Hi David,

How much significant is the assumption of short selling in Black Scholes?

Anil

It is *critical* as the key assumption under Black-Scholes is the ability to dynamically hedge. If you look at Euro formula:
call option = (stock price)(delta = N(d1)) - (discounted strike price)(N(d2) = probability of striking).

The call option is equal to the fractional (shares) purchased with borrowed cash (discounted strike = SHORT the bond). And then going forward in time, this assumes the hedger can rebalance *dynamically* with long/short adjustments (the hedger must be able to short to replicate the option with a riskless portfolio). So, from the perspective of the analytical derivation of the B-S, the ability to short is critical as it allows the hedger to self finance.

Piles of research have been done around relaxing the assumptions, some can be relaxed more than others. I’m not current on this literature necessarily, so I may not be aware of some proof that this assumption can be relaxed (but it would surprise me)...David