# Optimal delta-hedging frequency when gamma scalping

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?

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!

• Would you be able to write the formulas in LaTex? That would improve / ease the readability of your answer. Jun 7 at 10:05
• Thanks for fixing this amdopt. Jun 7 at 11:57