Math that may Explain the Skew in the Interest Rate Markets
September 4, 2013
Interest Rates - or, short rates - have a very complex dynamics that is captured by equally complex mathematics and there is a mind numbing array of math models covering a very wide spectrum from the Vasicek / CIR equilibrium models to non-Markovian Heath Jarrow Morton (HJM) model to the more recent Libor Market Model (LMM) class of models. Any good and robust mathematical model should do two things explicity: (1) be able to calibrate the model with the observed market prices of options and (2) be able to explain the volatility skew observed in the market. If a model cannot equip a trader to better hedge her positions then that model is practically useless.
One of the key properties of the interest rates is that when there are external shocks the underlying (the short rate) does not react in a way that is fully proportional to the level at which the underlying is trading. What do we mean by this?
For example, if a stock is trading at $100 and the market crashes then the stock will fall greatly, maybe by 20% which may take the stock to $80; the same stock trading at 10 will also fall when the market crashes but the fall will be such that it in absolute terms this fall will much less than $20 even though the percentage fall could be 20%. This is elementary mathematics - given the fact that percentage decline and the decline in the absolute value of the asset are two different things - and this elementary mathematics is the core of a lognormal random walk for equities. Not so for interest rates, though. Interest rates - which are themselves expressed in basis points or percentages - tend to move in absolute values; but then again not always. Thus, it is unclear as to whether short rates have a normal dynamics ("normal" as in Gaussian) or lognormal dynamics ("lognormal" as in natural log of the asset price following a Gaussian distribution). It seems, and that is borne out by empirical research and observation that short rates have dynamics that is half way between normal and lognormal. The assumptions that the quants make while modeling interest rates is that the short rate skew is such that the underlying reacts to shocks - or, extreme market movements - in way that is in between a proportional reaction (percentage moves) and an absolute reaction.
One of the stochastic models that can explain such behavior interest rates is the shifted lognormal model or the displaced diffusion model first proposed by Rubinstein in 1893. This is perhaps the simplest possible local volatility model which can be used for real hedging purposes. A shifted lognormal model has the following dynamics
Where, the underlying has
been shifted by a constant amount. The
good thing about this model is that it allows a closed form solution for
valuation of options and hence working with the model is easy. The other good
thing is - as given by Marris (1999) - is that it is quite easy to calibrate
the model to ATM options. However, a remarkable property of this model is that
the above mathematical equation can be derived from the dynamics of both a pure
lognormal process (geometric Brownian motion) and a normal process.
For more on the derivation of the above equation from
lognormal and normal processes and calibration of the model to ATM options see
CFE Lecture notes.
Massimo Morini (2011).
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