Why Stock Vega Holds the Current Price Fixed

Summary

The document clarifies a partial derivative used in option pricing. Vega measures how an option’s value changes as volatility changes, while other inputs to the pricing function—including the current stock price—are held fixed. Under Black–Scholes assumptions, calls and puts with matching terms have equal vega, consistent with put-call parity.

The apparent contradiction comes from treating the current stock price at a fixed time as though it were being recalculated from the entire stochastic price process while volatility changes. In the derivative defining vega, the current observed price is an input value, not a function being differentiated with respect to volatility. Thus the partial derivative of that input with respect to volatility is zero in this calculation. The explanation is conceptual and relies on distinguishing a model’s random process from the arguments of an option pricing function; it does not derive the Black–Scholes vega formula or address alternative models and sensitivities.

Key ideas

• Vega is the sensitivity of an option’s value to volatility, with other pricing inputs held fixed.
• The current stock price is treated as a fixed input when taking the partial derivative for vega.
• A stock’s stochastic evolution does not make its already observed value a function of volatility in this partial derivative.
• Under Black–Scholes assumptions, matching calls and puts have equal vega by put-call parity.

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# Question about the vega of a stock


# Question about the vega of a stock












In Black-Scholes model with constant parameters, a call and a put with the same characteristics have the same vega: https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model#The_Greeks

Using call-put parity yields $\frac{\partial S_t}{\partial \sigma} = 0$. This result is weird because we have: $S_t = S_0.e^{(r-\frac{\sigma^2}{2})t+\sigma W_t}$.

How can we justify this result? Thank you in advance for your answers.

## Answer by RRL (score 5, accepted)

https://quant.stackexchange.com/a/38082

Vega is the partial derivative of the option price (as a function of parameters -- current stock price $S_t$, strike price $K$, implied volatility $\sigma$, etc.) with respect to $\sigma$ -- holding other parameters fixed:

$$vega = \frac{\partial}{\partial \sigma} V(S_t,K,\tau,r,\sigma) $$

You are confusing the stochastic process with the parameter $S_t$ which is a constant when $t$ is fixed.

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