Scaling Black–Scholes Greeks to Percentage Moves

Summary

The document asks how to express option sensitivities as dollar gains or losses for percentage changes in the underlying price and implied volatility. It focuses on delta and gamma for spot moves, vega and volga for volatility moves, and vanna when both variables change. The proposed setting is the Black–Scholes model, with a request for analytic adjustments and numerical methods.

The responses suggest using finite differences to estimate sensitivities for the chosen move size, then multiplying the resulting per-option change by the position held to obtain position-level P&L. They also show that a spot move’s relative option-price change can be calculated directly by repricing the option at the shocked spot and comparing it with the original price. The discussion does not provide formulas for all requested Greeks, a worked numerical example, or guidance on volatility conventions and interaction effects, so these are methods to build on rather than a complete implementation recipe.

Key ideas

• Finite differences can estimate option price sensitivity for a specified spot or volatility shock.
• Multiply per-option sensitivity by the position quantity to estimate position-level P&L.
• A percentage spot shock can be evaluated by repricing the option at the changed underlying price.
• The responses do not fully specify the requested volatility and cross-sensitivity calculations.

Tags

# Option greeks: sensitivity to 1% move


# Option greeks: sensitivity to 1% move












In a Black&Scholes framework how can I compute the following sensitivities:

- to 1% move in the underlying price

- to 1% move in implied volatility

I would like the greeks to tell me how many dollars I lose/gain if the underlying/implied volatility moves by 1%. In particular, I would like to calculate the delta and gamma (to 1% move in underlying price) and vega and volga (to 1% move in implied volatility).

For the vanna I would like to consider a 1% move in both underlying and implied volatility.

Can you please suggest how to modify Black&Sholes greeks and also how to compute the sensitivities numerically?

A reference would also be very welcome. Thank you.

## Answer by Tulio Carnelossi (score 1)

https://quant.stackexchange.com/a/16317

Try Finite Differences to calculate your Greeks, it will give all the greeks for that specific underlying moviment. In order to back out the dollar change in your pnl just multiply each greek by the amount held in that position.

## Answer by AFK (score 0)

https://quant.stackexchange.com/a/16328

In the BS model, everything is explicit.

If your spot increases by $h\%$, the price will increase by $\Delta_{rel,h}\%$ where $$ \Delta_{rel,h} = \frac{C_{BS}(S(1+h),T,K,\sigma)}{C_{BS}(S,T,K,\sigma)} - 1 $$ That's high school math.

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