In tempranillo1/IVsurfaceApp: Shiny app to modeling s&P 500 implied volatility surface.

Option's price in Black-Scholes model is function $$C = f(S_t, K, r, d, \sigma, \tau) $$ where $S_t$ is underlying asset, $K$ strike price, $r$ interest rate, $d$ dividend yield, $\sigma$ volatility, $\tau$ time to maturity.

Every variable, except volatility $\sigma$ can be observe in the market...

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$\dots$but we have options price from the market, so we can find such $\sigma$, which satisfy equallity beetwen market and model price. Lets define $g$ as $$g(\sigma) = C^{\rm mkt} - f(S_t, K, r, d, \sigma, \tau)$$ and using numerical methods find $\sigma$ which satisfy $$g(\sigma) = 0$$ Obtained $\sigma$ is called implied volatility.

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Each day, in the market there are quoted many options with different strike prices $K$ and time to maturity $\tau$. After computng implied volatility for all avaible optios we can put our results in matrix, like below $$ \begin{pmatrix} \sigma^{\rm imp}{T_1,K_1} & \sigma^{\rm imp}{T_1,K_2} & \cdots & \sigma^{\rm imp}{T_1,K_N} \cr \sigma^{\rm imp}{T_2,K_1} & \sigma^{\rm imp}{T_2,K_2} & \cdots & \sigma^{\rm imp}{T_2,K_N} \cr \vdots & \vdots & \ddots & \vdots \cr \sigma^{\rm imp}{T,K_1} & \sigma^{\rm imp}{T,K_2} & \cdots & \sigma^{\rm imp}_{T,K_N} \cr \end{pmatrix} $$

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To obtain regular volatility surface we have to use interpolation between nodes, extrapolation outside avaible nodes and same smoothing techniques.

Between each time periods $T_i < T < T_{i+1}$ (fixed $K$) linear interpolation in terms of not annualized variance is used

$$ \sigma(T) = \frac{1}{\sqrt{T}} \sqrt{\sigma^2_i T_i + \tau\left( \sigma^2_{i+1} T_{i+1} - \sigma^2_{i} T_{i} \right)} $$ where $\tau = \frac{T - T_i}{T_{i+1} - T_i}$.

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Interpolating between strike prices (fixed $\tau$) is a bit more tricky. We have to use more sophisticated methods like:

• polynomial regression stats::poly()
• local polynomial regression (loess) stats::loess()
• natural cubic splines splines::ns()
• smoothing splines stats::smooth.spline()
• B-spline splines::bs()

tempranillo1/IVsurfaceApp documentation built on May 31, 2019, 8:33 a.m.