Volatility Results

A GARCH BEKK (1,1) model is constructed to model the spot and future rate volatil- ity and assess if there is any spillover between the two series. The GARCH model coefficients for the 5 sub periods are given in table 4.8. There are two equations for

each sub period representing the spot rate and future rate conditional variance. h11

represents the conditional variance for the future rate and h22 the conditional variance

for the spot rate, h12 is representative of the covariance between the future and spot

rates. 2₁and 2₂represent the shocks or information entering the future and spot market

respectively. The cross product of the error terms 1, 2 represents the shocks in both

4.4 Results

In the first time period 2001-2003, the future rate volatility is affected directly by both shocks in the futures market and in the spot market. The spot rate volatility is only affected by indirect effects, the covariance and to a lesser extent the cross error term.

From 2003-2006 all coefficients in the GARCH equation for the future conditional vari- ance are significant. Direct shocks from the future and spot market cause an increase in volatility in the futures market. The futures market is also indirectly affected by the spot market through the cross error term. The volatility is also caused by the lagged future conditional variance and lagged spot conditional variance. Spot volatility is sig- nificantly affected by its own lagged conditional variance and indirectly by the lagged conditional covariance.

During the third period from 2006-2008 the volatility in the future rate is significantly affected by shocks in the future market and cross error terms causing an increase in the volatility. Direct shocks in the spot price, however, significantly lower the volatility in the futures markets. This likely demonstrates that the direct news reduces uncertainty in the market and so decreases the future volatility. So the changes in the future rate volatility are affected both directly and indirectly by shocks in the spot market. In addition to shocks in the spot market the volatility of the spot rate is indirectly caused by the futures market as seen by the significant cross error terms and the covariance term which accounts for the majority of the volatility.

During 2008-2011 volatility spills over from the spot market into the futures markets both directly and indirectly as evidenced by the significant coefficients on the shocks in the spot market and the error product terms. The majority of the volatility in the future market is caused by news entering its own market and also from its own lagged volatility. For spot market volatility none of the coefficients except for the constant are significant. Showing that the volatility during this period is not driven by either the shocks in the spot or future market.

All GARCH coefficients in the future conditional variance equation in the final period 2011-2013 are again significant. Showing that there is spillover from the spot market again. The volatility, however, is mainly affected by shocks in its own market and by its own lagged volatility. The volatility in the spot market is in largely driven by its own lagged volatility, lagged future volatility and the lagged covariance. This is the first period where there is a direct spillover from the futures market to the spot as evidenced by the significant coefficient on the lagged future conditional variance. In summary, with exception to the 2008-2011 period, spot rate volatility is largely

driven by the covariance term h12which is evident from the highly significant covariance

coefficient. Insignificant 2₁terms show that there are no direct volatility spillovers from

the futures market into the spot market. In the final period, 2011-2013 the spot rate volatility is also driven by its own lagged volatility and the lagged future volatility. Future rate volatility is driven by shocks in both the future and spot market, this is

evident from the significant 2₁ and 2₂ coefficients. In addition to information shocks,

both lagged future and spot conditional variance are significant in explaining the change in the change in the future volatility during the three periods, 2003-2006, 2008-2011 and 2011-2013.

Figure 4.3 presents the conditional correlations between the Euribor future and the Euribor Spot rate. These graphs demonstrate the level of integration and conditional correlation between the two markets. The average conditional correlations for each sub period range from a minimum of 0.049 during the 2006-2008 period to a maximum of 0.238 during the 2011-2013 period.

As expected the conditional correlation between the future and deposit rate is generally positive with a positive mean conditional correlation for all five sub periods. There are several periods of negative conditional correlation this is evident mainly during the period 2006-2008 where the markets became less integrated. Generally the trend is

4.4 Results

Table 4.8: GARCH BEKK Model

c ε2_1,t−1 ε2_2,t−1 ε1,t−1ε2,t−1 h11,t−1 h22,t−1 h12,t−1 01/01/2001-31/12/2013 h11,t 0.073*** 3.525*** 0.099* 0.001 0.000 0.001 0.001 h22,t 0.066 0.015 -0.037 0.023** 0.010 0.195 0.941** 31/07/2003-24/04/2006 h11,t 0.012*** 0.824*** -0.239*** 0.017*** 0.044** 0.008*** 0.000*** h22,t 0.001 0.381 0.022 0.000 0.314 -1.129*** 1.014*** 25/04/2006-02/10/2008 h11,t 0.013*** 0.139*** -0.165*** 0.049*** 0.000 -0.001 0.001 h22,t 0.000 0.076 0.127** 0.053** 0.001 0.059 0.950*** 03/10/2008-04/08/2011 h11,t 0.000 0.461*** 0.080*** 0.003*** 0.360*** -0.056*** 0.002*** h22,t 11.604*** 0.628 -0.121 0.006 0.008 -0.040 0.050 05/08/2011-31/12/2013 h11,t 0.037*** 0.182*** -0.099*** 0.013*** 0.295*** 0.010*** 0.000* h22,t 0.627*** 0.030 -0.001 0.000 0.574*** 1.463*** 0.933***

Note: GARCH BEKK Model coefficients and significance. Where h11,t represents future

market conditional variance and h22,trepresents the spot market conditional variance. ***,

**, * represent significant coefficients at 1%, 5% and 10% levels respectively.

that of decreasing conditional correlation from 2001- 2008, after 2008 the conditional correlation has increased showing that the two markets have begun to move together again.

