Editor’s Letter

It is the morning after the U.S. election and the American people have spoken loud and clear! Well … LOUD, anyway—maybe not so clear. We find ourselves today in a place of great economic confusion and uncertainty. Confusion because we are facing economic conditions worse than any seen in the U.S. since the 1930s—persistent high unemployment and slow growth despite interest rates so low that one can only call it a liquidity trap—not to mention the ongoing collapse of the housing market and the financial infrastructure connected to it. We are way off the track, and our standard tools to get back on it—expansionary monetary policy and fiscal deficits—seem disturbingly ineffective. Economists are confused and in sharp disagreement with one another about what to do, especially, internationally.

This makes for tremendous uncertainty about the future. Actually, the term “uncertainty,” while I believe it to be quite accurate, sounds a little strange as a description of the tenor of the times. The country is more polarized today than it has been in a long time, and each side feels no uncertainty at all that its view of the world is right and the best way forward is obvious. They only appear to agree on two things: that the current situation is intolerable, and that the ideas of the people on the other side are completely and dangerously wrong.

For the last two years, one political party strained to combat the worst economic downturn since the 1930s with Keynesian remedies and, at the same time, to make major changes in the way large sectors of the economy operate, including health care, the financial system, energy production and consumption, and more, while the other party tried to block them in every way. Long-term planning for businesses and investors became very difficult because the future economic environment was so unsettled. The next two years will be different. Now the two parties will both be pulling as hard as they can, only in opposite directions. It is hard to think that this will reduce the level of uncertainty.

We will cross our fingers and hope for the best. In the meantime, let us turn our attention to more positive subjects, such as this issue of The Journal of Derivatives.

Each of our first three articles offers a new approach to modeling and pricing interest-dependent derivatives. Daglish leads off with a new way to build trinomial interest rate trees. Black, Scholes, and Merton introduced the finance profession to the mathematics of a continuous-time and continuous-state world and derived fundamental partial differential equations (PDEs) for contingent claims pricing. But only in the simplest cases do such PDEs yield closed-form solutions. Numerical solution techniques discretize the time and state dimensions either by finite difference approximations to the PDE or by building binomial or trinomial trees for the evolution of the underlying price or interest rate. The standard approach is to start at expiration and work backward through the lattice to the initial date and price. Daglish shows that a trinomial tree based on solving the PDE in the forward direction is feasible, just as accurate for a given size of time step, and considerably more efficient in terms of computation speed than current methods.

Next is an article by Wu and Chen that bridges a gap in modeling the dynamics of interest rates. In standard term structure theory, the yield on a long maturity bond is derived from the current short rate and its expected future evolution. Twenty-five years ago, Ho and Lee pointed out that interest rate models need to be built so that the assumed dynamics does not produce mispricing and internal arbitrage opportunities within the model. This principle applied in continuous time led to the Heath–Jarrow–Morton (HJM) model, a brilliant theoretical advance but one that is very challenging to practically implement because the probability distributions for future rates are intractable. For practical purposes, HJM has been supplanted by “market” models, such as the LIBOR Market Model (LMM), in which short-term rates at specific future dates are assumed to be lognormal but explicit modeling of their evolution from one date to the next must be abandoned. The LMM allows much simpler pricing of interest rate contracts using the Black model, but a new problem arises. Some interest-dependent instruments, such as caps and floors, can be priced easily in the LMM because the cash flows can each be valued individually and then added. But the fixed rate in a swap contract blends together the expected rates on all future payment dates, so if those are all lognormal, the swap rate cannot be. The market’s solution was the Swap Market Model in which the future swap rates are lognormal but the short rates on individual payment dates are not. Plainly, the two modeling approaches are mutually incompatible. This is where Wu and Chen come in, with an approach that connects the two modeling frameworks by developing lognormal approximations to the actual distribution of a future swap rate within the LMM.

Our third interest rate article breaks out of the standard approach in a different way by abandoning lognormal diffusion dynamics for the underlying interest rates. Observing bond price changes at high frequency reveals that the price changes do not move with the continuous smooth dynamics of a diffusion but rather tend to be more sporadic and jumpier. The standard lognormal diffusion is a special case in a broader class known as Lévy processes in which other members allow for jumps and also diffusive movement with time-varying volatility. Hainaut and MacGilchrist show how to embed a more general Lévy process, known as the normal inverse Gaussian (NIG), in an interest rate tree. Pentanomial branching is required to match the additional flexibility of the NIG, but the result consistently beats the lognormal diffusion in simulations.

The next article considerably changes gears and looks at the market for derivative contracts tied to electricity. A key feature of this market is that electricity needs to be used immediately, as soon as it is generated, but “shipping” it from the producer to the consumer can only be done through the grid. Network transmission rights must be acquired in advance, which significantly affects arbitrage across regional markets and imparts optionality to their values. Bunn and Martoccia explore how this works in two Northern European electricity markets.

Finally, we end with an article on foreign currency options. The contracts are very “plain vanilla,” but the market practice in quoting prices is markedly different from what is familiar to traders in other option markets, to the extent that it is easy to become confused about the actual terms of a contract. Reiswich and Wystup take us through the intricacies of the quotation conventions for FX options.

One final thought about the U.S. economy going forward: Professional option traders and market makers say that high volatility is much better for them than calm markets. In other words, if you are an options professional, maybe a couple of years of uncertainty will suit you just fine. The rest of us will be hanging onto our hats.

Stephen Figlewski

Editor

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