Analytic formula for option margin with liquidity costs under dynamic delta hedging

TL;DR

This paper derives an analytical formula to calculate the expected liquidity costs during delta hedging of European options, enabling faster computation compared to Monte Carlo simulations.

Contribution

It introduces a new analytical formula for option liquidity costs that incorporates delta process and supply curve parameters, improving computational efficiency.

Findings

Analytical formula accurately estimates liquidity costs.

Faster computation than Monte Carlo methods.

Numerical distributions of costs in special cases provided.

Abstract

This study derives the expected liquidity cost when performing the delta hedging process of a European option. This cost is represented by an integration formula that includes European option prices and a certain function depending on the delta process. We first define a unit liquidity cost and then show that the liquidity cost is a multiplication of the unit liquidity cost, stock price, supply curve parameter, and the square of the number of options. Using this formula, the expected liquidity cost before hedging can be calculated much faster than when using a Monte Carlo simulation. Numerically computed distributions of liquidity costs in special cases are also provided.