Elastic String Models of Forward Rates and Psychological Time
The n-month forward rate is the rate at which one can borrow money in the future, the loan starting in n-months for the three subsequent months. The forward rate *curve* (FRC) is the collection of all such rates. It moves all the time, due to supply and demand. However, its one dimensional, bead-on-string nature enforces a special kind of correlation between different maturities.
Twenty-five years ago, several authors proposed to model the FRC as an *elastic string* along which idiosyncratic shocks propagate.
In particular, Belal Ehsan Baaquie and myself (BB) proposed in 2004 a “stiff” elastic string field theory that accounted remarkably well for the full correlation matrix of the daily FRC returns during the period 1994-1996. Unfortunately, our paper (which I was very proud of at the time) fell into almost complete oblivion…
Fortunately, Victor Le Coz (a PhD candidate at the #Econophysics X-CFM Chair) decided to revisit this model with new ideas and new data. We reformulated the BB model in a way that makes its micro-foundation more transparent: our model can be interpreted as capturing the effect of market forces that set the rates of nearby tenors in a self-referential fashion. The model is parsimonious and with only two parameters we accurately reproduce the whole correlation surface of the FRC (represented below) over the time period 1994-2023, with a very small error around 1.5%. The parameters are very stable over time except perhaps during the Quantitative Easing period 2009-2014. Furthermore, the dependence of correlation on time resolution (also called the Epps effect) became faithfully reproduced within the model. This allows us to estimate a cross-tenor information propagation time of ca. 10 minutes.
Finally, and perhaps most interestingly, we confirm that the time perceived by interest rate market participants is a strongly sub-linear, logarithmic function of real time, as surmised by BB 2004.
In fact, our new results are fully compatible with “hyperbolic discounting”, in line with the recent behavioural literature. Arguably, interest rate markets offer the best possible laboratory to test such ideas. From our calibration to the data, we find that the cross-over time between normal time flow and logarithmic time flow occurs around 2 months in the future. This also means that a year ten years from now is perceived by the bond markets as one week in real time! This is quite an extreme distortion of future time that reflects the extremely myopic nature of financial markets.
Congratulations Victor! The model is elegant, math is beautiful and results are very compelling. For more details, see the paper here : Revisiting Elastic String Models of Forward Interest Rates by Victor Le Coz, Jean-Philippe Bouchaud :: SSRN
Written by Jean-Philippe Bouchaud (Top 10 Quants On Wall Street 2023)
