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MCarreira
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Joined: July 14th, 2002, 3:00 am

FX Volatility smile in delta space vs. excercise price space

February 2nd, 2011, 2:30 pm

breeden-litzenberger.pdfColumn L can be calculated as the butterfly price divided by h^2.So, in fact, to calculate the derivative you need to divide by $G$17^2.But, to convert it to a pdf, you need to consider which interval you are using.Think that the 2nd derivative at a particular point is the same whether you used 0.01 or 0.001.In order to calculate the integral/sum of the pdf, you'll need to multiply each value you calculated by the size of the range that this value represents (pdf values are not calculated for a point, they are calculated for a dK range).So multiplying each value by 0.01 (size of range) and then adding you'll have 1 as expected.
 
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struggler
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Joined: March 4th, 2008, 11:42 pm

FX Volatility smile in delta space vs. excercise price space

February 2nd, 2011, 5:55 pm

I did multiply by the size of range (0.01) in column M but the total was 0.958 cell M118. why is that the case and how to adjust to get a total of 1
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MCarreira
Posts: 1724
Joined: July 14th, 2002, 3:00 am

FX Volatility smile in delta space vs. excercise price space

February 2nd, 2011, 6:49 pm

With a simple example: S=1, r=q=0, vol=20%, t=1, step 0.01, range 0.99 delta to 0.01 (strikes from 0.64 to 1.62) I get 0.98 even with smaller steps and wider ranges.Even with delta the numerical delta is not equal to the bs delta.I would consider it a discretization error and consider the goal of the whole exercise achieved.
 
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bouncer
Posts: 27
Joined: October 20th, 2002, 12:45 am

FX Volatility smile in delta space vs. excercise price space

June 4th, 2011, 9:49 pm

I think that you get less than one partly because you do not have enough strikes, especially on the downside. Look, the value approximation for the density function on the downside goes down to .23 (that is your lowest value on the left of your density function estimate). On the upside, you have 15 or so strikes which yield a value lower than that. Effectively, you have chopped off part of the left wing of the density function. Hence, the area will be less than on. By the way, in your excel sheet I added up the values in column L and multiplied the sum by .01 (G17). That gives .85 as opposed to .95 when you multiply each value (as you do in col M and then add up). Isn't rounding error amazing...
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