Balancing Delta-Hedge Frequency, Gamma Scalping, and Trading Costs

Summary

The document outlines a framework for choosing how often to rebalance a delta hedge during gamma scalping. Under geometric Brownian motion, it approximates the hedge adjustment as gamma times the underlying price move, then estimates trading costs by multiplying the traded notional by a proportional transaction cost. Because expected absolute price movement is positive, more frequent hedging raises cumulative costs; in the limiting case of a vanishing hedge interval, the stated cost estimate diverges.

The response points to the Hoggard, Whalley, and Wilmott model as a starting point and describes translating transaction costs into an effective volatility adjustment. It then gives an expected P&L expression and a suggested scaling for P&L volatility to help derive a Sharpe ratio. However, it does not complete the optimization or provide a practical threshold. The result depends on the model assumptions and cost specification, so it is a rule of thumb rather than a universal prescription.

Key ideas

• Under geometric Brownian motion, a small delta hedge adjustment is approximated using gamma and the underlying price move.
• Proportional transaction costs make expected hedge costs positive even when the expected price move is zero.
• The stated framework implies that hedging costs rise as the rebalancing interval shrinks.
• An effective volatility adjustment can represent the transaction cost impact in the model.
• The response suggests deriving a Sharpe ratio but leaves the optimal hedge frequency unresolved.

Tags

# Optimal delta-hedging frequency when gamma scalping


# 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?

## Answer by Newquant (score 5)

https://quant.stackexchange.com/a/75791

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!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.