delta-threshold hedging strategy

A delta-threshold hedging strategy is a discrete rebalancing rule where the trader rehedges (buys or sells the underlying) whenever the change in net portfolio delta exceeds a predetermined threshold $D$. This is the hedging model underlying the Hedging Valuation Adjustment framework of Benedict Burnett.

The strategy works as follows: the trader monitors the portfolio delta $\Delta ={\hat{V}}_x+a$ (where $a$ is the hedge holding). When the delta change $\delta \Delta =\Gamma \delta X$ exceeds $\pm D$, the trader transacts $D$ units of the underlying to flatten their position. Between rebalancing events, the portfolio is not perfectly hedged, creating both P&L variance and friction costs.

The key statistical properties of this strategy are:

• Expected time between rehedges: $\mathbb{E}[\delta t]\approx (D/\sigma \Gamma {)}^2$ (time for a random walk to hit $\pm D$)
• Expected delta move per rehedge: $\delta X^2\approx (D/\Gamma {)}^2$ to leading order
• Expected friction cost per rehedge: $\psi (D,X_t)$

For discrete monitoring (e.g., daily position checks), the effective threshold is adjusted using the Siegmund overshoot correction: $D^{\prime }=D+\beta |\Gamma |\sigma \sqrt{\Delta t}$, where $\beta =-\zeta (1/2)/\sqrt{2\pi }\approx 0.5826$ and $\zeta$ is the Riemann zeta function.

Key Details

• The threshold $D$ is measured in “delta units” — e.g., for FX, $D_s=50,000$ domestic currency units per percent
• In the multi-asset extension, each asset $i$ has its own threshold $D_i$ based on asset class and volatility
• The strategy is suboptimal compared to e.g. Zakamouline (2005) but is realistic and widely used in practice
• The choice of $D$ is a trade-off: larger $D$ reduces frequency (and cost) of rehedging but increases P&L variance

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method