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Key Takeaways
- Lambda converts dollar sensitivity into local percentage sensitivity: delta × stock price / option price.
- The option and delta inputs must use the same contract and valuation inputs.
- For the worked Black-Scholes chain, call lambda is 22.52 at strike 110, 14.61 at 100 and 8.49 at 90.
- A vanilla put has negative lambda; calculate its magnitude directly from delta and premium.
- Use full repricing for a finite price scenario and expiration payoff for the final trade result.
What is lambda (omega) in options?
Lambda, also called omega or option elasticity, is the option Greek that measures the local percentage change in an option’s value for a 1% change in the underlying. It equals delta times the underlying price divided by a positive option price, and the contract multiplier cancels when the inputs use consistent units. A lambda of +12 maps a small +1% stock move to a first-order +12% option-price estimate.
At stock $100, an option with delta 0.20 and price $0.40 has lambda 50. At price $2 with the same delta, lambda is 10. The ratio connects the dollar sensitivity with the amount paid for the option.
Formula
Lambda (Omega) = Delta × (S / V) = percentage change in option value per 1 percent change in the underlying- S is underlying price, V is a positive option price and delta is the derivative of V with respect to S.
- The standard contract multiplier cancels when price and delta use consistent units.
- A lambda of +12 maps a small +1% underlying change to a first-order +12% option-price estimate.
- For a positive-price vanilla Black-Scholes call with nonnegative strike, S × delta is at least V, so lambda is at least one.
- Vanilla put lambda is negative. Its absolute value is below one whenever |delta| × S is below the put price.
- Recalculate the ratio with current valuation inputs and use full pricing for the complete scenario.
Lambda example: one option chain
XYZ is $100. Price European calls with Black-Scholes using 31% volatility, 30/365 years, zero rates and zero dividends. Use unrounded prices and deltas to compute lambda, then round the displayed results.
- 110 call: price $0.674846, delta 0.151980, lambda 22.52
- 100 call: price $3.544403, delta 0.517722, lambda 14.61
- 90 call: price $10.486936, delta 0.890640, lambda 8.49
- First-order percentage estimates for a +1% stock move: +22.52%, +14.61%, +8.49%
- Full repricing at stock $101, with other inputs held fixed: +24.55%, +15.24%, +8.59%
- At expiration with stock $100, the 110 and 100 calls pay zero; the 90 call pays $10 per share
Lambda ranks local percentage sensitivity across the same chain. Full repricing at $101 includes curvature and produces the separate finite-move results shown. Expiration payoff uses the final stock price and strike directly.
For a +1% stock move, those are the first-order percentage estimates. Repricing the same calls at $101 produces +24.55%, +15.24% and +8.59%. Gamma explains the curvature beyond the initial linear sensitivity.
Lambda sign and magnitude for calls and puts
For a positive-price vanilla Black-Scholes call with nonnegative strike, S times delta is at least the call value. Its lambda is therefore at least one. For a vanilla put, delta and lambda are negative.
A put’s absolute lambda is below one when its value exceeds |delta| times stock price. An at-the-money example already satisfies this: S = K = $100, one year, 100% volatility and zero carry gives price $38.29, delta −0.31 and lambda −0.81.
How traders use lambda
Compare lambda alongside entry premium, share-delta and the position’s payoff. For a local single-factor beta calculation, multiply lambda by the underlying beta; 22 times 1.2 equals 26.4 for that modeled component.
Lambda is a first-order estimate, so reprice the actual move in the options profit calculator, which shows the option’s value at the new price and date beside its expiration payoff. Calculate the final outcome from the stock price, strike and paid premium. Expiration losses and time decay are different calculations from spot-price elasticity.
Limitations of lambda
- Reprice finite moves in full. Lambda is a local, first-order estimate: the 110 call’s +22.52% estimate becomes +24.55% at $101.
- Track theta and vega beside lambda. It holds time and implied volatility fixed, while time decay and volatility changes move the option’s value even when the stock does not.
- Check the expiration payoff behind a high lambda. A high lambda comes from a small premium: in the example, the highest-lambda 110 call and the 100 call both pay zero at expiration with the stock at $100.
- Recalculate lambda with current inputs before each comparison. The ratio changes with price, time and volatility.
Bottom line
Lambda turns delta into percentage leverage, showing how much an option’s value moves for each 1% move in the stock relative to what the option costs. Use it to compare leverage across strikes, then reprice finite moves and check the expiration payoff, because a high lambda rests on a small premium that can expire worthless.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- OIC: Options Glossary (Lambda)
- MathWorks Financial Toolbox: blslambda (Black-Scholes elasticity)
- OIC: Delta
How entries are sourced, calculated and updated: editorial methodology.