Optimal delta-hedging frequency when gamma scalping

Question

Is there a practical way to calculate a delta threshold for rebalancing when gamma scalping?

I know it does not effect expected P&L, but what about optimizing for P&L sharpe ratio after transaction costs?

Answer 1

The model I quite like as a base-case/rule of thumb is the Hoggard, Whalley, and Wilmott (1994) model.

Assuming GBM - the number of shares, $N$, per interval is:

$$N = Δ(S+dS,t+dt)- Δ(S,t)≈ Γ*dS$$

Expanding $dS$:

$$N ≈ Γ * σ * S * dW$$

The incremental cost is given by the number of shares, $N$, multiplied by the share price, multiplied by the transaction cost fraction, $α$:

$$Cost=|NS| * α$$

$$Cost=|Γ* σ * S^2 * dW| * α$$

Under expectations, $|dW|$ is greater than 0 (it's $\sqrt(2/π$). Thus, costs are expected to be non-zero. Rewriting $dW$ as $Z\sqrt dt$, for an option with $T/dt$ number of hedges over the life $T$, total costs will scale with $T/dt∙\sqrt dt=T/\sqrt dT$, meaning that as the hedging interval $→ 0, costs → ∞$.

Working through, you find that the effective volatility (or theoretical bid/ask) becomes $~= σ +- α\sqrt(2/(π*dt))$.

• for the ask, - for the bid.

For the P/L:

$$E[P/L] = 0.5 * Γ * S^2 * ((σ_r^2 - σ_i^2)*dt - a * σ_r * \sqrt(2/(π*dt^3)))$$

Then you can work out the Sharpe ratio making some assumptions about P/L volatility. I believe Derman wrote it as:

$Vega * σ_r / \sqrt(n)$ which in our model (using $dt$ instead of $n$) results in:

$Vega * σ_r * \sqrt(dt/T)$.

I'll leave it to you to work out the optimal Sharpe from there!