Understand and evaluate the costs associated with discrete-time rebalancing.

Policyholders want downside protection on their separate-account value at maturity, so insurers embed a Guaranteed Minimum Maturity Benefit (GMMB), economically a long put written by the insurer. A variable annuity (VA) writer sells a 10-year GMMB and hedges the embedded put via delta replication. In theory, continuous rebalancing perfectly offsets the liability. In practice, you rebalance daily or weekly, and the gap between theory and practice has a price tag.

A delta hedger holds ${\Delta }_t=\partial V/\partial S$ units of the underlying at each rebalance time. Between rebalances, $S$ moves and the hedge is no longer delta-neutral. A Taylor expansion of the option value over one period gives:

$$\Delta V\approx \Delta \cdot \Delta S+\frac{1}{2}\Gamma (\Delta S{)}^2+\theta \Delta t$$

The hedger captures the $\Delta \cdot \Delta S$ term. The residual is the gamma-theta P&L. Under Black-Scholes pricing, $\theta \Delta t$ is the expected piece and $\frac{1}{2}\Gamma [(\Delta S{)}^2-{\sigma }^2{S}^2\Delta t]$ is the mean-zero noise. Discrete rebalancing turns that noise into realized variance.

Common mistakes

• Forgetting the square-root rule. Doubling $N$ cuts hedging error SD by $1/\sqrt{2}$, not by half. Halving SD requires four times the rebalances.
• Confusing variance and SD scaling. Variance scales linearly with $\Delta t$; SD scales with $\sqrt{\Delta t}$. Exam questions deliberately swap these.
• Ignoring the role of ${\Gamma }^2$. Hedging error variance depends on ${\Gamma }^2$, so short and long gamma positions leak identical variance. Magnitude, not sign, drives the cost.

Bottom line

• Discrete hedging is imperfect because delta drifts between rebalances; it is exact in continuous time but leaky in discrete time, with gamma as the leakage source.
• Boyle-Emanuel hedging error variance is approximately $\frac{1}{2}\int ({\sigma }^2{S}^2\Gamma {)}^2\Delta tdt$; per-period variance scales with $(\Delta t{)}^2$ and total error standard deviation scales with $\sqrt{\Delta t}$.
• Transaction costs scale with $1/\sqrt{\Delta t}$; halving the rebalance interval multiplies expected trading cost by $\sqrt{2}$.
• Per-rebalance trade size scales with $\sqrt{\Delta t}$ because delta drifts proportionally to $\sqrt{\Delta t}$.

Exam shortcut

If a question gives two rebalance frequencies and asks for the SD ratio, take $\sqrt{N_1/N_2}$. No formulas needed. DECISION: Variance question, linear in $\Delta t$. Standard-deviation question, $\sqrt{\Delta t}$. Cost question, $1/\sqrt{\Delta t}$. Memorize the three scalings; most exam questions test exactly one. For optimal frequency under proportional costs, set marginal variance reduction equal to marginal cost increase: $a=b/\Delta t^{\ast 2}$, and solve $\Delta t^{\ast }=\sqrt{b/a}$.

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Learning objectives

• 7d

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