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

摘要

An option market maker incurs funding costs when carrying and hedginginventory. To hedge a net long delta inventory, for example, she pays a fee toborrow stock from the securities lending market. Because of haircuts, she postsadditional cash margin to the lender which needs to be financed at herunsecured debt rate. This paper incorporates funding asymmetry (borrowed cashand invested cash earning different interest rates) and realistic stockfinancing cost into the classic option pricing theory. It is shown that anoption position can be dynamically replicated and self financed in the presenceof these funding costs. Noting that the funding amounts and costs are differentfor long and short positions, we extend Black-Scholes partial differentialequations (PDE) per position side. The PDE's nonlinear funding cost termscreate a free funding boundary and would result in the bid price for a longposition on an option lower than the ask price for a short position. Aniterative Crank-Nicholson finite difference method is developed to computeEuropean and American vanilla option prices. Numerical results show thatreasonable funding cost parameters can easily produce same magnitude of bid/askspread of less liquid, longer term options as observed in the market place.Portfolio level pricing examples show the netting effect of hedges, which couldmoderate impact of funding costs.