Lambda

What is lambda?

Lambda (λ), often called option elasticity or leverage, measures the percentage change in an option’s price for a 1% change in the underlying asset’s price. Put another way, it compares the leverage the option provides relative to holding the underlying security.

Formula:
λ = (∂C / C) / (∂S / S) = (S / C) × Δ = ∂ ln C / ∂ ln S

where:
* C = option price
S = underlying price
Δ (delta) = ∂C / ∂S

So lambda is simply delta multiplied by the ratio of the stock price to the option price.

Why lambda matters

Lambda quantifies how much an option amplifies moves in the underlying:
* A high positive lambda means the option’s value moves much more, in percentage terms, than the underlying — useful for leveraged exposure.
* A low lambda implies less percentage sensitivity relative to the underlying.
* Lambda complements the standard Greeks (delta, gamma, vega, theta) by expressing leverage in percentage terms, which helps compare risk across different option strikes and expirations.

Example

Stock price S = $100
At-the-money call price C = $2.10
Delta Δ = 0.58

λ = 0.58 × (100 / 2.10) ≈ 27.62

Interpretation: a 1% rise in the stock (~$1) would correspond (approx.) to a 27.62% increase in the option’s dollar value. For example, five contracts (each contract = 100 shares) at $2.10 cost $1,050; after a $1 stock rise the option price would rise by about $0.58 to $2.68, making the five contracts worth $1,340 — about a 27.6% gain.

Lambda, volatility and vega

Lambda and vega are related but distinct:
* Vega measures an option’s sensitivity to changes in implied volatility (absolute change in option price per one-point change in IV).
* Lambda measures percentage sensitivity of option price to the underlying (elasticity).

Implied volatility affects option prices and therefore changes lambda indirectly. If implied volatility rises and option premiums increase (higher C), the ratio S/C may fall and thus lambda can decrease — though delta will also change, so the net effect depends on moneyness, time to expiry, and the size of the volatility move.

Both lambda and vega vary with moneyness and time to expiration. Vega typically increases with longer expirations; lambda’s level and behavior depend on option price and delta and therefore also vary with these factors.

Practical uses of lambda

• Volatility‑neutral (lambda‑neutral) strategies: Traders can balance long and short options so net lambda is near zero, reducing sensitivity to percentage moves in the underlying while keeping other exposures (delta, vega) under control.
• Delta hedging refinement: Lambda helps anticipate how delta will scale in percentage terms as prices move, aiding more dynamic hedge sizing and rebalancing decisions.
• Volatility speculation: Positions with high positive lambda can amplify gains from expected upward moves in the underlying; negative lambda exposures (more common with some puts) can be used when expecting declines in implied leverage or volatility.
• Multi‑leg spreads (butterflies, iron condors): Considering lambda helps understand how the risk‑reward profile of complex spreads will react to different market moves and changing volatility.

Calls vs puts

• Call options typically exhibit positive lambda: as the underlying rises, the option’s percentage return tends to be positive and amplified relative to the stock.
• Put options often show negative lambda in terms of sign convention (percentage moves behave differently as the underlying rises), but the magnitude and sign depend on moneyness and time to expiration.

Lambda and the volatility smile

The volatility smile — differences in implied volatility across strikes — influences option prices and deltas across strikes. Since lambda depends on both delta and the option price, variations in implied volatility across strikes will affect lambda, helping explain differences in percentage sensitivity across moneyness levels.

Key takeaways

• Lambda = (S / C) × delta; it measures option elasticity (percent change in option price per 1% change in underlying).
• It provides a clear view of the leverage embedded in an option relative to holding the underlying.
• Lambda and vega are connected through implied volatility, but they measure different sensitivities. Use lambda alongside other Greeks (delta, gamma, vega, theta) for fuller risk management and strategy design.