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ronm
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Joined: June 8th, 2007, 9:00 am

Forward swap rate

April 18th, 2015, 2:55 pm

Let say F is Forward Swap Rate and Sigma is its Volatility. I have come across an approximate formula likeSigma-Normal ~ F * Sigma-Log-NormalCan expert here help me to understand this formula, what is the underlying Model (i.e. Data generation process) ? What does this approximate formula mean.Really appreciate for your pointer.Thanks and regards,
 
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bearish
Posts: 5186
Joined: February 3rd, 2011, 2:19 pm

Forward swap rate

April 18th, 2015, 3:07 pm

There are at least a couple of ways to look at it. First you can inspect the SDEs: [$] dF=\mu_N dt + \sigma_N dW [$] vs [$] dF=\mu_L F dt + \sigma_L F dW[$]. Second, if you take the price of an at-the-money-forward swaption and back out implied volatilities from the standard Bachelier and Black models, you will find that, indeed, [$] \sigma_N \approx \sigma_L F [$].
 
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ronm
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Joined: June 8th, 2007, 9:00 am

Forward swap rate

April 18th, 2015, 3:25 pm

Hi bearish,Thanks for your pointer. However can you please elaborate more on the 2nd approach i.e. price of an at-the-money-forward swaption and back out implied volatilities from the standard Bachelier and Black models.Can you please refer any workout (like any document/materials on internet) on how can I **see** the derivation of that approximate formula. Thank you.
 
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mtsm
Posts: 78
Joined: July 28th, 2010, 1:40 pm

Forward swap rate

April 18th, 2015, 5:01 pm

use the search function and look for a couple of notes/papers written by pat hagan. or the Lehman note by Fei Zhou for example. I believe that these papers are all on this or the other site.