Call and Put Vega Equality from Put-Call Parity

Summary

The note shows how to establish that a call and put have the same vega when they share the same forward underlying, strike, maturity, and volatility inputs. It starts from put-call parity, which expresses the call-minus-put price difference as the forward price minus the strike.

Because the right-hand side does not depend on volatility, differentiating both sides with respect to volatility makes the call and put vega terms equal. This argument is concise and does not require deriving separate option pricing formulas. Its scope is the parity relation as written; adjustments to parity for discounting, dividends, or other contract conventions are not discussed.

Key ideas

• Put-call parity relates call and put prices through the forward price and strike.
• The parity right-hand side is independent of volatility under the stated setup.
• Differentiating parity with respect to volatility proves equality of call and put vegas.
• The derivation relies on the specified parity convention and does not cover other contract adjustments.

Tags

# i have an option derivative question


# i have an option derivative question












please show that the call and the put share the same vega, i.e., please prove the following equality.

do we derive the call and put equations ?

## Answer by Canardini (score 4)

https://quant.stackexchange.com/a/50008

Use the call put parity :

$$C(t,F_{t,T},T,\sigma,K)-P(t,F_{t,T},T,\sigma,K)=F_{t,T}-K$$ where $F_{t,T}$ is the forward rate(underlying), $K$ is the strike, $t$ the valuation date, $\sigma$ the model volatility, $T$ is the maturity.

Differentiate the equation with respect to $\sigma$, and you will get the result wanted.

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