Gamma Pnl vs Vega Pnl

Question

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?

Answer 1

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 $\nu (\sigma_{Re} - \sigma)$.

Answer 2

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.