Extending the Black-Scholes Option Pricing Theory to Account for an Option Market Maker's Funding Costs

TL;DR

This paper extends the Black-Scholes model to include funding costs for option market makers, resulting in nonlinear PDEs that better explain bid-ask spreads and are solved using a finite difference method.

Contribution

It introduces a novel extension of the Black-Scholes PDE to incorporate asymmetric funding costs and develops a numerical method for pricing options under these conditions.

Findings

Funding costs can explain observed bid-ask spreads.

The extended model produces realistic option prices with funding costs.

Portfolio netting can mitigate funding cost impacts.

Abstract

An option market maker incurs funding costs when carrying and hedging inventory. To hedge a net long delta inventory, for example, she pays a fee to borrow stock from the securities lending market. Because of haircuts, she posts additional cash margin to the lender which needs to be financed at her unsecured debt rate. This paper incorporates funding asymmetry (borrowed cash and invested cash earning different interest rates) and realistic stock financing cost into the classic option pricing theory. It is shown that an option position can be dynamically replicated and self financed in the presence of these funding costs. Noting that the funding amounts and costs are different for long and short positions, we extend Black-Scholes partial differential equations (PDE) per position side. The PDE's nonlinear funding cost terms create a free funding boundary and would result in the bid price for a long position on an option lower than the ask price for a short position. An iterative Crank-Nicholson finite difference method is developed to compute European and American vanilla option prices. Numerical results show that reasonable funding cost parameters can easily produce same magnitude of bid/ask spread of less liquid, longer term options as observed in the market place. Portfolio level pricing examples show the netting effect of hedges, which could moderate impact of funding costs.