Short selling as a hidden assumption for forward price

Hi there,

Having a non-dividend paying share of current price S_0 with and a risk-free investing/borrowing force of interest r the general arbitrage-free forward price is S_0 * e^(rT) (T is the term of the forward agreement).

My simple question is why?

As I think that without the assumption of short-selling we can state just that fwd price <= S_0 * e^(rT). For the reversed inequality we need to sell a share without owning it (or without having any knowledge whether we are owning it or not).

My other question is that why in general the short-selling assumption is that Ok to assume? I mean in practice / in life one can not sell something without owning it first (or not borrowed it first and depositing some money to the lender - i.e. car-ski rentals and so on - I know they are not shares...). My point here is that with this pricing strategy (allowing short selling) in fact, we can quite misprice some products. What do you think?

Thx in advance!

Sándor

 

Hi Steve,

Thx for your quick reaction! Really appreciated.

In both prooves presented in the core reading there is short selling.

Proof a): page 17 around the middle of the page where the reversed part is assumed and a contradiction is found by an investment strategy directly including short selling.

Proof b): the principle of no-arbitrage could not be used as presented. There are two portfolios

A: one long fwd contract (strike K, expiry time T);
B: borrowing Ke^(-rT) & buying one share. ​

How would you find an arbitrage without short-selling in case of an assumption K < S_0 * e^(rT) ?

Thx in advance,

Sándor