Delta hedging frequency, Gamma PnL

Question

I want to know the expectation and the variance of the Gamma PnL for different hedging frequencies.

Let's say the return of the underlying follow a normal process: $dr= \sigma*dW$, the market trades 24hours and there is no transaction cost. I consider I have a constant dollar $\Gamma$ position (-r% or +r%, r $\in$ $\mathbb{R}$), provide the same $\Gamma$) with a stripe of options. I also assume, there is no change in implied vol during the day, therefore the vega PnL is zero.

The daily volatility is $s = \frac{\sigma}{\sqrt{365}}$ therefore $dr \sim \mathcal{N}(0,s^2)$.

What is the Gamma PnL if we hedge every seconds, hours ... every time period $t$?

If we hedge every time $t$ (to simplify I normalize $t$ to a day, for instance every hour would be $t$ = 1/24), using the properties of Brownian motions, I can consider I have $1/t$ independent return processes $dr_{t} \sim \mathcal{N}(0, s^2*t)$, the Gamma PnL process is then for a day:

$PnL_{\Gamma}= \frac{1}{t}*\Gamma*\frac{dr_{t}^2}{2}$

and then follows $PnL_{\Gamma}\sim \chi^2_{1}$, with mean $E =\Gamma*s^2*t/t = \Gamma*s^2$ and variance $V = \Gamma^2*s^4*t^2/t^2 = \Gamma^2*s^4$

What I do not understand is that there is no dependency of the Gamma PnL to the hedging frequency. As we increase the hedging frequency we should have less variance and less return on the gamma and the reciprocal should be true? Where is my math failing? I do not see it.

Answer 1

From a quant perspective (I am sure more quants will chime in), delta hedging is a physical representation of a stochastic integral, which is inherently independent of the step size (in both time and price dimension). All you can really say there is that the closer to continuous hedging you get, the lower your hedging error will be.

From a traders perspective re-balancing frequency will indeed come into play if you leave the perfect BS world. For example, in real life, you are usually trying to optimize your hedging for a balance of (a) smoothness of P&L, (b) transaction costs, (c) vol dampening or amplifying (d) mean reversion found in the asset you're trading and (e) your risk limits.

Answer 2

The problem with what I do above is that I change the unit of time, as opposed to keeping the same unit of time and comparing the different frequencies.

We want to compare the returns over different hedging frequencies, the easy way is to use log return over different frequencies:

$dr_{t}=-\frac{s^2*t}{2} + s*\sqrt{t}*\epsilon∼N(-\frac{s^2∗t}{2},s^2∗t)$

If you do the rest of the math you will get the effect of the expected Gamma PnL and its variance.