I'm reading Hagan's 2002 paper Managing Smile Risk originally published on the WILMOTT magazine, and got something confusing on his comment on Dupire's local volatility model.
The set up: Consider a European call option on an asset $A$ with exercise date $t_{ex}$ , and strike $K$. If the holder exercises the option on $t_{ex}$, then on the settlement date t_set he receives the underlying asset A and pays the strike $K$.
Dupire's Local Volatility model is $$d\hat{F}=\sigma_{loc}(\hat{F}) \hat F dW,\quad \hat F(0)=f $$
The paper comments on it starting to quote the implied volatility as equation (2.8)
$$\sigma_B(K,f) = \sigma_{loc}\left(\frac{f+K}{2}\right) \left\{ 1 + \frac{1}{24} \frac{\sigma_{loc}''\left(\frac{f+K}{2}\right)}{\sigma_{loc}\left(\frac{f+K}{2}\right)}(f-K)^2 + \cdots \right\}$$
Suppose today the forward price is $f_0$ and the implied volatility curve seen in the market is $\sigma_B^0(K)$, calibrating the model to the market leads to equation (2.9)
$$\sigma_{loc}(\hat F) = \sigma_B^0 (2\hat F - f_0)\left\{1 + \cdots\right\}$$
Now that the model is calibrated, suppose that the forward value changes from $f_0$ to some new value $f$, from (2.8), (2.9) we see that the model predicts that the new implied volatility curve is equation (2.10)
$$\sigma_B (K, f) = \sigma^0_B (K+f-f_0)\left\{1+\cdots\right\}$$
for an option with strike $K$, given that the current value of the forward price is $f$.
Suppose that today's implied volatility is a perfect smile as equation (2.11a) $$\sigma_B^0(K) = \alpha + \beta (K-f_0)^2$$ around today's forward price $f_0$. Then equation (2.8) implies that the local volatility is as equation (2.11b) $$\sigma_{loc}(\hat F) = \alpha + 3\beta (\hat F - f_0)^2 + \cdots $$
As the forward price $f$ evolves away from $f_0$ due to normal market fluctuations, equation (2.8) predicts that the implied volatility is as equation (2.11c)
$$\sigma_B(K,f) = \alpha + \beta \left[ K - \left(\frac32 f_0 - \frac12 f\right)\right]^2 + \frac34 \beta (f-f_0)^2 + \cdots $$
I'm lost here on (2.11b) and (2.11c).
My calculation is as equation (A)
$$\sigma_{loc}(\hat F) = \sigma_B^0 (2\hat F - f_0) = \alpha + \beta \left((2\hat F - f_0) - f_0\right)^2 = \alpha + 4\beta (\hat F - f_0)^2$$
and equation (B)
$$\sigma_B(K,f) = \sigma^0_B (K+f-f_0)\left\{1+\cdots\right\} = \alpha + \beta \left((K+f-f_0)-f_0\right)^2 = \alpha + \beta \left[K-(2f_0-f)\right]^2$$
So my result (A) is different from (2.11b) and (B) is different from (2.11c).
Could someone pls enlighten me, where did I get wrong?
