A Dynamic Model for Hard-to-Borrow Stocks

We study the price-evolution of stocks that are subject to restrictions on short-selling, generically referred to as hard-to-borrow. Such stocks are either subject to regulatory short-selling restrictions or have insufficient float available for lending. Traders with short positions risk being “bought-in”, in the sense that their positions may be closed out by the clearing firm at market prices. The model we present consists of a coupled system of stochastic differential equations describing the stock price and the “buy-in rate”, an additional factor absent in standard models. The conclusion of the model is that short-sale restrictions result in increased prices and volatilities. Our model prices options as if the stock paid a continuous dividend, reflecting a modified form of Put-Call parity. Another consequence is that stocks that do not pay a dividend may have calls subject to early exercise. Both features are in agreement with empirical (market) observations on hard-to-borrow stocks.