In theory, as time till expiration increases, ITM and OTM options should asymptotically approach absolute value of 0.5 (half unity). I am looking at DJX Dec 2008 put options. Today, 5/18/2007, DJX is at 135.57. The ITM put with strike of 136 has delta of -0.253; and the 147 put has delta of -0.498. My question is, why does the 136 ITM put have delta less than 0.5 in magnitude, and why it take such far depth in-the-money to have delta close to |0.5|? What am I missing? Something else strange, too. I recall a few days ago, extreme ITM call on something (SPY or some index) had theta of close to negative one (-1), and extreme ITM put had theta close to zero, 0. I had expected both extreme ITM call or put to have theta close to zero. It was on IB Option Trader. I am unable to explain why. Any comments?

You need to look at the forward value of the DJX, not the cash value. The delta values will then make sense.

Thanks for the suggestion. By "forward value," I assume you mean the value at T=(Dec 2008), which is what everybody thinks DJX will be at time = T. If T=1.5 (year from today), DJX today is at 135.57 and has IV of 14.67% today, then 135.57exp(0.1467*1.5) = 169 (value too high) Using risk-free interest rate of 5.25% instead of 14.67% IV, the above expression is 146.68, which is the DJX value that gives delta of about 0.5. This is mathematics without macroeconomics. DJX increased by about 10 points last month or so, or about the same distance from 135 to 146.67. I don't trade futures. But I imagine Dec 08 Dow futures will be at a value corresponding to 146 of DJX. Comments?

You are right. You should use the risk-free interest rate 5.25% for the calculation, and the fair value of Dec 08 Dow future should be 146.68. To be exact, you should use the future risk-free interest rate. Since it is unknown, most traders will use their own prediction of the interest rate or the current interest rate.

Forward value = cash + risk free interest rate - future dividends. But estimating future divs on the index constituents ain't easy ! It depends what accuracy you're striving for, but I would suggest that if you're going out as far as Dec08 then just using the DJIA yield as a substitute for actual $ dividends will do. For example, if the risk free rate is 6% and the DJIA is yielding 4% then set interest rates in your model / calculations to (6%-4%) 2%. Ideally, if futures are available for the month you're interested in then take the futures value as the forward value - and remember to set interest rates to 0% any model you use.

I don't understand this sentence. From the subjectline I suppose you mean the delta. The options themselves asymptotically approach max(strike-exp-value, 0). The delta's will become 1 and 0 resp. Ursa..

Not a typo. As time or volatility increases, ITM and OTM option deltas approach 0.5, whereas ATM delta remains at 0.5. The intuitive way I think about it is: as volatility (entropy) increases, it is easier for option price to jump to ITM or OTM territory across the ATM boundary; if volatility is zero, all particles are frozen, and the probability of any moves is zero, and thus, ITM remains ITM; OTM remains OTM. In extreme heat (volatility), a particle can be anywhere; it is as likely to be at location x as at y. Thus, the 0.5 delta.