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Why does Gamma Pnl have exposure to realised volatility, but Vega Pnl only has exposure to implied volatility? I am confused as to why gamma pnl is affected (more) by IV and why vega pnl isnt affected (more) by RV?

Essentially how do you show what gamma pnl will be mathematically and how do you show what vega pnl will be? I believe that gamma pnl is spot x (vega x IV - RV)

Also does gamma pnl usually dominate (in $ terms) the vega pnl of an options, as most literature is on gamma pnl?

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2 Answers 2

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For an option with price $C$, the P$\&$L, with respect to changes of the underlying asset price $S$ and volatility $\sigma$, is given by \begin{align*} P\&L = \delta \Delta S + \frac{1}{2}\gamma (\Delta S)^2 + \nu \Delta \sigma, \end{align*} where $\delta$, $\gamma$, and $\nu$ are respectively the delta, gamma, and vega hedge ratios. Then it is clear the vega P$\&$L has exposure to the change of the implied volatility $\sigma$. Note that, for the gamma P$\&$L, \begin{align*} \frac{1}{2}\gamma (\Delta S)^2 = \frac{1}{2}\gamma S^2 \frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2\Delta t, \end{align*} where $\frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2$ is the realized variance, and $\sqrt{\frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2}$ is the realized volatility. To see why $\sqrt{\frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2}$ is the realized volatility, we assume that, heuristically, \begin{align*} dS_t = S_t\left(r dt + \sigma_{Re} dW_t \right), \end{align*} where $\sigma_{Re}$ is the realized volatility and $\{W_t, \, t \ge 0\}$ is a standard Brownian motion. Then \begin{align*} \sqrt{\frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2} \approx \sigma_{Re}. \end{align*}

Consider the delta neutral portfolio $\Pi=C-\frac{\partial C}{\partial S}S$. Assuming that the interest rate and volatility are not change during the small time period $\Delta t$. The P$\&$L of the portfolio is given by \begin{align*} P\&L_{\Delta t}^{\Pi} &= \frac{1}{2}\gamma (\Delta S)^2 + \theta \Delta t, \end{align*} where $\theta$ is the theta hedge ratio. For small interest rate, which we assume to be zero, $\theta \approx -\frac{1}{2}\gamma S^2 \sigma^2$ and $\gamma = \frac{\nu}{S^2\sigma T}$; see, for example, Black–Scholes model. Then \begin{align*} P\&L_{\Delta t}^{\Pi} &\approx \frac{1}{2}\gamma S^2 \frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2\Delta t - \frac{1}{2}\gamma S^2 \sigma^2 \Delta t\\ &\approx \frac{1}{2}\gamma S^2 \sigma_{Re}^2 \Delta t - \frac{1}{2}\gamma S^2 \sigma^2 \Delta t\\ &= \frac{1}{2}\gamma S^2 (\sigma_{Re} + \sigma)(\sigma_{Re} - \sigma) \Delta t\\ &\approx \gamma S^2 \sigma (\sigma_{Re} - \sigma) \Delta t \hspace{1in} (\text{assuming that } \sigma_{Re}\approx \sigma)\\ &=\frac{\nu}{T}(\sigma_{Re} - \sigma) \Delta t. \end{align*} The cumulative P$\&$L, over the interval $[0, T]$, is then $\nu (\sigma_{Re} - \sigma)$.

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  • $\begingroup$ Im stil confused about the gamma pnl $\endgroup$
    – Trajan
    Commented May 17, 2018 at 7:02
  • $\begingroup$ how could gamma pnl be gamma pnl is spot x vega x (IV - RV)? $\endgroup$
    – Trajan
    Commented May 17, 2018 at 7:15
  • $\begingroup$ @Permian: See the above updates. Where did you get this? Can you please provide us the source? I would like to check the context. $\endgroup$
    – Gordon
    Commented May 17, 2018 at 18:10
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    $\begingroup$ best explanation of getting from gamma term in ito expansion to the gamma pnl formula i've seen & frankly maybe the only one. $\endgroup$ Commented Mar 14 at 2:54
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    $\begingroup$ @KaiSqDist: that would be another question. The approximation here is related to the realized volatility. $\endgroup$
    – Gordon
    Commented Mar 27 at 17:09
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Not sure this is a valid question! Gamma p/l is by definition the p/l due to realized volatility being different from implied. Vega p/l is by definition the p/l due to moves in implied volatility.

The second part of the question you have answered yourself. Short dated options have more gamma exposure, long dated options have more vega exposure.

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    $\begingroup$ Its not clear to me at all how this is both by definition $\endgroup$
    – Trajan
    Commented May 7, 2018 at 12:10
  • $\begingroup$ What other explanation is possible ? $\endgroup$
    – dm63
    Commented May 7, 2018 at 22:15
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    $\begingroup$ The connection between gamma and realised volatility, probably mathematically $\endgroup$
    – Trajan
    Commented May 8, 2018 at 9:06

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