Leland's Model For Hedging With Costs

Delta hedging means constantly buying and selling shares to track an option's changing delta, and every one of those trades crosses a bid-ask spread. Black-Scholes assumes this trading is free, fine as a first approximation but wrong in a specific, systematic way: every rebalance leaks a little money, and a gamma-heavy option gets rebalanced often. Hayne Leland's 1985 paper found a strikingly clean fix — keep using Black-Scholes exactly as it is, but feed it a slightly wrong, inflated volatility, chosen so the extra premium it produces exactly covers the expected cost of hedging.

Padding a budget for the tolls you know you'll pay

Imagine pricing the fuel for a road trip, then adding a fixed percentage on top to cover tolls you know you'll hit, without recalculating the whole route. You just pad the one number that needs padding, by an amount proportional to how many tolls you expect. Leland's adjustment does exactly this to the volatility input: it pads $\sigma$ upward proportional to the trading costs you expect to pay while hedging, then lets ordinary Black-Scholes do the rest.

The adjusted volatility

Leland's adjusted volatility, used in place of the true volatility $\sigma$ inside the ordinary Black-Scholes formula, is

$$\sigma_L^2 \;=\; \sigma^2 \left(1 + k \sqrt{\frac{2}{\pi \sigma^2 \Delta t}}\right), \qquad k = \frac{2 \times \text{(proportional cost per trade)}}{\sigma\sqrt{\Delta t}} \cdot \frac{\sigma\sqrt{\Delta t}}{2},$$

which is usually written more simply in terms of the round-trip proportional cost $\kappa$ (kappa) and the rebalancing interval $\Delta t$ as

$$\sigma_L^2 \;=\; \sigma^2 \left(1 + \sqrt{\frac{2}{\pi}} \cdot \frac{\kappa}{\sigma\sqrt{\Delta t}}\right).$$

In plain English: the adjustment grows with the transaction cost $\kappa$ (bigger spread, more padding), and grows as $\Delta t$ shrinks (more frequent rebalancing means more trades, more total cost to pad for, even though each hedging error shrinks). Plug $\sigma_L$ into ordinary Black-Scholes and hedge as normal; the higher premium from a higher $\sigma_L$ turns out, in Leland's approximation, to almost exactly cover the expected trading costs over the option's life.

Worked example 1: computing the Leland-adjusted volatility

A desk hedges daily ($\Delta t = 1/252 = 0.00397$), true volatility $\sigma = 20\%$, and pays a round-trip transaction cost of $\kappa = 0.1\%$ of the trade's notional (bid-ask spread plus fees).

$$\frac{\kappa}{\sigma\sqrt{\Delta t}} = \frac{0.001}{0.20 \times 0.0630} = \frac{0.001}{0.0126} = 0.0794.$$ $$\sigma_L^2 = (0.20)^2\left(1 + \sqrt{\tfrac{2}{\pi}}\times 0.0794\right) = 0.04\left(1 + 0.7979 \times 0.0794\right) = 0.04 \times 1.0634 = 0.04254.$$ $$\sigma_L = \sqrt{0.04254} = 0.2063, \text{ i.e. } 20.63\%.$$

A true volatility of 20% becomes an effective hedging volatility of 20.63% — a modest but real 63 basis point bump, entirely to cover expected trading friction.

Worked example 2: what the bump is worth in premium

Take a one-year at-the-money call, S=K=\100$,$r=0$. Its Black-Scholes vega at$\sigma=20%$is roughly$S\phi(d_1)\sqrt{T} \approx 100 \times 0.3969 \times 1 = 39.69$(per 1.00 of volatility, so per 1 percentage point it is$0.3969$). The premium increase from a$0.63$ percentage-point volatility bump is approximately

$$\Delta V \approx \text{vega} \times \Delta\sigma = 0.3969 \times 0.63 = \$0.25.$$

So the option's fair price rises by about 25 cents to pre-fund the desk's expected hedging costs over the year — the buyer of the option is, in effect, paying the transaction costs of their own hedge upfront, bundled invisibly into the quoted premium.

What this means in practice

Leland's model is used as a quick, back-of-envelope adjustment to fold expected transaction costs into a quoted price without building a full cost-aware pricing engine — a desk keeps every existing Black-Scholes tool and simply feeds it a bumped volatility. It's most reliable near-the-money at moderate gamma; for very gamma-heavy or long-dated options the approximation degrades, since it assumes a fixed rebalancing schedule and proportional costs, neither of which holds exactly for a real book.