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Estimating Option Vega from a Call-and-Put Replication

Article Quant Q&A · Author: Cassey_Adams
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Summary

The note addresses how to measure an option payoff’s sensitivity to changes in the underlying stock’s volatility, expressed as the derivative of option value with respect to volatility. It proposes first representing the payoff as a linear combination of calls and puts, then calculating the position’s Vega from the Black–Scholes Vegas of those component options. The approach turns a potentially unfamiliar payoff into a sum of standard option exposures, provided the replication is valid.

The discussion gives no payoff details, derivation, worked example, or numerical evidence, so it does not establish whether a specific payoff can be replicated or how to choose the component strikes and weights. Its guidance is a suggested method rather than a complete calculation. The resulting sensitivity would also rely on the assumptions behind the chosen pricing model and on the replication remaining appropriate for the payoff being analyzed.

Key ideas

  • A nonstandard option payoff may be expressible as a combination of vanilla calls and puts.
  • The volatility sensitivity of a replicated position can be built from the Vegas of its components.
  • Black–Scholes Vega supplies the component sensitivities in the suggested approach.
  • The note does not provide enough payoff detail to verify or implement a specific replication.

Tags

OptionsVolatilityDerivatives pricing
Full text
# Sensitivity to changes in the volatility


# Sensitivity to changes in the volatility












I have a sample payoff function shown below:

How do I find a formula which gives the sensitivity to changes in the volatility of the underlying stock. In other words, I want to find a formula for $\frac{\partial V}{\partial \sigma}$ for the option V(s,t).

I am a beginner in Quant Finance so please excuse me for asking such simple questions

## Answer by d_797 (score 2)

https://quant.stackexchange.com/a/59256

For a payoff like this, it may be possible to replicate it by a linear combination of calls and puts. From there you can get the option's Vega by using the BS Vega of the calls and puts in your replication.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.