This paper studies European call option pricing problem under a hard-to-borrow stock model where stock price and buy-in rate are fully coupled. Avellaneda and Lipkin (2009) proposed a simplified solution approach with an independence assumption, and then derived a semi-explicit pricing formula. However, such an approach has limited its application to more general cases. In this paper, we propose a partial differential equation (PDE) approach for pricing European call options, regardless of the independence assumption. A two-dimensional PDE is derived first with a set of appropriate boundary conditions. Then, two numerical schemes are provided with different treatments of the jump term. Through our numerical results, we find that the semi-explicit formula is a good approximate solution when the coupling parameter is small. However, when the stock price and the buy-in rate are significantly coupled, the PDE approach is preferred to solve the option pricing problem under the full hard-to-borrow model.
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