Deriving Dupire Local Volatility in Log-Moneyness Coordinates
Summary
The document asks how to transform Dupire’s local-volatility formula from strike and maturity variables into a representation based on log moneyness and total implied variance. It starts from the call-price formula, substitutes a Black–Scholes price with strike- and maturity-dependent implied volatility, and outlines the chain-rule derivatives needed to change coordinates. The target expression uses derivatives of the transformed variance function with respect to time and log moneyness.
The author’s partial derivation shows how differentiating the Black–Scholes terms directly can quickly become unwieldy, and asks for a cleaner route or useful simplifications. No completed derivation, numerical example, or validation is provided, so the document is best read as a mathematical problem statement rather than a usable implementation guide. It is relevant to extracting local volatility from an implied-volatility surface, but the stated formula and variable conventions would need to be checked carefully before coding, especially the definitions of forward moneyness, time origin, and total variance.
Key ideas
- Dupire’s formula recovers local volatility from derivatives of call prices across strike and maturity.
- A log-moneyness transformation can express the result using derivatives of a transformed total-variance function.
- Applying the chain rule directly to the Black–Scholes price creates many terms.
- The document poses the derivation problem but does not supply a solution or numerical evidence.
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Full text
# Changing parameterization in Dupire's Formula
# Changing parameterization in Dupire's Formula
In chapter 2 of Bergomi's book on stochastic volatility, we have dupire's formula given as $$\sigma(S, t)^2 = \left|\frac{\frac{\partial C}{\partial T} + (r-q)K\frac{\partial C}{\partial K} + qC}{\frac{1}{2}K^2\frac{\partial^2 C}{\partial K^2}}\right|_{K=S, T=t}$$ where $C$ is the price of a call option under the local volatility model $dS_t/S_t = (r-q)dt + \sigma(t,S_t)dW_t$.
With the following change of variables to log moneyness, $y = \log \frac{K}{F_t} = \log \frac{K}{S} - (r-q)t$ and $f(t,y) = (t - t_0)\widehat{\sigma}_{K,T}^2$ and using the Black Scholes formula Bergomi states that the above formula becomes $$\sigma(S, t)^2 = \left|\frac{\frac{\partial f}{\partial t}}{\left(\frac{y}{2f}\frac{df}{dy} - 1\right)^2 + \frac{1}{2}\frac{d^2f}{dy^2} - \frac{1}{4}\left(\frac{1}{4} + \frac{1}{f}\right)\left(\frac{df}{dy}\right)^2}\right|_{y = \log \frac{S}{F_t}}$$
I would like help deriving this change in second formula given the first one. I am having trouble changing variables as the expressions I am getting are quite messy.
I started with the fact that we have $$C(K,T) = C_{BS}(K, T, \widehat{\sigma}_{K,T})$$ We have to change from $C$ to $C_{BS}$ and also from $(K,T)$ to $(y,T)$. I started with this but it got a bit too hairy and I'd like a little help with it.
EDIT: I'll add a bit of the direction I was taking. Let $$C_{BS}(y,t) = Se^{qt}\left(N(d+) - e^{y}N(d-)\right)$$ where $$d_{\pm} = \frac{-y}{\sqrt{f(t,y)}} \pm \frac{\sqrt{f(t,y)}}{2}$$ This is exactly the black scholes formula $C_{BS}(K,T, \widehat{\sigma}_{K,T})$ with variables changed to $y, t$. So we have $C(K,T) = C_{BS}(y(K,T),T)$
To get the first formula we need to compute things like $\frac{\partial C}{\partial K}$. We have $$\frac{\partial C}{\partial K} = \frac{\partial C_{BS}}{\partial y} \frac{\partial y}{\partial K}$$ and $$\frac{\partial C_{BS}}{\partial y} = Se^{qt}\left[N'(d+)\left(\frac{\sqrt{f} + \frac{y}{2\sqrt{f}}\frac{df}{dy}}{f} + \frac{1}{2\sqrt{f}}\frac{df}{dt}\right) - e^yN(d-)-e^yN'(d-)\left(\frac{\sqrt{f} + \frac{y}{2\sqrt{f}}\frac{df}{dy}}{f} - \frac{1}{2\sqrt{f}}\frac{df}{dt}\right)\right]$$
As you can see, it is getting really hairy already with just one term. I don't know if I am doing something wrong or missing some obvious simplifications, but I would like some help with this.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.