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Suppose I model the forward swap rate lognormal

$$dS_t = \sigma_{ln}S_tdW_t$$

On the other hand we could model it simply by a normal assumption:

$$dS_t = \sigma_{n}dW_t$$

I would like to know if there is a relationship for the volatilities $\sigma_n,\sigma_{ln}$? A friend told me, that he saw the approximation

$$\sigma_n\approx \sigma_{ln}S_t$$

However, neither my friend nor I was able to come up with a justification of this approximation. So is this a valid approximation? If so, why and if not, how else can I relate the two volatilities?

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1 Answer 1

For any normal variable, you have $$aX\sim N(a\mu,a^2\sigma^2).$$ So a linear transformation preserves the distribution type (note that $dW_t\sim N(0,dt)$).

When you want to approximate $dS_t$ by setting $dS_t=dS_t$, canceling the $dW_t$ you get:

$$\sigma_{ln}S_t=\sigma_n$$

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I believe Pat Hagan did some work like this for his SABR model too, wilmott.com/pdfs/021118_smile.pdf formula A.64 may be relevant. –  experquisite Aug 20 at 23:03

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