Problem: CONVEXITY OF LONG BONDS, SWAPS, AND ARBITRAGE

The yield of a long bond tells you how much you can earn from this bond. Correct? Wrong. You can earn more. The reason is that long bonds and swaps have convexity. If there are two instruments, one linear and the other nonlinear, and if these are a function of the same risk factors, we can form a portfolio that is delta-neutral and that guarantees some positive return. This is a complex and confusing notion and the purpose of this case study is to clarify this notion a bit. At first, the case seems simple. Take a look at the following single reading provided on an arbitrage position taken by market professionals and answer the questions that follow. The more sophisticated traders in the swaps market-or at least those who have been willing to work alongside their in-house quants-have until recently been playing a game of one-upmanship to the detriment of their more naive interbank counterparties. By taking into account the convexity effect on long-dated swaps, they have been able to profit from the ignorance of their counterparties who saw no reason to change their own valuation methods. More specifically, several months ago several leading Wall Street US dollar swaps houses-reportedly JP Morgan and Goldman Sachs among them-realized that there was more value than met the eye when pricing LIBOR-in-arrears swaps.

According to London traders, they began to arbitrage the difference between their own valuation models and those of "swap traders who still relied on naive, traditional methods" and transacting deals where they would receive LIBOR in arrears and pay LIBOR at the start of the period, typically for notional amounts of US$100m and over. Depending on the length of the swap and the LIBOR reset intervals, they realized that they could extract up to an additional 810 bp from the transaction, irrespective of the shape of the yield curve. The counterparty, on the other hand, would see money "seep away over the life of the swap, even if it thought it was fully hedged," said a trader. The added value is only significant on long-dated swaps-typically between five and 10 years-and in particular those based on 12-month LIBOR rather than the more traditional six-month LIBOR basis. This value is due to the convexity effect more commonly associated with the relationship between yields and the price of fixed income instruments. It therefore pays to be long convexity, and when applied to LIBOR-in-arrears structures proved to be profitable earlier this year. The first deals were transacted in New York and were restricted to the US dollar market, but in early May several other players were alerted to what was going on in the market and decided to apply the same concept in London.

One trader expressed surprise at the lack of communication between dealers at different banks, a fact which allowed the arbitrage to continue both between banks directly and through swaps brokers. Also, "none of the US banks active in the market was involved in trying to exploit the same opportunities in other currencies," he said, adding "you could play the same game in sterling-convexity applies to all currencies." In fact, there was one day in May when the sterling market was flooded with these transactions, and it "lasted for several days" according to a sterling swaps dealer, "until everyone moved their prices out," effectively putting a damper on further opportunities as well as making it difficult to unwind positions. Further, successful structures depend on cap volatility as the extra value is captured by selling caps against the LIBOR-in-arrears being received, in addition to delta hedging the swap. In this way value can be extracted from yield curves irrespective of the slope. "In some cap markets such as the yen, volatility isn't high enough to make the deal work," said one dealer. Most of the recent interbank activity has taken place in US dollars, sterling, and Australian dollars. As banks have become aware of the arbitrage, opportunities have become rarer, at least in the interbank market. But as one dealer remarked, "the reason this [structure] works is because swap traders think they know how to value LIBOR-in-arrears swaps in the old way, and they stick to those methods." "Paying LIBOR in arrears without taking the convexity effect into account," he added, "is like selling an option for free, but opportunities will still exist where traders stick to the old pricing method." Many large swap players last week declined to comment, suggesting that the market is still alive, although BZW in London, which has been active in the market, did say that it saw such opportunities as a chance to pass on added value to its own customers. (Thomson Reuters IFR issue 1092, July 29, 1995)