ZeroBonus Wrote: ------------------------------------------------------- > by long T I meant >10yrs because small price > changes have almost little to no impact on the > option value. Again, I’m not a specialist in options, but I believe that having it long-dated means that the TIME VALUE doesn’t change very much. However, the INTRINSIC VALUE still changes normally. At the money, the intrinsic value changes from slope=0 to slope=1, so if it is truly at the money, the best estimate for the slope is 0.5 (mathematically it is undefined, but empirically, you’d estimate it as 0.5). So time value is more-or-less constant slope=0, intrinsic value changes with slope=0.5. Option price = intrinsic value + time value ==> slope of option value = 0.5 = delta
nvm I am assuming its an European option instead of American and so you can’t exercise until expiration date.
if the call option were any cheaper, it wouldn’t make sense for the seller to sell it as they would make better risk-free profits by selling their SPY position and investing directly in risk-free.
bchadwick Wrote: ------------------------------------------------------- > ZeroBonus Wrote: > -------------------------------------------------- > ----- > > by long T I meant >10yrs because small price > > changes have almost little to no impact on the > > option value. > > > Again, I’m not a specialist in options, but I > believe that having it long-dated means that the > TIME VALUE doesn’t change very much. However, the > INTRINSIC VALUE still changes normally. > > At the money, the intrinsic value changes from > slope=0 to slope=1, so if it is truly at the > money, the best estimate for the slope is 0.5 > (mathematically it is undefined, but empirically, > you’d estimate it as 0.5). > > So time value is more-or-less constant slope=0, > intrinsic value changes with slope=0.5. > > Option price = intrinsic value + time value ==> > slope of option value = 0.5 = delta I am thinking realistically. I would not see a value of a long term (>10yrs) call option change much if the stock price went up by 1pt.
even if it was a european option, it doesn’t make sense for an option that has a t=70. when the expected return on the SPY is 20,000%+, downside inter-day, inter-week, inter-month, inter-year and inter-decade moves are negligable. if the options only moves 0.5 as much as the SPY, after 35 years and the SPY is up 10,000%, and the call is only up 5,000%, it would be incredibly undervalued.
True, the price would not change very much. It would change by 1/2 a point.
JohnThainsLimoDriver Wrote: ------------------------------------------------------- > ZeroBonus is close. The delta for a long-dated > at-the-money option will never be close to 1. not true according to black-scholes. the delta of a call is the N(d1) term of the BS formula. For at the money option, d1 is approximately equal to 1/2*sigma*Sqrt(T) delta = N(d1) will only be 0.5 of d1 is close to zero. for long maturitities (T>10,20, 30 yrs), the d1 factor is significantly different than zero (unless your volatility is negligible). the delta becomes much bigger than .5 whether black-scholes is good enough to value long-term options, thats another story. i’d say not
MattLikesAnalysis Wrote: ------------------------------------------------------- > even if it was a european option, it doesn’t make > sense for an option that has a t=70. > > when the expected return on the SPY is 20,000%+, > downside inter-day, inter-week, inter-month, > inter-year and inter-decade moves are negligable. > if the options only moves 0.5 as much as the SPY, > after 35 years and the SPY is up 10,000%, and the > call is only up 5,000%, it would be incredibly > undervalued. obviously the delta is going to change as the option nears a more ‘normal’ time value. But as of today it won’t be more than .5
MattLikesAnalysis Wrote: ------------------------------------------------------- > but for example, using $100 SPY options as our > case, the range between Dec 2009 and Dec 2010 is > $6-$11. Dec 2010 to Dec 2011 is $11-$15. Obviously > each additional year will result in less of a > premium for that extra year. > > i guess i’ll be the only one who actually does a > calculation. > > using black-scholes assuming vol = .17, RF = > inflation = 3%, t = 70 years; S and K both equal > 100. > > = 100 N(d1) - 100 e^-(0.03)(70) N (d2) > > N(d1) and N(d2) both equal 0.999. this is where > most of the difference lies between a 1 and 2 year > option for example, but in this example, 70+ yrs, > it makes no difference. > > = 99.9 - 12.221 = $87.679 > > if S = $100 and the call option = $87.679, they > would move almost entirely in step as every > incremental gain made by the stock is inherently > an incremental gain in the call option, assuming > its american. > > thus, delta = 0.98+ i get N(d1) = .9857… which is equal to call option delta… this would change if you factored in dividends. then: d1 = ((ln(s/k) + (rfr - div rate + .5*volatility^2)*T)/(volatility*T^(1/2)) then call option delta would equal = N(d1) * e^-(T* div rate)
Matt and IheartMath remind me the guys on Big Bang Theory
so you’re saying that if you buy an $87 call option on a $100 stock price (technically they are perfectly correlated when the ER is 20,000%+, the price movements won’t be close over a 70 year period? the delta over the first 5-20 years may be around 0.5, but 20yrs from now when the SPY hits 3000, 300 in option tense, the correlation will be almost 1. and 50 years of it being almost at 1, will skew the delta over the life of the option. I think the upside correlation will be high starting at day 1, but if the SPY dropped 50% the option would barely move as there are 69 more years to recoup. over the life of the option, the delta will be .98+ for an american and much higher than 0.5 for a european, so long as you are not using daily movement which would be immaterial when judging a 70 year option anyway. all previous discussion on this by me assumed that we are talking about an american option b/c who buys european options anymore?
ZeroBonus Wrote: ------------------------------------------------------- > Matt and IheartMath remind me the guys on Big Bang > Theory i’ll take that as a compliment so long as i’m not sheldon.
^Rookie options mistake, you neglect to account for changes in gamma as the price of the underlying changes. Delta is not a linear function at large price movements.
ok, ill be sheldon after my sex change operation.
Matt: I was talking about what should you expect the delta to be today, not the average delta over the life of the option. Today the delta will be very low and increase over time.
ZeroBonus Wrote: ------------------------------------------------------- > Matt: > > I was talking about what should you expect the > delta to be today, not the average delta over the > life of the option. Today the delta will be very > low and increase over time. For at-the-money options the delta will not really change until right before expiration. A graph of delta vs. time would look something like a long straight line followed by a near vertical line at the end.
MattLikesAnalysis Wrote: ------------------------------------------------------- > all previous discussion on this by me assumed that > we are talking about an american option b/c who > buys european options anymore? you seemed to also be under the assumption that we werent considering dividends, in which case an american and european call option would be priced the same according to B-S anyway. (in your example, if we did come up with some continuous dividend return then delta would be much much lower due to the time frame) further, B-S definately cannot be used to price an american option anyway… especially with our ridiculous time horizon.
MattLikesAnalysis Wrote: ------------------------------------------------------- > all previous discussion on this by me assumed that > we are talking about an american option b/c who > buys european options anymore? BRK sold 20 year ATM puts in 2007 - so someone must be buying them!
JohnThainsLimoDriver Wrote: ------------------------------------------------------- > ZeroBonus Wrote: > -------------------------------------------------- > ----- > > Matt: > > > > I was talking about what should you expect the > > delta to be today, not the average delta over > the > > life of the option. Today the delta will be > very > > low and increase over time. > > > For at-the-money options the delta will not really > change until right before expiration. A graph of > delta vs. time would look something like a long > straight line followed by a near vertical line at > the end. Thats what I thought It’ll just look like an exponential graph that doesn’t rise until the very end of the option time frame
My mate WaBu has a good discussion of this stuff here: http://www.berkshirehathaway.com/letters/2008ltr.pdf on page 20