Smile-Dependent Volatility Changes Option Gamma and Vanna
Summary
The document examines a derivation of option Greeks when implied volatility varies with moneyness. It compares the reported gamma and vanna expressions from a smile dynamics paper with calculations that treat the Black–Scholes price as depending on the smile-implied volatility, and identifies a discrepancy in the expansion terms.
The response explains that differentiation with respect to the underlying must account for the dependence of implied volatility on moneyness. In particular, the chain rule adds a term involving the sensitivity of implied volatility to the underlying price; including it is said to recover the paper’s expressions. The reply is brief and does not show the full derivation or resolve every convention behind the expansions, so the result should be checked against the paper’s definitions of moneyness and Greeks.
Key ideas
- When implied volatility depends on moneyness, an option price depends on the underlying through both its direct price input and the volatility surface.
- Applying the chain rule adds derivatives of implied volatility to the usual Black–Scholes Greek calculations.
- Ignoring smile dependence can change the gamma and vanna expansion coefficients.
- The brief explanation does not provide the complete derivation or all convention details.
Tags
Full text
# smile dynamics IV appendix 4
# smile dynamics IV appendix 4
I am having difficulty in recovering some result in smile dynamics of Bergomi https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1520443, the paper gives $(1-3\alpha x +(6\alpha^2 - \frac{5}{2}\beta)x^2)$ and $(x - (2\alpha-\sigma_0\alpha')x^2)$ part, while my own calculation gives $(1-\alpha x +(\alpha^2 - \frac{1}{2}\beta)x^2)$ and $(x - (\alpha-\sigma_0\alpha')x^2)$ respectively, which will give different SSR ratio for this model. So I think something wrong with my calculation.
Quickly sum up, in the paper it states given a model of smile as a function of moneyness, $$\hat{\sigma}(x) = \sigma_0(1+\alpha(\sigma_0)x+\frac{1}{2}\beta(\sigma_0)x^2)$$ one can calculate below greeks of option $Q$, whose BS price is $P^{BS}(\hat{\sigma}(x))$, as: $$\frac{1}{2}S^2\frac{d^2Q}{dS^2}=\frac{1}{2}\frac{SN'(d)}{\sigma_0\sqrt{T}}(1-3\alpha x +(6\alpha^2 - \frac{5}{2}\beta)x^2)$$ $$S\sigma_0\frac{d^2Q}{dSd\sigma_0}=\frac{SN'(d)}{\sigma_0\sqrt{T}}(x - (2\alpha-\sigma_0\alpha')x^2)$$
Here is how I approached it. By checking the context I suppose calculation of $\frac{d^2Q}{dSd\sigma_0}$ it should be using BS greeks at $\sigma=\hat{\sigma}(x)$ and link $\frac{d}{d\sigma_0}=\frac{d\hat{\sigma}}{d\sigma_0}\frac{d}{d\hat{\sigma}}$. Giving a go based on that, I get gamma theta as: $$\frac{1}{2}S^2\frac{d^2P^{BS}(\hat{\sigma})}{dS^2}=\frac{1}{2}\frac{SN'(d)}{\sigma_0\sqrt{T}}\frac{\sigma_0}{\hat{\sigma}(x)}$$ and doing taylor expansion at order 2 in x and order 0 in T, $$\frac{\sigma_0}{\hat{\sigma}(x)}:=f(x)=\frac{1}{1+\alpha(\sigma_0)x+\frac{1}{2}\beta(\sigma_0)x^2}\sim 1-\alpha x + (\alpha^2-\frac{1}{2}\beta)x^2$$
Similar to vanna theta, I get $$S\sigma_0\frac{d^2P^{BS}}{dSd\sigma_0}=S\sigma_0\times vanna^{BS}\times\frac{d\hat{\sigma}}{d\sigma_0}=\frac{SN'(d)}{\sigma_0\sqrt{T}}\sigma_0^2\sqrt{T}\frac{x+\frac{1}{2}\hat{\sigma}^2T}{\hat{\sigma}\sqrt{T}}\frac{1}{\hat{\sigma}}(\frac{\hat{\sigma}}{\sigma_0}+\sigma_0(\alpha'x+\frac{1}{2}\beta'x^2))$$, where $\alpha'=\frac{d\alpha}{\sigma_0}$. And finally still using $f(x)$ above I get $$S\sigma_0\frac{d^2P^{BS}}{dSd\sigma_0}=\frac{SN'(d)}{\sigma_0\sqrt{T}}(x - (\alpha-\sigma_0\alpha')x^2)$$
## Answer by ooooolivia (score 0)
https://quant.stackexchange.com/a/79722
I think in paper's setting, $P_{BS}$ depends on $\sigma (x)$, which means when taking derivative, we need to consider $\frac{dP}{dS} = \frac{\partial P}{\partial \hat{\sigma}} \frac{\partial\hat{\sigma} }{\partial S}$. With the additional term included, we can the derive the same results as the paper~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.