I'm trying to build a crude model for the effects of delta hedging on major indices like the S&P 500. My background is more in pure mathematics so a lot of this stuff is new to me. That said I would like to know if there is a function $f$ such that $vanna=f(gamma)$, where these greeks are from the black-scholes equation. I would also appreciate it if someone pointed me to some good research papers that cover equity index gamma and vanna modeling. Thanks in advance.