Exercise: Leland's formula for optimal rebalancing

In the presence of proportional transaction costs

k

(half the bid-ask spread, e.g.,

k = 0.001

for 20 bps), Leland (1985) showed that delta-hedging a short call with rebalancing frequency

1/\Delta t

becomes asymptotically free-of-error in the expected sense if the hedger prices the call at an inflated volatility

\sigma_\ell := \sigma\sqrt{1 + \sqrt{\tfrac{8}{\pi}}\cdot \tfrac{k}{\sigma\sqrt{\Delta t}}}.

This is known as the Leland-adjusted volatility.

Tasks

Derive (or verify) Leland's formula by balancing the expected transaction cost per rebalance against the expected gamma-adjusted variance gain. Specifically, show that:

a. Expected absolute change in delta per rebalance of length $\Delta t$ is approximately $\mathbb{E}[|\Delta_{t+\Delta t} - \Delta_t|] \approx \Gamma\sigma S \sqrt{2\Delta t/\pi}$ .

b. Expected transaction cost per rebalance is therefore $k \cdot S \cdot \Gamma\sigma S\sqrt{2\Delta t/\pi}$ .

c. Total rebalances in $[0, T]$ : $T/\Delta t$ , so total transaction cost is $T \cdot k\sigma S^2 \Gamma / \sqrt{2\pi\Delta t / (2)} = k\sigma S^2 \Gamma T \sqrt{2/\pi}/\sqrt{\Delta t}$ .

d. Leland matches this against the gamma-P&L term $\tfrac12 (\sigma_\ell^2 - \sigma^2) S^2 \Gamma T$ from the hedging-error formula. Solve for $\sigma_\ell$ .
For $\sigma = 0.20$ , $k = 0.001$ (20bp half-spread), and $\Delta t \in \{1/252, 1/52, 1/12\}$ , compute $\sigma_\ell$ .
Plot $\sigma_\ell$ as a function of $\Delta t$ for $\Delta t \in [0.001, 0.1]$ . What happens as $\Delta t \to 0$ ?
Leland's adjusted vol is used to price the option (charge more premium) and to compute the hedging delta (use a different $\Delta$ ). Why both? What would break if you only did one?
Interpret Leland's formula in plain English. Why does the trade-off lead to an inflation of the hedge vol rather than a deflation?

The $\sqrt{2/\pi}$ factor comes from the mean absolute value of a standard normal: $\mathbb{E}|Z| = \sqrt{2/\pi}$ .