The surprise element: jumps in interest rates

The surprise element: jumps in interest rates Sanjiv R. Das Journal of Econometrics 106 (2002) 27–65 Presented by Yuzhen Ding

2009/03/11

The surprise element: jumps in interest rates

Framework

Introduction Data Description Model Specification Estimation Some Applications Concluding comments

The surprise element: jumps in interest rates Introduction

Introduction

This research examines the role of jump-enhanced stochastic processes in modeling the Fed Funds rate. The paper offers three distinct sets of contributions: (1) develop an analytical modeling framework for jumps in fixed-income markets. (2) establish that modeling surprises with jump based models provides a better statistical characterization of the short interest rate than is possible with complex Gaussian models. (3) present a range of applications of the model.

The surprise element: jumps in interest rates Data Description

Data Description

We use the Fed Funds Rate during the period January 1988–December 1997. The data is daily and is obtained from the Federal Reserve. The total number of observations is 2609. From a statistical point of view, there are three features of the short rate process that consistently exist in bond markets. (1) Higher moment behavior: changes in interest rates demonstrate considerable skewness and kurtosis. (2) Volatility behavior. (3) Autocorrelation and mean reversion.

The surprise element: jumps in interest rates Data Description

This table presents summary statistics for the short rate of interest.

The surprise element: jumps in interest rates Data Description

The surprise element: jumps in interest rates Model Specification

Model Specification The following is the mean-reverting process for interest rates employed in this paper: dr = k (θ − r) dt + vdz + Jdπ (h) θ is the central tendency parameter for the interest rate r,which reverts at rate k. The variance coefficient of the diffusion is v 2 and the arrival of jumps is governed by a Poisson process π with arrival frequency parameter h, which denotes the number of jumps per year. The jump size J may be a constant or drawn from a probability distribution. The diffusion and Poisson processes are independent of each other, and independent of J as well.

The surprise element: jumps in interest rates Estimation

Estimation

In this section,we use a simple discrete-time approach. This allows us to estimate a model where the jumps are normally distributed. Since the limit of the Bernoulli process is governed by a Poisson distribution, we can approximate the likelihood function for the Poisson–Gaussian model using a Bernoulli mixture of the normal. In discrete time, we express the process as follows:
∆r = k (θ − r) ∆t + v∆z + J µ, γ 2 ∆π (q)

The surprise element: jumps in interest rates Estimation

Estimation The transition probabilities for the interest rate following a Poisson–Gaussian process are written as (for s > t): f [r (s) |r (t)] = ) ( − [r (s) − r (t) − k (θ − r (t)) ∆t − µ]2
q exp 2 vt2 ∆t + γ 2 1 ×q
2π vt2 ∆t + γ 2 ( ) 1 − [r (s) − r (t) − k (θ − r (t)) ∆t]2 p + (1 − q) exp 2 2vt ∆t 2πvt2 ∆t

The surprise element: jumps in interest rates Estimation

Estimation This approximates the true Poisson–Gaussian density with a mixture of normal distributions. Estimation involves maximizing the function L, where: L=

T Y

f [r(t + ∆t)|r(t)]

i=1

The above equation can be written as: max

Ω=[k,θ,v,µ,γ 2 ,q]

T X i=1

log {f [r (t + ∆t) |r (t)]}

The surprise element: jumps in interest rates Estimation

Estimation In order to compare different processes for the short rate, we estimated four nested models on the data set. The models estimated are (i) a pure-Gaussian model (h = 0) (ii) the Poisson–Gaussian model (iii) an ARCH–Poisson–Gaussian model, which consists of the Poisson–Gaussian model with the variance of the Gaussian component following an ARCH(1) process: 2 vt+∆t = a0 + a1 [∆rt − E(∆rt |rt−∆t )]2

(iv) a pure ARCH–Gaussian model.

The surprise element: jumps in interest rates Estimation

Estimation There is a sharp drop in Gaussian volatility !!

