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}

Vanna market impact

The figure below shows the value of vanna for a $100 strike option. As discussed in the delta section, vanna is positive for out of the money options and negative for in the money options. This implies that an increase in volatility will increase the delta for OTM contracts, and decrease it for ITM contracts, which has interesting implications for market maker hedging dynamics.

vanna