Comparing Option Vega Across Strikes with Black-Scholes

Summary

The document outlines how to compare option vegas across strike prices for a fixed maturity. Vega measures how an option price changes as implied volatility changes. The proposed method is to choose a pricing model, differentiate its option price with respect to volatility, and then evaluate the resulting expression while varying strike and holding the other inputs fixed.

Black-Scholes is given as an example, with vega expressed using the underlying price, time to maturity, dividend yield, and the model's intermediate d1 quantity. Since d1 depends on strike, the formula describes how vega varies across strikes and can be plotted for comparison. The discussion provides a modeling procedure rather than empirical evidence. Comparisons depend on the chosen pricing model and the inputs held constant; no particular strike ranking or numerical illustration is supplied.

Key ideas

• Vega is the sensitivity of an option's value to volatility in a specified pricing model.
• Comparing strikes requires evaluating the model's vega while holding other inputs constant.
• In Black-Scholes, vega depends on strike through the intermediate d1 term.
• Plotting the resulting values can show how vega changes across strikes and maturities.
• The comparison is model-dependent and the document supplies no numerical example.

Tags

# How can you compare Vega between strikes?


# How can you compare Vega between strikes?












Given a specified maturity is there a way to compare Vegas between different strikes? Surely the Vega of an ATM option will be very different from the same Vega of an OTM option

## Answer by Piyush Gupta (score 1)

https://quant.stackexchange.com/a/70190

Vega is the measure of the rate of change of option's value with every percentage change in volatility, i.e., dV/dσ.

- First of all, we need an option valuation model to get the function of option price (V) wrt underlying security price, strike price, time to maturity, risk-free interest rate & volatility.

- Once we have an option pricing model, take the partial derivative of option price (V) wrt volatility (σ) to get vega (ν).

- If we consider Black Scholes model for valuation, Vega (ν) = ∂V/∂σ = e−qTS√T φ(d1), where d1 is a intermediate parameter in black scholes model.

- On putting all the values in the function of Vega (ν) except Strike price, we get a function describing the dependency of vega for different strike prices. We can even plot the graph like the below example to compare vega for different strike prices & different times to maturity.

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