# Box Spreads!

The nuts and bolts of financial decision making are returns and probability.

Returns can be negative or positive and possibly very large.

Probability is usually limited between one and zero.

Returns are measurable, ex post, the dent or jump in your bank account balance is there for all to see.

Probability is hidden before and after, it's never directly observable and many people never explicitly have to calculate a probability.

It only began to be investigated in the last 500 years due to its slippery nature.

In finance, quants make use of probability's slippery nature by using it as a fudge factor. Whatever the real probability of anything is, they set the expected average return of every investment to equal the return of an investment which is 'certain' (usually the return on treasury bills).

There's a bunch of ways to calculate this 'certain' or 'risk free' rate of return.

Without getting too meta-physical, we can see what treasury bills are yielding right now (6 month t-bill yield is 0.14% right now).

Or we can over-complicate a little and remove all uncertainty from a bunch of options to calculate the risk free rate.

Using S&P 500 index options expiring in January I calculated 'box spreads'.

A box spread first replicates a long future using put call parity (C-P=F) with strike X1.

Then a short future with strike X2, where X1 < X2.

We are now long and short and guaranteed a payoff of X2-X1!

Of course, forgetting about fees the cost of the positions will be almost X2-X1. But *how* close?

Pretty close using option mid prices.

The lowest rate is -0.5% and highest is 0.81%, nevertheless the average is 0.16%.

Quite gratifying to tame usually wild equity markets to see theory work out pretty much as expected.

The most striking feature is that ATM options contribute to all the volatility in the return figure, whereas the wings are pretty darn steady.

The standard deviation is around 0.15 or so - so we shouldn't get too carried away.

Another nice feature is that index options are European, so you won't be left hanging.

Bid-ask spreads are so fat however that it's hard to make any money with this in practice.

I have seen opportunities with non-index options, but American exercises complicates things a little.