Gamma PnL when hedging with implied volatility - where is the mark to market PnL?

Question

It is well known that hedging with implied volatility involves a PnL:

$0.5*(σ^{2}_r−σ^{2}_i)S^{2}*Γ_{i}dt$

In the Wilmott paper (http://web.math.ku.dk/~rolf/Wilmott_WhichFreeLunch.pdf), they imply that the collective PnL from such a strategy is the integral of above expression across time.

However, this seems to assume that the market implied volatility stays constant at $σ_i$. Otherwise, one would also encounter the mark-to-market PnL governed by the sensitivity of the option to implied volatility among other terms:

$C_{σ}* (dσ)+.....$

Why is the mark to market PnL not accounted for in the above analysis?

Answer 1

Consider any function $f(S(t),K,t,T,\{x_i(t)\})$ with payoff $(S(T) - K)_+$ when $t=T$, where $\{x_i(t)\}$ are other variables/parameters so that at $t=0$ you are able to choose (i.e. calibrated) these so that your function matches the market price of the option: $f(S(0),K,0,T,\{x_i(0)\}) = C^{market}(t=0)$.

As the payoff of the option does not depend on $\{x_i(T)\}$, if you decide to look only at the option value at maturity, then you are free to keep these other variables fixed and only hedge changes in $S_t$. In this case, according to your chosen `reality' (this is 'marking to model' as opposed to 'marking to market') the change in option value is $$ df = \theta dt + \Delta dS + \frac{1}{2} \Gamma (dS)^2 $$ since you have chosen all the others variables/parameters to be constant. $dS$ is whatever change in stock price is observed.

However, if you decide to / or are forced to 'look' at the option value in the market before expiration, then your delta-hedge P/L will equal: $$ P\&L = C^{market}(t=0) + \int_0^u \left( \theta_t dt + \frac{1}{2} \Gamma_t (dS_t)^2 \right) - C^{market}(t=u) $$

Answer 2

If you assume that the vols $\sigma_r,\sigma_i$ are deterministic functions of time their formula (1) still holds $$\tag{1} dV(t)=\frac{1}{2}(\sigma^2_r(t)-\sigma_i^2(t))\,\Gamma^i(t)\,dt. $$ Integrating gives the accumulated hedge PnL $$ V(t)=\frac{1}{2}\int_0^t(\sigma^2_r(s)-\sigma_i^2(s))\,\Gamma^i(s)\,ds. $$ One could extend the derivation to the case of stochastic vol $\sigma_r(t)$ by applying Ito's formula to the call price with two state variables $C(t,S(t),\sigma_r(t))\,.$ I am however not sure how useful such a general result will be in practice. Formula (1) holds approximately for small time intervals when $\sigma_r(t)$ can be assumed to be nearly deterministic.

Answer 3

The sigma i is the implied volatility you locked when u bought the call! It never moves. The only thing that move is sigma realised that you build during rebalancing… understood?