Convertible Arbitrage and Delta Hedging in 1931

Warning: Long post, worth reading if you are really into convertible bonds.

Somewhere before 2000 I believe I read in a footnote of an equity derivatives research piece written by Alexander Ineichen that there existed a book which explained convertible arbitrage published in 1931 (I have found since that Sheen Kassouf had also referred to it in “Evaluation of Convertible securities” first published in 1962). I was quite interested but had no way of finding it so left it at that. I have seen the book quoted a few times since but no review of what is actually in the book, so I finally procured a copy a couple of years back.

Thanks to a recent forced horizontal break for a couple of days I actually read it, and even though I knew what I was likely to find, I was still amazed.

The book is

Meyer H Weinstein, Arbitrage in Securities, Harper and Brothers, 1931.

No, none of this is new

Some of you will rightly say: “All right but options have been in existence for more than a 100 years, and Bachelier’s thesis showing a correct pricing formula for options dates back to 1901, so nothing really new here no?”. That’s true and If you’d like to read more about that I encourage you to read Scott Mixon’s Research which shows that Option Implied Volatility behaved in a similar way to today in the 19th century (Yes you read that well).

My answer, though, would simply be that convertible bonds are a different animal. For starters they are a corporate bond plus an option to exchange that corporate bond for a stock, so it’s an option with a changing strike. Secondly, almost all traded options, and this was true also in the 19th century, had much shorter maturities than convertible bonds. A long maturity means you rarely own the option for its whole life and you can rarely find out thus if your hedging policy was more or less expensive that the option premium you paid. Finally these convertible bonds were already exotic options: some seemed to have, for example, a conversion ratio that came down as the number of bonds outstanding decreased, so you weren’t even sure what ratio applied when you sent bonds for conversion.

So what’s in the book?

Most of the book is devoted to explaining in great details how to execute all of the standard arbitrages: arbitraging between places of listing, arbitrage of rights issues, and even some mentions of what happens in company refinancing with debt exchanges. Each situation is described either with a simple formula when feasible or at minimum a mock trading scenario showing what the correct implementation is.

Weinstein introduces what we now call "Equity Derivatives" as "Equivalent Securities" and his method with regards to these essentially consists in doing the necessary calculations to summarize the behaviour of each type of security into a relationship table table between the stock and the security. I can still remember a time not long ago when live P&L and greeks in your bank's front office software still relied on on a bunch of tables for the prices and greeks of that had been pre computed at close of business the day before and stored.

While the details can be boring you have to admire how methodical Meyer Weinstein is. His world is not one where you play a game a million times with a tiny edge, but one where you read the prospectuses of securities and you try to understand what will happen. No artificial intelligence involved. It contains a myriad of good comments about what can go wrong.

I don’t want to say the book should be required reading, but my view is that there are more people who know about the Black & Scholes formula than there are who understand what happens with simple corporate actions like dividends, rights issues etc, and it shouldn’t be the case.

Convertible arbitrage in the 1930s

Now the most interesting thing for me: His definition of how to arbitrage convertible bonds. I take the liberty of reproducing two pages from the book so you can see for yourself. I believe that because I am reviewing the book and this is in the context of a historical comparison this reproduction is covered by “fair use”:

A few remarks:

• In the general case his initial position is on a 50% delta, and expects to make a small profit either way: he has basically explained delta neutrality in plain English. On that second page he explains that if the convertible is very close to the bond floor you may want to start with a 25% delta, so there’s no doubt he understands the delta is related to the conversion probability.
• “by taking advantage of trading turns gradually reduces the premium and when conversion actually takes place he will have sold short the remainder of the stock” I think Weinstein explains here that if you gamma hedge properly up to the end of the life of the option, your gain will be of the same order as the premium you paid for the option. Maybe you will say I’m reading too much here, but I do think this is an amazing insight for 1931. Having said that his book is about arbitrage, not quasi-hedging, most of his trades are classic “almost risk-free” arbitrage trades, so I would expect him to view all trades in the context of the cost of replicating an instrument.
• There is no explicit mention of credit or default as a risk to the trader, but an acknowledgement that "sometimes, ...,the convertible may fall as fast as the stock into which it is convertible". My impression overall from the book is that while not saying so explicitly the author considers debt almost as an equity derivative.

A couple of things that will still ring a bell today

Some remarks will make you smile at how close the arbitrageur's preoccupation are still with us. Take this sentence on stock borrow: "If the loaning rates are 'flat', in the absence of a special situation, the arbitrageur is inclined to the belief that the security is scarce and that a rather large short interest may be harbored in it.". In this case 'flat' means that the lender of the stock will NOT pay interest for the margin initially left with him by the borrower, so the fee paid by the borrower is basically the interest rate.

I am sure quite a few of convertible trading readers remember all the timing issues between conversions made before or after interest payments and dividend ex-dates. Well that's a very very old issue: here's the 1931 warning telling you to be careful:

A more general view of arbitrage

My last point is that his view of dynamic hedging which I find very prescient in convertible (there are probably earlier references in the options world) is great, but only takes a few pages of the book, and I find that very good. Why? Because as Weinstein himself explains you can get the hedge size wrong. So a lot of time is spent on less risky arbitrage trades, which makes sense.

If you dig into his tables you will see that it is only in convertible arbitrage trades that he shows some losses. The average return looks higher than in other instruments, but the median is probably not that far and there are a couple of position with large losses in terms or return on capital invested. This a nice implied early warning that option replication via delta hedging is not risk free and that probably if you have actual risk-free trades you should start there.

Conclusion

I am not going to talk about what could have been in the book, but in conclusion I think it does two things really well, in practice better than most recent math packed books:

• It explains in plain English how replicating corporate instruments of all kinds is ACTUALLY done, and most of his remarks about settlement issues etc are still true today. You still cannot find these remarks in most textbooks today, by the way.
• It shows results of a good number of actual trades and what happened with good or bad results. Again don’t think I have seen that anywhere else. I myself am at pains to remember what positions the desk I worked at had in this big or small situation. I don’t think much of that real life stuff is taught, or at least not to that extent.

Finally I like python, I am very interested in statistics, maths and models, I have programmed convertible pricing models in VBA and C a few years ago, but I still think understanding corporate instruments and how to replicate them is a better starting point to trading than derivatives maths. This is just what that book does, and I don’t know any modern equivalent to it, but feel free to mention some in your comments.

Final "Finally": The book should be in the public domain in about 5 years time by my count if you are patient and do not want to buy it from a collector. There haven't been any reprints that I am aware of since 1931 unfortunately.

References:

Weinstein, Meyer, H. (1931) Arbitrage in Securities, Harper and Brothers.

Ineichen, Alexander, In Search of Alpha - Investing in Hedge Funds, UBS Warburg, October 2000

Kassouf, Sheen T. (1962) Evaluation of Convertible Securities (New York: Analytic Investors)

Mixon, Scott, Option Markets and Implied Volatility: Past Versus Present (August 1, 2008). Journal of Financial Economics (JFE), Forthcoming, Available at SSRN: https://ssrn.com/abstract=889543