4.5

Conclusion

This chapter investigates the dynamic relationship between the 3 month Euribor spot deposit rate and the 3 month Euribor future rate over the 13 year sample period from 2001-2013. Using econometric models the relationship between both the rate and the rate volatility of the spot and future market is analysed. Additionally, structural break tests are applied to assess the stability of this relationship.

The structural break tests results show the instability of the relationship between the spot and future rate over the 13 year period. Multiple structural breaks are identified and divide the sample into 5 separate subperiods. This changing relationship is also reflected in the tests for price discovery and volatility linkages across the two markets. Only 2 of the 5 subperiods are found have a cointegrating relationship. This is surprising as the nature of the future and spot relationship would suggest a strong mean reverting basis to be present in all subperiods throughout the sample. Cointegrating relationships appear to hold during periods when both markets are relatively stable and break down in the time leading to and after the financial crisis.

Where cointegration is present, the results suggest that the spot rate generally leads future in the price discovery. However, the future rate does provide some information for the spot rate in the short run and bi-directional Granger causality is shown in 4 of the 5 subperiods.

It is interesting to note that the Eurodollar and the Euribor interest rate markets appear to function differently. Cheung and Fung (1997) suggest that the futures market has a greater impact on the spot market than the spot on the future. Fung and Leung (1993) also find that the futures market plays a greater role in price discovery than our results find in the Euribor market. The reason for this difference may be due to timing differences, there is no mention if the lagged deposit rate includes the rate which is

4.5 Conclusion

announced in the morning when explaining the change in the end of day futures rate. If this is the case it is not the most recent change in the deposit rate that is being used in the equation.

The multivariate GARCH model demonstrates that there is significant volatility spillover between the markets and the conditional correlation shows the changing strength of the volatility co movements between the deposit and future markets throughout the sam- ple. The results show direct volatility spillover from the spot market to the futures market in all five sub periods. Volatility in the deposit market is mainly driven by the lagged covariance term. The conditional correlation shows the co movement of volatility strongest at the beginning and end of the sample with very low and sometimes negative correlations evident during the period 2006-2008 which encompasses the beginning of the global financial crisis.

In summation, these results show that although there is bidirectional flow of information through both the rate and volatility of the two markets, the spot rate would appear to provide greater information for the future rate. Findings show this relationship breaks down leading up to the financial crisis during the period 2006-2008. Over this period results show that the futures contract is no longer is influenced by changes in the spot market. This suggests a lack of confidence in the information provided by the spot market. Coincidently, this period coincides with ’the Euribor fixing scandal’ where four Euribor reporting institutions colluded to manipulate the Euribor rate. As such, the lack of causality from the spot to the future market during this period may reflect a perceived lack of credibility in the information provided by the spot market.

Figure 4.3: Conditional Correlation 01/01/2001-30/07/2003 Average correlation 0.223025 Date Correlation 2001 2002 2003 -0.50 -0.25 0.00 0.25 0.50 0.75 31/07/2003-24/04/2006 Average correlation 0.217330 Date Correlation 2004 2005 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 25/04/2006-02/10/2008 Average correlation 0.048843 Date Correlation 2006 2007 2008 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 03/10/2008-04/08/2011 Average correlation 0.146975 Date Correlation 2009 2010 2011 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 05/08/2011-31/12/2013 Average correlation 0.237679 Date Correlation 2012 2013 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8

Chapter 5

Jumps in Euribor and the effect

of ECB monetary policy

announcements

5.1

Introduction

There is strong empirical evidence of jumps in interest rates. The main tool that the ECB uses to influence monetary policy is through the short term refinancing rate, a change in the short term interest rate can in turn cause the whole yield curve to shift. In addition to central bank announcements, interbank rates such as Euribor are also influenced by various macroeconomic events, liquidity in the money markets and the perceived credit worthiness of financial institutions.

This chapter examines the jump characteristics of the 3 month Euribor futures contract and its corresponding futures option contracts. Firstly, the futures rate is modelled using a jump diffusion model with a Bernoulli jump distribution similar to that of

Ball and Torous (1983). This allows the estimation of jump parameters using historical futures rates and provides an insight of the jump behaviour of Euribor futures contracts. Secondly, option implied parameters are estimated using a jump diffusion process with Poisson distributed jumps in the spirit of the Merton (1976) jump diffusion model. Jumps provide an important information channel for the market. Using jump diffusion models to extract jump parameters inherent in the futures rate allows the identification of jump risk and the market’s expectation of jumps in the rate. Finally, this chapter looks at what effect ECB refinancing rate monetary policy announcements have on the 3 month Euribor and the jump parameters.

