Reading Time and Moneyness in Dupire Local Volatility
Summary
The document presents Dupire's local volatility formula in two forms: one using derivatives of vanilla call prices and another using the total implied variance surface. In the latter, log-forward moneyness is expressed as the logarithm of strike relative to the forward price, while total implied variance combines maturity with implied volatility. The central question is how to interpret the maturity variable and where the surface values and derivatives are evaluated.
The author is trying to reconcile option maturities with the time index used in a Monte Carlo simulation, asking whether the formula uses calendar time, time to maturity, or a remaining horizon from the calibration date. They also question whether the quantities should be evaluated at the simulation time or at option expiry. The document provides the formulas and identifies the ambiguity, but it contains no answer or worked example resolving the time conventions. Readers therefore need an accompanying derivation or model convention before implementing the expressions.
Key ideas
- Dupire local volatility can be expressed through derivatives of call prices or through an implied variance surface.
- The implied variance representation uses log-forward moneyness and total implied variance.
- The document raises a distinction between option maturity and simulation time without resolving it.
- Correct implementation requires consistent time coordinates and evaluation points for the surface derivatives.
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Full text
# 62014
# Local volatility (dupire equation) for monte Carlo, discretisation - need help understanding spatial and TEMPORAL dimensions
As proven in Gatheral notes (or in this discussion)
The equations of the local volatility as a function of the vanilla calls can be written as $$ \sigma^2(T,K) = \frac{\frac{\partial C}{\partial T} + (r - q)K \frac{\partial C}{\partial K} + qC}{ \frac{1}{2} K^2 \frac{\partial^2C}{\partial K^2}} $$
And or as a function of implied volatility surface as $$ \sigma_{\mathrm{Dup}}(T,K)^2 = \frac{ \frac{\partial w}{\partial T} }{1 - \frac{y}{w} \frac{\partial w}{\partial y}+ \frac{1}{4}\left( - \frac{1}{4} + \frac{1}{w} + \frac{y^2}{w^2} \right) \left(\frac{\partial w}{\partial y}\right)^2 + \frac{1}{2}\frac{\partial^2 w}{\partial y^2} } $$ with $y = \ln(K/F_0^T)$ and $w(T,y) = T\Sigma^2(T,y)$
However, I would require a precision on the time-dimension variables meaning (as I feel a bit confused with all these variables, especially the $w$ and $y$, that in their writings show a dependance to a $T$ variable.
In short version:
- What is $T$ in these writings ( $y = \ln(K/F_0^T)$ and $w(T,y) = T\Sigma^2(T,y)$ ) ?
- What points are evaluated the variables and derivatives ?
In confused version (this may help provide more elements of my misunderstanding):
- What is the $T$ in Dupire local volatility ?
- is it a $ T= t $ ?
- For instance a simulation step between such that $t>t_0 $ ? where $t_0$ is the spot calibration date ? But in such case, at what grid point are evaluated $w, y,$ and all the partial derivatives ?
- For instance, $w$ is defined as $w(T,y) = T\Sigma^2(T,y)$, in such case, if $T=t$, taking $t\Sigma^2(t,y)$ for $w$ does not make sense as it should be defined as a total implied variance from a maturity $T$ point ? Or should it be $T-t$ instead ?
- I have the same questions for the variables and derivatives in the formula of local volatility from vanilla calls. Should the calls be evaluated at $T$, $t$ or $T-t$ ?
- I don't think $T$ in Dupire formula represent a time to maturity as the local volatility is supposed to be specific maturity independent. But, I would require some explanations for a better understanding I guess...
Note: I help writing my question with use of different pdf and some quant.stackexchange pages such as this oneShown 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.