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Implied Probabilities and Structured Products – a rational approach

This post aims to summarise and extend the discussions we had at the excellent CLS investment offsite 2013. Also see this related piece.

The price of an equity linked, defined-payoff structured product can be explained in terms of an "implied probability distribution", which helps to understand in a quantitative way what investment view you express by buying it.

For example, let's take a 5 year long auto-call on the FTSE index paying an X% coupon with 50% "soft protection" European barrier (The auto-call, or kick-out bond, is probably the most popular structured product in the UK - we will assume here that the payoff is well understood).

Like all risky investments, it comes with potentially high returns, in exchange for the risk of losses: in this case at least 50% capital loss, and no coupon, if the FTSE is 50% lower in 5 years AND has never been above the auto-call barrier on the relevant previous dates.

Buying the product means betting against that outcome ("lay", in betting parlance).

Let's compare it to betting on the horses - to be rational, you would ask two questions:

• What odds is the bookmaker giving for that horse winning? (the "implied probability")
  
• What is your personal estimate of the odds of that horse winning? (your "subjective probability")

If these odds differ, you will then either "back" or "lay" that horse.

Similarly, a rational investor analysing the auto-call above would ask (among other questions):

• what is the "implied probability" of the FTSE being below the barrier at expiry? 
• what is your "subjective probability", i.e. what chance do you give to the FTSE being there at the end?

You will then consider buying the product if your estimate is much lower than the implied one (strongly simplifying here for the sake of conciseness... to go more in details would take some quality time e.g. our course), avoiding it otherwise.

How to come up with your "personal" view, the "subjective probability"?

I will shirk that question here, but a rational investor cannot do that! Somehow, every time he decides on an investment, he has a view, however unconsciously, which might be based on past history, on macro economics views, gut feeling, back aches, the exploration of chicken innards or whatever else one does to call the market.

How to find out the "implied probability"?

It is possible, but not trivial, to work out approximately some relevant implied probabilities just from the price and the terms of a particular product. On the other hand, we know that the prices of vanilla options are one of the key inputs used by the issuing banks to price the product, and we also know how to get the implied probabilities from those (see here for some numbers).

This type of analysis helps understand the success and popularity of auto-calls:

The implied probabilities of the equity market dropping a large amount over a long term are very often much larger than the realised frequency of such events: e.g. a drop of 50% in 5 year is often priced at 15-20% probability by the option markets (and hence the structured products), while the frequency of such events has been very much lower.

This very basic fact explains why the auto-calls have on average performed very well, though of course it is no guarantee that any particular one will do so in the future - as the voice rapidly informs us at the end of adverts for financial products.

But... but... why are the implied probabilities of downside drops so large, compared to historical? Are the traders involved in option markets inveterate bears, continuously losing money? Or is there a rational explanation to that?

From the other side of the argument: buying this autocall involves selling a put... and you might have heard disparaging comments on the "selling put" strategy deployed by many hedge funds (or Warren Buffett). How does it relate? Continued 

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