Time Derivatives in the Implied-Volatility Dupire Formula
Summary
The document asks how to differentiate an option price expressed through the Black–Scholes function when deriving local volatility from implied volatility. It presents the Dupire equation in log-moneyness and total variance coordinates, then questions whether the maturity derivative should include a chain-rule term for the maturity-dependent forward and log-moneyness.
The key issue is distinguishing explicit maturity dependence in the Black–Scholes representation from dependence carried through the chosen coordinates. The post cites Gatheral’s explanation that the option price’s explicit dependence on maturity enters through the forward, but it does not provide a resolution to the question. As presented, this is a focused mathematical question rather than a complete derivation, so readers should consult the underlying derivation to verify the coordinate conventions and chain rule.
Key ideas
- The post frames local volatility in terms of implied volatility using the Dupire equation and transformed coordinates.
- It questions whether differentiating with respect to maturity requires a term for the changing log-moneyness.
- The issue depends on how explicit maturity dependence and coordinate dependence are separated.
- The document raises the mathematical question but does not answer it.
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Full text
# $\frac{\partial C_{BS}}{\partial T}$ in local volatility derivation in terms of implied volatility
# $\frac{\partial C_{BS}}{\partial T}$ in local volatility derivation in terms of implied volatility
In Gatheral's book, in the derivation of local volatility in terms of implied volatility, we use the regular Dupire formula
$$ \frac{\partial C}{\partial T} = \frac{1}{2} \sigma^{2}K^{2}\frac{\partial^{2}C}{\partial^{2}K} + (r_{T}-q_{T}) \left ( C - K \frac{\partial C}{\partial K} \right ) - r_{T}C $$
and reparametrize with $y = \ln \frac{K}{F_{T}}$ and $w = \Sigma(K,T)^{2}T$ where $\Sigma(K,T)$ is the implied volatility and get:
$$ \frac{\partial C}{\partial T} = \frac{v_{L}}{2} \left [ \frac{\partial^{2}C}{\partial^{2}y} - \frac{\partial C}{\partial y} \right ] + \mu_{T}C $$
where $v_{L}$ is the local variance.
Next, we set the market price equal to the B-S price to write the formula above in terms of derivatives of the B-S function:
$$ C(S_{0}, K, T) = C_{BS}(S_{0}, K, \Sigma(K,T), T) $$
So far so good.
But in the next step when we write $\frac{\partial C}{\partial T}$ in terms of the B-S call function, we get:
$$ \frac{\partial C}{\partial T} = \frac{\partial C_{BS}}{\partial T} + \frac{\partial C_{BS}}{\partial w}\frac{\partial w}{\partial T} $$
However, I would expect it to be
$$ \frac{\partial C}{\partial T} = \frac{\partial C_{BS}}{\partial T} + \frac{\partial C_{BS}}{\partial w}\frac{\partial w}{\partial T} + \frac{\partial C_{BS}}{\partial y}\frac{\partial y}{\partial T} $$
since a shift in maturity implies a shift in the forward and thus in the log-moneyness $y$. Gatheral's explanation is that it "follows from the fact that the only explicit dependence of the option price on T [...] is through the forward price". Why do we not also take into account the log-moneyness shift here?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.