  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.
