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# Derivative of Delta with Respect to Theta?

Discussion in 'Options' started by deltahedge, Jan 13, 2011.

1. ### deltahedge

Hey everyone,

Invariably I use 2-4 month OTM call options to speculate on equities I believe will rise in the near term. However as we all know the delta of those options decreases as time passes (assuming vol and stock price are constant). Does anyone know if there's a second derivative greek of delta with respect to time (i.e. d (delta)/ d (time)) and where I could find a program that could calculate such a lovely greek?

Thanks!

2. ### Rodney King

"Charm" is the term for d(delta)/d(time)... which is not d(delta)/d(<i>theta</i>). http://optiontradingtips.com/greeks/charm.html

3. ### deltahedge

Sir King,

Thanks for the prompt response. So the question then remains where I can find a program where Charm is calculated?

4. ### Rodney King

I don't think canned software is likely to include charm or other 'exotic' Greeks (speed, volga, ... ). They're computation-intensive and of little practical value except to marketmakers. You could roll your own -- take the code snip below from http://quantnet.com/forum/showthread.php?p=18456#post18456 and translate it (which would be straightforward) into Excel or R, as suits your needs.

a = Vola*sqrt(t);
d1 = (log(S/K) + (r - q + (Vola^2)/2)*t)/a;
d2 = d1 - a;
n1 = normcdf(d1);
n2 = normcdf(d2);
p = sqrt(2*pi);
p1 = exp(d1^2/2)/p;
p2 = exp(d2^2/2)/p;
eq = exp(-q*t);
er = exp(-t*r);
Price = exp((-1)*q*t)*S*n1-exp((-1)*r*t)*K*n2;
delta = eq*n1;
gamma = eq*p1/(S*a);
vega = eq*S*p1*sqrt(t);
theta = eq*S*p1*Vola/(2*sqrt(t))-r*K*er*n2+q*S*eq*n1;
rho = K*t*er*n2;
volga = S*eq*p1*sqrt(t)*d1*d2/Vola;
vanna = -eq*p1*d2/Vola;
charm = -eq*q*n1+p1*eq*(2*(r-q)*t-d2*a)/(a*2*t);
color = -eq*p1*(2*t*q+1+d1*((2*(r-q)*t-d2*a)/(a*2*t)))/(2*S*t*a);
duald = -er*d2;
dualg = er*p2/(K*a);

5. ### tomk96

slightly off topic...

are you hedged or just long calls? if just long calls, why does it matter?

i would think your change in delta due to vol is more important. you might be able to back out your delta decay from that.

6. ### deltahedge

Though my name says deltahedge it's probably a misnomer as I'm usually just long calls without any short stock or short call positions.

You're probably right that I'm losing some of my delta due to vol changes as measured by vanna. However, I do believe (and someone correct me if I'm wayyy off base here) but just like vega's effect on the price of the options increases as one goes further out in time, my guess is that vanna does not bear a significant amount on delta since I'm trading 2-4 month calls as would charm.

Keep in mind I'm assessing this from a relative perspective in terms of the implied volatility profile of the calls whose stock I purchase (i.e. purchasing calls on IBM whose IV isn't significantly high nor does it drastically change as opposed to a buying a call on NFLX or CMG) where if I were to buy NFLX/CMG I would entertain and agree that vanna's value would have a far more meaningful impact than charm on delta. In a similar vein if I were to purchase IBM LEAPs, I would concede that vanna would have a greater baring on my delta than charm given how changes in IV further out have a much larger ripple effect.

7. ### deltahedge

Time to go learn about R and see if I can program these in thinkscript.

8. ### Rodney King

That's Matlab code, but the syntax of R is not too different. In any case, learning R is always a good idea.

9. ### abdibile

What is the best way to learn R in your view?

Any recommendations on a easy/itiots guide introduction?

I never reallay could ask google as the search term "R" does not really give good results

10. ### Rodney King

I would suggest working through Smith & Venables, free online http://cran.r-project.org/doc/manuals/R-intro.pdf or in paperback http://search.barnesandnoble.com/An-Introduction-To-R/William-N-Venables/e/9780954161743 -- it's brief and well-written. And there are many other books, monographs, reference cards & etc available nowadays.

Re searching -- have a look at http://www.r-project.org/search.html

#10     Feb 9, 2011
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