Pricing European Call Options Under a Hard-to-Borrow Stock

Abstract

In 2009, Avellaneda and Lipkin (A&L) proposed a dynamic model for hard-to-borrow stocks, in which the stock price and the buy-in rate, an additional factor introduced by them, are full coupled. In order to obtain a semi-explicit pricing formula for European call options, A&L had to make an independence assumption which has facilitated the derivation of their formula. However, it has also limited its application to more general cases. When such an assumption is removed, a more general approach is needed. A PDE approach is presented in this paper, taking the pricing of European call options as an example to demonstrate that it is a broader approach than A&L’s semi-explicit pricing formula. The PDE system governing the value of European call options together with a set of appropriate boundary conditions is first established, followed by numerical results obtained with two carefully chosen methods, the difference of which lies in the treatment of the jump term.

Through this work, it is verified that the formula presented by A&L is a good approximate solution, provided that the independence assumption is indeed a reasonable one. However, in the event that this is not the case, the full model needs to be solved with the PDE approach as demonstrated in this paper.

Keywords: Hard-to-borrow stock; Option pricing; Short-selling; Buy-in