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The other answers have given a good qualitative description of what the VIX measures. In this answer, I will try to give a comprehensive quantitative overview of how the VIX formula works. What is the VIX? The CBOE Volatility Index (VIX) is an index which measures the stock market’s expectation of the future 30-day volatility of the S&P-500 index. This ...

2

Writing $M_T=\max_{0\le t\le T}S_t\,,\,\, m_T=\min_{0\le t\le T}S_t$ the option payoff is \begin{align} (K-S_T)^+\underbrace{1_{\{m_T\le L\}}}_{\text{KI}}\underbrace{1_{\{M_T< U\}}}_{\text{KO}}=(K-S_T)^+(1-1_{\{m_T>L\}})1_{\{M_T<U\}}\,. \end{align} In other words, the KI-KO-option is a portfolio of a long positon in a single barrier KO option and a ...

1

Consider any function $f(S(t),K,t,T,\{x_i(t)\})$ with payoff $(S(T) - K)_+$ when $t=T$, where $\{x_i(t)\}$ are other variables/parameters so that at $t=0$ you are able to choose (i.e. calibrated) these so that your function matches the market price of the option: $f(S(0),K,0,T,\{x_i(0)\}) = C^{market}(t=0)$. As the payoff of the option does not depend on \$\{...

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