Phenomenology of the interest rate curve

Abstract

This paper contains a phenomenological description of the whole U.S. forward rate curve (frc), based on an data in the period 1990-1996. We find that the average frc (measured from the spot rate) grows as the square-root of the maturity, with a prefactor which is comparable to the spot rate volatility. This suggests that forward rate market prices include a risk premium, comparable to the probable changes of the spot rate between now and maturity, which can 1 be understood as a 'Value-at-Risk' type of pricing. The instantaneous frc however departs form a simple square-root law. The distortion is maximum around one year, and reflects the market anticipation of a local trend on the spot rate. This anticipated trend is shown to be calibrated on the past behaviour of the spot itself. We show that this is consistent with the volatility 'hump' around one year found by several authors (and which we confirm). Finally, the number of independent components needed to interpret most of the frc fluctuations is found to be small. We rationalize this by showing that the dynamical evolution of the frc contains a stabilizing second derivative (line tension) term, which tends to suppress short scale distortions of the frc. This shape dependent term could lead, in principle, to arbitrage. However, this arbitrage cannot be implemented in practice because of transaction costs. We suggest that the presence of transaction costs (or other market 'imperfections') is crucial for model building, for a much wider class of models becomes eligible to represent reality.

Back to basics: historical option pricing revisited

“…Furthermore, the fact that the risk-neutral and historical probabilities need not be the same is often an excuse for not worrying when the parameters of a specific model deduced from derivative markets are very different from historical ones. This is particularly obvious in the case of interest rates [19]. In our mind, this rather reflects that an important effect has been left out of the models, which in the case of interest rates is a risk premium effect [19].…”

“…This is particularly obvious in the case of interest rates [19]. In our mind, this rather reflects that an important effect has been left out of the models, which in the case of interest rates is a risk premium effect [19]. We believe that a more versatile (although less elegant from a mathematical point of view) theory of derivative pricing, such as the one discussed above, allows one to use in a consistent and fruitful way the empirical data on the underlying asset to price, hedge, and control the risk the corresponding derivative security.…”

Quantum field theory of forward rates with stochastic volatility

“…(3) is not ruled out by no arbitrage [14], and an empirical study [11] provides strong evidence for this term in the evolution of the forward rates.…”

“…al. [14] analyzed the future contracts for the forward rates. Both references concluded that many features of the market, and in particular the (stochastic) volatility of forward rate curve, could not be fully explained in the HJM-framework.…”

Ab initio yield curve dynamics

“…The consistency of the general shape of the eigenmodes derived from empirical yield curve data and the explanatory power of the truncated expansions in those eigenmodes is surprisingly robust over time and largely independent of the country in which the interest rates are set [19,20,21]. While this analysis motivated the use of yield-curve modes by fixed-income strategists and risk managers some time ago, an explicit link between yield-curve modes and dynamics appeared in comparatively recently research demonstrating the eigenstructure to be consistent with both the existence of a line tension along the yield curve and a diffusion-like equation of motion for yield curves [22]. This notion of a line tension along the yield curve has found further expression in descriptions of the yield curve as a vibrating string [23].…”