1.4.6 Vanna

Vanna is the first derivative of delta with respect to IV. This quantity describes how sensitive a delta hedging scheme is to changes in IV. A portfolio consisting of mostly far out of the money options that is delta hedged will be much more sensitive to changes in IV than a portfolio consisting primarily of options near the money or in the money.

Summary of Terms

$V$ = Contract Value

$\Delta$ = Contract Delta

$\nu$ = Contract Vega

$\Psi$ = Contract Vanna

$S$ = Spot Price

$K$ = Strike Price

$\sigma$ = Implied Volatility

$\tau$ = Years to Expiration

$r$ = Risk Free Rate

$q$ = dividend yield

Calculation

\Psi = \frac{\partial^2 V}{\partial S \partial \sigma} =\frac{\partial \Delta}{\partial \sigma} = \frac{\partial \nu}{\partial S}

\Psi = {Se^{-q\tau}\sigma d_- \over 2\sqrt{2\pi\tau}}e^{-{1 \over 2}d_+^2}

d_+ = {1 \over \sigma \sqrt{\tau}}\bigg[\ln\bigg({S \over K}\bigg) + \bigg(r - q + {\sigma^2 \over 2}\bigg)\tau\bigg]

d_- = d_+ - \sigma\sqrt{\tau}

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