Deriving Gamma PnL from Itô’s Lemma
Summary
The note explains how the option portfolio’s gamma contribution to price change follows from the second-order term in Itô’s lemma. It distinguishes the instantaneous variance term in the pricing equation from realized movement in the underlying price.
Using the diffusion model for the underlying, the squared price increment is approximately σ²S²dt for a small time interval, with higher-order terms ignored. Thus gamma PnL can be written as one half gamma times the squared realized price change; volatility is reflected in that movement rather than multiplied into it again. The derivation assumes a small time step and the stated diffusion model, and does not address transaction costs or other sources of PnL.
Key ideas
- The second-order price effect contributes approximately one half gamma times the squared underlying price change.
- The squared price increment has an expected diffusion component of σ²S²dt over a small interval.
- Volatility is embedded in the underlying price increment, so multiplying by it again would double count it.
- Higher-order time terms are ignored in the small-interval approximation.
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Full text
# Gamma PnL from Itô's Lemma derivation
# Gamma PnL from Itô's Lemma derivation
The change in a call portfolio ($f$), derived from Itô's Lemma, is: \begin{align*} \left( \frac{\partial f}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2}\right)\mathrm{d}t &=r\left( f-rS\frac{\partial f}{\partial S}\right)\mathrm{d} t, \\ \implies\frac{\partial f}{\partial t}+rS\frac{\partial f}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2} -rf&=0 \end{align*}
where $\frac{\partial f}{\partial t}$ denotes theta, $\frac{\partial f}{\partial S}$ denotes delta and $\frac{\partial^2 f}{\partial S^2}$ denotes gamma.
So gamma's PnL would be $\frac{1}{2}\Gamma \sigma^2 \mathrm{d}S^2$, where $\mathrm{d}S^2$ is the underlying price's change.
But why is gamma's PnL in reallity $\frac{1}{2}\Gamma \mathrm{d}S^2$, and not the previous formula? Why shouldn't volatility be included gamma's PnL?
## Answer by ir7 (score 9, accepted)
https://quant.stackexchange.com/a/55336
$$ \frac{1}{2} \frac{\partial^2 f}{\partial S^2} dS^2 \approx \frac{1}{2} \sigma^2 S^2\frac{\partial^2 f}{\partial S^2} dt$$
(for small $dt$, ignoring $(dt)^2$ terms )
$\sigma$ is embedded in $dS = \mu S dt + \sigma S dW$ and $$ dS^2 = \mu^2 S^2 dt^2 + 2\mu \sigma S^2 dt dW + \sigma^2 S^2 dt \approx \sigma^2 S^2 dt$$
You picked up $1/2\Gamma \sigma^2$ from the PDE and for some (unknown) reason you multiplied it by $dS^2$. You can only multiply it by $S^2$ as in the PDE (to get PnL per unit of time) or by $S^2 dt$ like in the SDE (to get dollar PnL).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.