imaginary volatility

Imaginary volatility is a striking feature of the hedged HVA discovered by Benedict Burnett in Burnett (2021). When the HVA itself is included in the value being hedged, the total derivative book value $\hat{V}$ satisfies a Black-Scholes-type PDE with an effective volatility:

${\hat{\sigma }}_{\hat{V}}^2={\sigma }^2\left(1-\frac{2{\hat{V}}_{xx}\psi }{D^2}\right).$

For options with large positive gamma (${\hat{V}}_{xx}>0$) and high friction costs ($\psi$ large relative to $D^2$), this effective volatility can become negative, meaning $\hat{\sigma }$ is imaginary.

The physical interpretation is profound: if the friction cost of hedging an option overwhelms the hedging benefit, the effective value of the option can become negative. The theta-gamma relationship ${\hat{V}}_t\approx -\frac{1}{2}{\hat{\sigma }}_{\hat{V}}^2{\hat{V}}_{xx}$ with imaginary vol implies positive theta, meaning the option value increases with time — the opposite of the usual case. This leads to a hockey-stick-shaped option value that dips below zero near at-the-money.

This can be viewed as an admission that the no-arbitrage hedging argument breaks down: if friction costs exceed the hedging benefit, a rational trader simply will not hedge, and the no-arbitrage value becomes indeterminate.

Key Details

• Arises only in the hedged HVA case (not the unhedged reserve case)
• Requires sufficiently large ${\hat{V}}_{xx}\psi /D^2$ — typically only relevant for high-gamma, illiquid positions
• The equivalent formulation for the HVA alone uses ${\hat{\sigma }}^2={\sigma }^2(1-4V_{xx}\psi /D^2)$, with the factor of 4 from expanding at higher precision
• Connection to nonlinear PDEs: the HVA PDE $U_t+rX{U}_x+\frac{1}{2}{\sigma }^2{U}_{xx}-rU=(\sigma (V_{xx}+U_{xx})/D{)}^2\psi$ is nonlinear; Burnett notes that the Deep BSDE Solver could potentially solve this exactly

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