Dynamic Hedging: Rebalancing Risk Without Eliminating It

Audit dynamic hedging with signed targets, executable rebalancing, a self-financing cash ledger, local Greek attribution, basis risk, and settlement inventory.

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

Dynamic hedging repeatedly changes signed hedge inventory as prices, time, volatility, model sensitivities, and portfolio composition change. A common implementation trades shares to keep portfolio Delta near a declared target and within a tolerance band, but a hedge can also use futures, ETFs, currencies, rates, or options after explicit unit and basis mapping.

It reduces one specified local exposure; it does not remove risk. Gamma, Charm, and Vanna move Delta, while jumps, volatility-surface changes, basis, borrow, funding, liquidity, assignment, settlement, and execution can create losses before or during the next rebalance.

Target, cash, and control process

Lock each claim, signed quantity q_i (long positive, short negative), multiplier M_i, live deliverable, style, settlement, expiration and event timeline. Freeze the model, surface, market-data timestamp, currency, and Greek units.
In one compatible stock-share coordinate, calculate D_opt,t=Σ_i q_iM_iDelta_i,t and G_opt,t=Σ_i q_iM_iGamma_i,t. Choose target D_target,t, signed hedge inventory h_t, and tolerance; for stock, h*_t=D_target,t−D_opt,t and order quantity is δh_t=h*_t−h_t⁻. Futures, ETF, FX, and rate hedges need point-value, beta, FX, duration, or basis conversion.
Specify time-, price-, Delta-band-, or hybrid triggers, plus rounding, minimum and maximum order size, limit-price logic, turnover, market-hours, stale-data, halt, borrow, partial-fill, rejection, buying-power, and end-of-day rules.
Use actual fills in a self-financing ledger. Immediately before a stock trade, W_t⁻=Σ_i q_iM_iV_i,t+h_t⁻S_mark,t+B_t⁻. After execution at S_exec,t with total cost TC_t, cash is B_t⁺=B_t⁻−δh_tS_exec,t−TC_t, so W_t⁺−W_t⁻=−TC_t−δh_t(S_exec,t−S_mark,t). Sale proceeds arrive with an equal stock liability; they are not profit.
Between trades, reconcile option repricing, old hedge inventory, cash, dividends, borrow, financing, collateral, fees, and taxes. A local explanation is ΔW≈(D_opt+h)ΔS+0.5G_opt(ΔS)²+Vega_posΔIV+Theta_posΔt+carry−TC+R; declare whether Vega is per volatility point and Theta is per day or year. Delta drift may include ΔD≈G_optΔS+Charm_posΔt+Vanna_posΔIV.
Compare the local explanation with full repricing, actual marks, fills, and cash flows. Investigate residual R; stress jumps, gaps, halts, surface recalibration, near-expiry nonlinearities, proxy basis, borrow recall, margin liquidation, and execution latency rather than mechanically increasing frequency.
Prewrite close, roll, holder exercise, writer assignment, expiration, physical delivery, cash settlement, adjusted-contract, dividend, contrary-instruction, and broker-cutoff branches. Finish with option, hedge, cash, collateral, fee, tax, and residual-inventory reconciliation using only information available at each historical timestamp.

Four worked examples

Target and executable cash. Ten long calls have M=100 and Delta=0.55, so D_opt=+550 shares. Existing inventory is h=−400; for D_target=0, the target is h*=−550 and δh=−150 shares. With stock mark $80, sell fill $79.98, cost $5, and opening cash $5,000, new cash is B⁺=5,000−(−150)×79.98−5=$16,992. Immediate wealth changes by −150×(80−79.98)−5=−$8; the sale cash itself is not a gain.
A complete rebalance loop. At t₀, twenty long calls have M=100, V₀=$6, and Delta=0.50; with S₀=$100, h=−1,000, and B=$88,000, wealth is 20×100×6−1,000×100+88,000=$0. At S=$103, V=$7.40, and Delta=0.62, pretrade wealth is −$200; selling 240 shares at $103 with $60 cost leaves h=−1,240, B=$112,660, and wealth −$260. When S=$100 and V=$5.55, pretrade wealth is −$240; buying 240 shares with another $60 cost leaves h=−1,000, B=$88,600, and terminal wealth −$300. The bridge is option −$900, hedge loop +$720, and costs −$120.
Local attribution and residual. Let D_opt=+600 shares, h=−600, G_opt=+40 shares/$1, Vega_pos=+$250/volatility point, and Theta_pos=−$120/day. For ΔS=+$2 and ΔIV=−1.5 points, the Delta term is zero, Gamma is +$80, Vega is −$375, and Theta is −$120, so local P/L is −$415 and Gamma-only Delta drift is +80 shares. If full repricing is −$455, then R=−$40. Selling 80 shares with $4.40 of slippage and fees makes the post-trade result −$459.40.
Physical and cash settlement leave different inventory. A writer is short 5 physically settled calls with K=$50, M=100, and long-unit Delta=0.70, and holds 350 shares, initially offsetting the signed option Delta of −350 shares. At expiration S=$54, assignment delivers 500 shares and receives $25,000; call intrinsic liability is $2,000, and stock inventory becomes −150 shares. Buying those shares the next day at $54.30 costs $8,145, which is $45 more than a $54 close. A comparable cash-settled claim instead pays $2,000 cash while the 350-share hedge remains; neither lifecycle automatically flattens the package.

Hedging checklist

Position signs can reverse the intended hedge.
Quantities, multipliers, and adjusted deliverables can be wrong or double-counted.
Per-share, per-contract, position, futures, and currency Greek units can be mixed.
Models, surfaces, marks, and quote timestamps can be stale or inconsistent.
Gamma, Charm, Vanna, skew, rates, dividends, and borrow move the target.
Jumps, overnight gaps, halts, and price limits prevent continuous rebalancing.
Near-expiry and pin behavior can make small moves highly nonlinear.
Early exercise and assignment alter option, stock, and strike-cash inventory.
Physical and cash settlement create different residual positions.
Rejected, delayed, partial, or out-of-order fills leave unintended exposure.
Bid-ask spread, slippage, impact, and adverse selection rise with turnover.
Stock borrow can be unavailable, repriced, recalled, or forcibly bought in.
Funding, restricted sale proceeds, collateral, margin, and liquidation constrain execution.
Dividends, payment in lieu, tax, and corporate actions change cash and deliverables.
ETF, index, futures, FX, and related-asset hedges retain basis and correlation risk.
Surface and model recalibration can reclassify P/L and change Greeks abruptly.
Higher frequency can reduce one tracking error while increasing total cost and impact.
Backtests can use hindsight Greeks, final curves, unavailable quotes, or impossible fills.
P/L residuals can hide missing cash flows, unit errors, or model failures.
Limits, data versions, rule changes, broker records, and remaining inventory need governance.

Common misconceptions

“Delta neutral means risk-free.” Higher Greeks, jumps, liquidity, basis, funding, and lifecycle risk remain.
“Cash received from selling the hedge is profit.” It accompanies an equal inventory liability.
“More frequent rebalancing is always safer.” Tracking error, spread, impact, and operational risk trade off.
“Gamma scalping or long Gamma guarantees profit.” Hedge gains must exceed option decay, volatility changes, carry, and costs.
“Closing Greeks, midpoints, and hindsight fills prove the strategy was executable.” Only contemporaneous data, orders, fills, cash, and inventory can do that.