Short selling as a hidden assumption for forward price

[Sandor Kelemen]

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

 

[Steve Hales]

Hi
Section 4.2 of Chapter 12 will explain why, and short-selling isn't required for this proof. There is a "short position", but that refers to agreeing to selling the asset in the future.
In general though, we do assume that assets can be held in any amount, which includes short selling.
Thanks
Steve

 

[Sandor Kelemen]

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

 

[Steve Hales]

For some reason I thought you were talking about short selling the derivative!

The proof would still work even if I owned some shares to start with; that way the selling wouldn't be "short" as I'd actually possess them. Imagine I have 100 shares which are currently priced at S_0. I sell one for S_0. So now I own 99 shares and S_0 in cash (which I invest at the risk-free rate). I enter into a forward contract to buy a share for price K<S_0*e^(rT). At time T I buy a share at price K, which means I now own a total of 100 shares plus the difference in cash between S_0*e^(rt) and K. That's a risk-free gain.

Even in general short selling is fine. I could sell you a car I don't currently own; we agree on the car and the price, which you then pay to me. I now have to fulfil my side of the deal, which I can do by finding a car (with the money you've already given me) to complete the trade.

I could start an eBay listing for an item that I'm not currently in possession of - that's short selling.

It's harder to short sell consumables. For example, I can't really short sell coffee to Starbucks because they'll want it immediately. Investment products (like shares) can easily be shorted.

Hope that helps.

Steve