MoreNotesToSelf

Technology, Finance, and Life

Synthetic tranches intuition for stock option guys

Posted by DK on January 9, 2010

I had a couple of interesting conversations comparing equity options to tranches, so I thought I'd develop some of the parallels here.

I'm assuming you're already familiar with equity options, however, so let me walk you through an example. Let's assume there is a stock index that, for argument's sake, can vary between $0 and $100. Now, consider the following series of call spreads on this index.

  • Call spread A = long call option with a strike of $0, short a call with a strike of $3.
  • Call spread B = long call option with a strike of $3, short a call with a strike of $7.
  • Call spread C = long at $7, short at $10.
  • Call spread D = long at $10, short at $15.
  • Call spread E = long at $15, short at $30.
  • Call spread F = long at $30, short at $100 (I know we've limited the stock to $100, but work with me here).

Let's say the index trades at around $1.50. Call spread A is most sensitive to changes in the index price (relative to the other call spreads) since it is "at-the-money" (ATM). In contrast, the $30-$100 spread offers little value since it is so far "out-of-the-money" (OTM). If the stock price increases to $5, call spread A has moved completely "in-the-money" (ITM) and is no longer as sensitive to moves in the underlying index (the maximum PnL for the spread has been realized). Call spread B is now the ATM option portfolio. As the index price moves, the value of each call spread will fluctuate depending on whether it is ITM, ATM, or OTM. Another way to look at it is in terms of option premium. If the index is trading at $1.50, I'll likely get much more premium by selling call spread A or B than call spread F.

Now let's consider the constituents of this index. Let's say it's made up of biotech companies that are highly dependent upon a certain upstream compound, pending FDA approval, for their businesses to succeed. If the compound is approved, these companies are going to make tons of money and the value of the index will likely approach $100. If it is not approved, the value of the index will approach $0. Your estimate of the compound's likelihood of approval will bias your estimate of call spread relative value. If you think approval is more likely than expected, you may be able to purchase the $30-100 call spread cheaply since it's OTM. If enough people agree with you, the premium associated with the $30-100 call spread will be driven higher until it reaches some equilibrium level. This reflects the binary nature of the approval process and the highly correlated expected returns of the index constituents.

The example would be much different if the index was made up of a well-diversified group of companies, spanning different sectors, etc. Some constituent stocks will go up and some will go down, but one might expect the distribution of potential index values to approach something more bell-curved than the binary outcome described in the biotech example. In this case, the value of the $30-100 call spread will remain low since the index probably won't generate those higher expected returns (again, relative to the biotech example).

Now stop. Replace the "$" signs in the example above with "%", generalize the "biotech vs. diversified" discussion in your head to correlated vs. uncorrelated, and substitute "expected loss" for "expected return." You officially understand standardized synthetic tranches. Tranches on the standard CDX index work in exactly the same manner. The expected loss of the index is tranched into 0-3%, 3-7%, etc., slices. If the index is implying a loss of 1.5%, for example, the 0-3% tranche is the ATM tranche. The intuition regarding the greeks, discussed in previous posts, follows naturally (delta, gamma, rolldown/theta, vega/correl01).

One common stumbling block is the whole expected return vs. expected loss business.  To be explicit, credit guys are primarily concerned with expected loss (default risk) whereas equity guys are focused on expected return. If I buy protection on the 0-3% tranche, I expect default risk to increase. When I buy the $0-3 call spread, I expect the stock price to increase. So remember, when you talk about CDS, you should talk explicitly in terms of buying and selling protection.

  • Buy protection = I expect things to get crappier (I want to short the credit)
  • Buy call option = I expect things to improve (I want to get long the stock)

So from a directional perspective (crappier <–> better), I suppose buying tranche protection is more like buying a put spread on a stock/index. For whatever reason, though, I prefer to think of it as buying a call spread on expected loss. This preference is driven by the quoting conventions of credit vs. stocks. CDS is quoted in spread (which reflects default risk) while stocks are quoted in terms of price.

The same term structure considerations are also applicable, though one should remember CDS maturities (e.g. 5, 7, 10y) are much longer than equity options.
 
Anyway, there are direct lines one can draw between stock options and standard synthetic tranches. Hopefully this helps bridge the gap.

And for something totally unrelated, here's a link to an oldie but goodie:

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