How Dupire Local Volatility Shifts the Implied Smile
Summary
The document asks how to interpret the smile dynamics implied by Dupire local volatility in a discussion of Hagan’s analysis. It presents an approximation linking local volatility to Black implied volatility, then considers calibrating that relationship to an initial market smile and repricing after the forward changes. The example uses a smile that is quadratic around the initial forward and compares the resulting local volatility and later implied volatility with the questioner’s direct substitutions.
The central lesson is that the approximation’s correction terms matter: simply replacing the strike in the initial smile misses terms that change the curvature and level of the updated smile. The document itself contains the question and formulas but no answer resolving the algebra, so it does not provide a derivation that can independently verify the displayed coefficients. It is useful as a focused prompt about local volatility’s implied smile dynamics, with the numerical expressions treated as claims from the cited paper rather than fully explained results.
Key ideas
- Dupire local volatility links the local volatility function to the implied volatility surface through an approximation.
- A calibrated local volatility model predicts how the smile changes as the forward moves.
- The quadratic smile example shows that updating strike alone omits correction terms.
- The post contains no answer deriving the stated coefficients, so those expressions are not independently justified there.
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# Hagan's 2002 SABR paper "Managing Smile Risk" on Dupire local vol model
# 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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.