The surprise element: jumps in interest rates Estimation

Estimation

This figure offers visual evidence for the better fit of the ARCH-jump model versus other specifications

The surprise element: jumps in interest rates Estimation

Estimation

We extend the basic Poisson-Gaussian model by allowing the jump-mean parameter to vary, now the specification is as follows:
∆r = k (θ − r) ∆t + v∆z + J µ, γ 2 ∆π (q) 2 vt+∆t = a0 + a1 [∆rt − E(∆rt |rt−∆t )]2

µt = α0 + α1 (θ − rt )

The surprise element: jumps in interest rates Estimation

Estimation

Not significant !!!

The surprise element: jumps in interest rates Some Applications

Some Applications

In this section we shall employ the model to examine various phenomena in the bond markets via the lens of the model.

The surprise element: jumps in interest rates Some Applications

Some Applications Day of the week effects © In this section we examine whether jumps are more likely to occur on specific days of the week, by introducing a modification to make the arrival intensity of jumps depend on the day of the week. © By using dummy variables,our specification is: q t = λ 0 + λ1 d 1 + λ 2 d 2 + λ3 d 3 + λ 4 d 4 λ0 is the arrival probability of a jump if the day is Friday. And λ1 to λ4 denote Monday to Thursday respectively.

The surprise element: jumps in interest rates Some Applications

Some Applications The table presents results of the estimation of the jump-diffusion model .

The surprise element: jumps in interest rates Some Applications

Some Applications Federal reserve activity © The FOMC (Federal Open Market Committee )meets eight times each year. There are two types of meetings of the FOMC: one-day meetings and two-day meetings. There are usually 2 two-day meetings and 6 one-day meetings every year. © We want to find whether the probability of a jump is linked in any way to the FOMC meetings. © The Fed undertakes more distinct activity at the two-day meetings than at one-day meetings. So the two-day meetings appear to have a greater information impact than one-day meetings.

The surprise element: jumps in interest rates Some Applications

Some Applications

Our specification is: qt = λ0 + λ1day f1t + λ2day f2t 2 vt+∆t = a0 + a1 ε2t + a1day f1t + a2day f2t

The surprise element: jumps in interest rates Some Applications

Some Applications

The surprise element: jumps in interest rates Some Applications

Some Applications

Overreaction in the bond markets © Examining the time series of the Fed funds rate shows that the market often overreacts, i.e. large moves in the interest rate are followed by speedy reversals. © The existence of overreaction would mean that the direction of the interest rate would be predictable after large moves. © To test this, we modify the jump model by making the jump intensity a function of the product of the current and prior change in interest rates.

The surprise element: jumps in interest rates Some Applications

Some Applications

Our specification is: qt = q0 + q1 max [0, (rt − rt−∆ ) (rt−∆ − rt−2∆ )] +q2 min [0, (rt − rt−∆ ) (rt−∆ − rt−2∆ )] = q0 + q1 Rt+ + q2 Rt− where Rt+ = max [0, (rt − rt−∆ ) (rt−∆ − rt−2∆ )] Rt− = min [0, (rt − rt−∆ ) (rt−∆ − rt−2∆ )] capture the asymmetry and magnitude of continuations and reversals in the data.

The surprise element: jumps in interest rates Some Applications

Some Applications

The surprise element: jumps in interest rates Some Applications

Some Applications

The coefficients q1 , q2 are both significant indicating that they impact jump intensity strongly. The average values of R+ and R− are 0.0145 and −0.039, respectively. The average jump intensity q in this model is q = q0 + q1 Rt+ + q2 Rt− = 0.3 Thus reversals account for approximately 75% of the jump intensity. Thus, there is strong evidence in favor of market overreaction. This also indicates that only 25% of the jumps actually persist.

The surprise element: jumps in interest rates Concluding comments

Concluding comments

This paper explores surprise elements in the fixed-income markets using Fed Funds data. We conclude that jumps are an essential component of interest rate models. Several examples of questions that may be explored using jump processes are provided in this paper.

The surprise element: jumps in interest rates Concluding comments