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JamesH83
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Joined: June 25th, 2003, 11:38 pm

risk neutral density

January 11th, 2006, 11:14 am

differentiating the BS equation twice with respect to strike yields the discounted risk neutral density.i keep trying to derive the formula but i keep going wrong somewhere!anyone can show me the equation or a link to a paper containing it?thanks
Last edited by JamesH83 on January 10th, 2006, 11:00 pm, edited 1 time in total.
 
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Paolos
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risk neutral density

January 11th, 2006, 12:42 pm

If I remember correctly John Hull book "Options, Futures, and Other Derivatives" should contain the derivation of the formula in the appendix of the chapter devoted to volatility smile. I don't have the book here to check itP.
 
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JamesH83
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risk neutral density

January 11th, 2006, 12:58 pm

wow, i didnt realise, i will check it outthanks!
 
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JamesH83
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risk neutral density

January 11th, 2006, 1:03 pm

oops i left it in chicago, guess i will have to buy another copy. in the mean time if anyone else could post the formula that would be much appreciatedJ
 
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player
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risk neutral density

January 11th, 2006, 2:44 pm

Write it in integral form..Then differentiate the payoff and then what you intergrate against
 
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JamesH83
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risk neutral density

January 11th, 2006, 3:18 pm

yes i know how to do that....i want the actual differential of the BS equation
Last edited by JamesH83 on January 10th, 2006, 11:00 pm, edited 1 time in total.
 
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AVt
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risk neutral density

January 11th, 2006, 5:12 pm

it is exp(-r*t)*diffN(dTwo(s,e,t,r,v))/(e*v*sqrt(t)), for e=excercise, v=vol
 
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mj
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risk neutral density

January 11th, 2006, 9:25 pm

see my book
 
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list
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risk neutral density

January 11th, 2006, 10:55 pm

Dear JamesH83where such definition of the density you found. In math. if you have integral then expression under integral call density of the integral with respect to measure represented by the differential used by the integral.
 
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JamesH83
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risk neutral density

January 12th, 2006, 6:59 am

Thanks MJ, will check tonightJH
 
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cosmologist
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Joined: January 24th, 2005, 8:08 am

risk neutral density

January 13th, 2006, 5:09 am

james no offence meant,rnd is a very basic concept in distributions. Secondly,to find rnd of any distribution you can differentiate the call expression with respect to strike twice.Double diferentiation of the call formula wrt the strike will yield the fomula.Interesting point of discussion would be the result if we differentiate the put expression. cheers
 
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JamesH83
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risk neutral density

January 13th, 2006, 6:56 am

Dont mean to be rude cosmologist, but why do you bother repeating what I said in my original post?
Last edited by JamesH83 on January 12th, 2006, 11:00 pm, edited 1 time in total.
 
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cosmologist
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risk neutral density

January 13th, 2006, 8:13 am

QuoteOriginally posted by: JamesH83Dont mean to be rude cosmologist, but why do you bother repeating what I said in my original post?sorry for that mess up. How do I send you the step by step differentiation? Typing it is impossible. But then the second point is interesting ,too. You might have noticed that when you do a detailed calculation of 'delta', a miraculous ,spectacular cancellation happens. I did it eight years back when Hull only gave N(d) formulas. I first could not believe myself. I thought it can not be true. But,then, a reference with the distribution chapters showed me that a simple approach to find N(d) first could result in the option value,too. That happens when you break the CALL into basic building blocks. the detailed derivation must have been done in some books. I shall find out and tell you.cheers
 
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Gerry
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risk neutral density

January 17th, 2006, 1:50 am

If the risk neutral density is p(x), isn't it just:d/dK d/dK int_K^\infty (x-K) p(x) dx = d/dk [ (K - K) p(K) + int_K^\infty -1 p(x) dx ] = d/dk int_K^\infty -1 p(x) dx = p(K) ?using the fundamental theorem of calculus and Leibniz's rule to differentiate through the integral?I'm just browsing, so I might be missing something (I'm clearly ignoring discounting).Cheers,Gerard
Last edited by Gerry on January 16th, 2006, 11:00 pm, edited 1 time in total.