is astronomically high and the far month is less high, you may still lose even if expiry of the near month takes place in the area planned, if the IV of the far month falls significantly (known as a volatility crush). This is particularly prevalent in, say, biotech stocks where trial results are known to be due on a specific date. The near month IV may be 300% and the far month "merely" 200%, but that doesn't make it a good calendar spread play because after the announcement the likelihood is that IV will plummet, and also the stock price may move dramatically away from the planned strike.
Have a look at GNTA on the nasdaq for Monday 030504. A drug test announcement was due that day, and in the days before, IV for May was over 300%, and based on prices at the time you could sell a calendar spread which gave a profit provided the underlying price was between $9 and $25 at May expiry. As you can see, the news was leaked on Friday 300404, and the price dropped over 2 days from $15 to $4.50.
Back months invariably have their IV hit after earnings - there will be loads of put buyers going into earnings as an insurance against bad numbers, and once the numbers are out the IV will come off as the demand for puts dries up.
For these calendars to work well, IV of the long position should be really low, so that it has nowhere to go. Just because it is low to where it has been is not enough if it is high in nominal terms. Also just because there is a big skew between front and back months doesn't make it a low risk trade. It's the crush on the far month IV that does the damage, and a crush on the near month as well makes little difference. The profit comes from the value left in the far month longs.
There are numerous strategies where you can use the skew in your favour, the principle being that you try to sell expensive options to fund the purchase of cheap ones.