Heston Vega: Converting Variance Sensitivity to Volatility Sensitivity

Summary

The document clarifies why Heston option vega may include a factor of two times spot volatility. In a stochastic volatility model, an option value is naturally expressed as a function of variance. If a trader instead wants sensitivity to volatility, the chain rule converts the derivative with respect to variance into a derivative with respect to volatility, since variance is the square of volatility. This produces the factor of two and a volatility multiplier.

The volatility based convention makes the reported measure resemble Black Scholes vega, which is defined with respect to volatility; the same chain rule can convert Black Scholes sensitivity to variance as well. The answer also notes that European calls and puts have equal vega under put-call parity. The explanation depends on keeping variance and volatility distinct, and on identifying whether the derivative is with respect to spot variance or spot volatility. It gives a conceptual derivation but no numerical example or implementation guidance.

Key ideas

• Heston option values are commonly parameterized by variance, while market vega is often stated with respect to volatility.
• The chain rule converts variance sensitivity into volatility sensitivity and yields the factor of two times volatility.
• The volatility convention makes the measure comparable to Black Scholes vega.
• European calls and puts have equal vega under put-call parity.
• Always identify whether a sensitivity is defined against spot variance or spot volatility.

Tags

# Calculating vega in Heston?


# Calculating vega in Heston?












I often see Vega in the Heston model specified as: \begin{align*} \nu & = \frac{\partial C}{\partial v} = \frac{\partial C}{\partial v_0} 2 \sqrt{v_0} \end{align*} where $v = \sqrt{v_0}$.

Why are we setting $v = \sqrt{v_0}$?

Where is the square-root and "$2$" coming from?

## Answer by Kevin (score 4, accepted)

https://quant.stackexchange.com/a/60468

Let $V$ denote the variance and $v$ the volatility, i.e. $V=v^2$. The natural argument for the option price under a stochastic volatility model is typically the variance, i.e. $C_\text{SV}=C_\text{SV}(S_0,V_0,...)$. However, using the chain rule, we can compute vega in terms of the volatility: $$\nu=\frac{\partial C_\text{SV}}{\partial v}=\frac{\partial C_\text{SV}}{\partial V}\frac{\partial V}{\partial v}=\frac{\partial C_\text{SV}}{\partial V}\frac{\partial v^2}{\partial v}=\frac{\partial C_\text{SV}}{\partial V}2v=\frac{\partial C_\text{SV}}{\partial V}2\sqrt{V}.$$ We do this in order to resemble the Black-Scholes vega which is the partial derivative of the Black-Scholes option price, $C_\text{BS}=C_\text{BS}(S_0,\sigma,...)$, with respect to $\sigma$. Of course, when we have $\frac{\partial C_\text{BS}}{\partial \sigma}$, we can easily infer $\frac{\partial C_\text{BS}}{\partial \sigma^2}$ using again the chain rule. This would be the Black-Scholes vega with respect to the variance.

The model-free put-call parity implies that European-style put and call options have the same vega (with respect to volatility or variance).

Writing $V_0$ and $v_0=\sqrt{V_0}$ emphasises that we talk about the spot variance and volatility.

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