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Implied Volatility

Next: Strike Given Delta Up: Retrieving the Arguments Previous: Retrieving the Arguments

Implied Volatility

Since $v_{\sigma}>0$ , then function $\sigma\mapsto v(x,K,T,t,\sigma,r_d,r_f,\phi)$ is

strictly increasing,
concave up for $\sigma\in[0,\sqrt{2\vert\ln f-\ln K\vert/\tau})$ ,
concave down for $\sigma\in(\sqrt{2\vert\ln f-\ln K\vert/\tau},\infty)$

and also satisfies

$\displaystyle v(x,K,T,t,\sigma=0,r_d,r_f,\phi)$	$\textstyle =$	$\displaystyle [\phi(xe^{-r_f\tau}-Ke^{-r_d\tau})]^+,$	(45)
$\displaystyle v(x,K,T,t,\sigma=\infty,r_d,r_f,\phi=1)$	$\textstyle =$	$\displaystyle xe^{-r_f\tau},$	(46)
$\displaystyle v(x,K,T,t,\sigma=\infty,r_d,r_f,\phi=-1)$	$\textstyle =$	$\displaystyle Ke^{-r_d\tau},$	(47)
$\displaystyle v_{\sigma}(x,K,T,t,\sigma=0,r_d,r_f,\phi)$	$\textstyle =$	$\displaystyle xe^{-r_f\tau}\sqrt{\tau}/\sqrt{2\pi}I\!\!I_{\{f=K\}},$	(48)

there exists a unique implied volatility $\sigma=\sigma(v,x,K,T,t,r_d,r_f,\phi)$ , which can be found by a Newton-Raphson method. However, the starting guess for employing this method should be chosen with care, because the mapping $\sigma\mapsto v(x,K,T,t,\sigma,r_d,r_f,\phi)$ has a saddle point at

$\begin{displaymath} \left(\sqrt{\frac{2}{\tau}\vert\ln\frac{f}{K}\vert},\phi\lef... ...eft(\phi \sqrt{2\tau[\ln\frac{K}{f}]^+}\right)\right\}\right). \end{displaymath}$

(49)

To ensure convergence of the Newton-Raphson method, we are advised to use initial guesses for $\sigma$ on the same side of the saddle point as the desired implied volatility. The danger is that a large initial guess could lead to a negative successive guess for $\sigma$ . Therefore one should start with small initial guesses at or below the saddle point. For at-the-money options, the saddle point is degenerate for a zero volatility and small volatilities serve as good initial guesses.

Next: Strike Given Delta Up: Retrieving the Arguments Previous: Retrieving the Arguments

2000-06-11

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