When your options are deep in the money

Shouldn’t they trade at a discount relative to options just at the money?

For example:

Two call options. Same expiration date.

Stock Price at 150

Option 1: strike at 100

Option 2: strike at 150

There is a lot of downside risk for the very in the money option (1) compared to option 2 because of the higher absolute cost.

Seems to me that they don’t usually trade that much more cheaply than the other. I’m guessing that I’m missing something in terms of hedging.

Do you have an example with option prices? The 100 strike call in your example should be worth about the intrinsic value of $50. The 150 strike call will be worth more than its intrinsic value. So, the OTM call has less “volatility value”. Is this what you mean?

Take these AAPL call options for May 11th. AMERICAN OPTIONS

530 - 36.85 price

565 - 8.60 price

535 is deep in the money ( in the money by 30 dollars roughly based on Friday’s close). Premium over what you could execute it for is only about 1.85

565 which is at the money costs about 8 dollars in premium.

Ok. So, it might help to look at this diagram:


Notice that far away from the strike, the option value (the dotted line) converges to intrinsic value (the solid line). Near the strike, option value has a premium over intrinsic value. This is because near the strike, the payoff is not symmetric - the gain from stock price going up is more than the loss from stock price going down.

don’t know if this is relevant to what you are talking about, but …


there’s a parameter in the black-scholes formula , sigma , the volatility,

>>In the Black–Scholes model, the theoretical value of a vanilla option is a monotonic increasing function of the Black–Scholes volatility.

>>at-the-money options tend to have lower implied volatilities than in- or out-of-the-money options

Volatility smile/skew might be a factor in this case, but it is not necessary in explaining why ATM options have a higher premium over intrinsic value. You would still see the ATM price premium if you assumed static volatility.

I would think that deep ITM options would be less liquid, explaining part of the volatility smile (though there is skew in the smile too, likely reflecting risk aversion that pushes up the implied-realized volatility gap on the put side.)

EDIT: rethinking, I guess if they can be dynamically hedged, then the options should have similar liquidity across most strike values… but I’d be interested in what more options-savvy people say.

Well, in general, far ITM or OTM options are indeed less liquid. Few people want to trade say, 80% ITM calls - might as well just buy stock. However, this has little to do with the actual volatility smile or skew. Illiquidity is generally ovserved through wide bid/ask spreads. There would be no reason for a *directional* bias from illiquidity, since both buying and selling are difficult for illiquid contracts.

Volatility smile or skew reflect risk aversion or potential for non-symmetric shocks. This is why gold has positive skew (OTM calls are more expensive than OTM puts), but risk assets like stocks generally have negative skew. There are some exceptions, obviously.

So puts on equities are normally more expensive than calls, Ohai?

Er, I guess I should be more specific. First of all, “expensive” for options refers to implied volatility. This helps you adjust the option prices for different strikes. Implied volatility explains why, for instance, a 10% OTM call option price can change even though the underlier price does not change.

Second, *OTM puts* are generally more “expensive” than *OTM calls*. The expensiveness depends on the strike, not based on whether or not it’s a call or put. Puts and calls with the same strike have the same implied volatility through put/call parity. And in general, implied volatility increases for equity options as strike decreases.

By extension to this logic, ITM puts are generally cheaper than ITM calls, since ITM puts have higher strikes than ITM calls.

Note that all this assumes negative volatility skew, which is normally the case for equity options. Some equity volatility surfaces can have positive skew. For instance, AAPL had positive skew a few weeks ago when people thought the price could keep spiking upwards.

IMO, it’s important to differentiate between equity index options and single stock names. Equity index options normally have negative skew because the skew prices the correlation of the index constituents - it’s not strictly because of supply/demand factors. This is different than skew observed in single stock names, which often times are earnings driven.

I’m not sure what you are trying to say. Correlation certainly affects the implied volatility of index options. However, it all gets derived from market prices at some point. The implied volatility of index options implies a market correlation, not the other way around.

I saw an interesting article somewhere which tried to compare the price of an index option with the prices of market cap weighted ATM options on the components of the index and concluded that the price difference represented the effects of correlation among assets. I thought it was an interesting idea, but I would not want to have to be busy hedging all these things to keep them in balance in practice. I think the transaction costs would be enormous.