# Hagan's 2002 SABR paper "Managing Smile Risk" on Dupire local vol model

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?

• Have you managed to find an explanation for this? I am stuck in the same place as you were then Commented Jan 26 at 10:32
• @Marius I suppose it's typo... Commented Jan 27 at 23:16