As interbank liquidity dried up during June 2008 the Euribor market experienced rate spikes pushing both futures contracts and the underlying spot rate to their highest since the inception of Euribor in 1999. Later in 2008 the market experienced a sudden reversal and the Euribor rate dropped significantly, in one day, the 13th October 2008 the 3 month Euribor near month expiry futures contract experienced a drop of 0.915%. Further out the yield curve other futures contracts also experienced corresponding drops with the second and third month contracts dropping 0.545% and 0.39% respectively.

These sudden changes or jumps in the interest rate result in a number of implications relating to risk, pricing and hedging. A large jump represents a discontinuity in the movement of the rate which is otherwise assumed to be continuous. These disconti- nuities in the rate result in a diversion from the normal distribution and can result in a leptokurtic distribution. Despite this, a lognormal distribution is often assumed to describe the path of the interest rate where the probability of a large change is too small to reflect what is empirically observed in the market. Without acknowledging this jump risk when the market is incomplete traditional hedges will fail to protect a risky position. The fear of jumps in the market can also be observed in the Euribor option implied volatility smile. Empirically, the volatility implied by Euribor options exhibits

5.1 Introduction

a smile shape across strikes with options deep in and out of the money demonstrating higher implied volatility than that at the money. If jumps are present in the market, incorporating a jump component into an option pricing model allows this smile to be captured and in turn options can be successfully priced and hedged more effectively. The jump diffusion model can accurately capture the implied volatility smile or skew that is often found in interest rate markets. Although a stochastic volatility model can to some extent model the implied volatility smile, if the smile or skew is exceptionally steep it may be unable to fully explain the skew.

The Euribor interest rate is the building block of European fixed income markets. Euribor futures and options are commonly used to hedge over the counter interest rate products and thus are two of most highly traded exchange traded products in Europe. Because of this, understanding Euribor future rate behaviour and identifying jumps if present is important for many market participants. Understanding the jump effect that the ECB has on the futures rate can provide important information as to how futures react to changes in monetary policy. Owing to this, this chapter provides an interesting contribution to the empirical literature on jumps in European interest rate markets.

The literature surrounding jumps in asset prices is wide ranging and an array of jump models have been proposed over the last 40 years. The original jump model used for financial asset pricing, a mixed Gaussian and pure jump distribution is first examined by Press (1967). Press proposes a model to describe the movement in stock price changes, noting that empirically security price distributions are generally skewed and leptokurtic. Despite this, stock prices are often assumed to follow a normal stable distribution, and because of this and the non-normal empirical nature of returns, Press proposes a model consisting of a Poisson distribution with Brownian motion. Press tests his model using monthly returns for 10 stocks listed on the Dow Jones index between 1926 and 1960.

In their seminal paper, Black and Scholes (1973) provided the market with the tools to price financial options. Yet the Black Scholes model is based on a normal distribution of asset returns which does not accurately describe the empirical distribution. The need for a distribution that is capable of accounting for the occurrence of jumps is desirable if one is to accurately model price dynamics in the market. To account for this Merton (1976) extends the Black Scholes option pricing model to price options where the underlying asset follows a discontinuous process. Similar to Press (1967), Merton assumes that the change in the underlying asset is described by a Gaussian diffusion process and a pure jump Poisson process.

Merton (1976) proposes that the change in the price of an asset is composed of two distinct parts, normal vibrations in price and abnormal vibrations representing discon- tinuous jumps in price. The jump diffusion model allows for the presence of jumps by incorporating a compound Poisson process with geometric Brownian motion. This results in the price path of the given asset to be predominantly continuous over time, modelled by the diffusion process, but also capable of experiencing occasional disconti- nuities, modelled by the jump component. This original model provides the foundation for many jump diffusion option pricing models in current literature.

Ball and Torous (1983) offer a simplified jump process using a Bernoulli jump process as an approximation for the Poisson process. The Bernoulli distribution is a discrete distribution and is based on a Bernoulli trial which is an experiment with only two outcomes, success or failure. Therefore, in the case of a jump model, during a small period of time there can be only two possibilities, a jump or no jump. This method allows the use of maximum likelihood estimation of the model parameters, which proves to be empirically tractable while successfully describing the movements of the stock returns in their sample. In Ball and Torous (1985) this model is compared to Black and Scholes (1973) to price call options and by allowing the underlying to jump they

5.1 Introduction

were able to eliminate some systematic empirical biases that the Black Scholes model exhibits.

Building on Merton (1976), Bates (1991) derives a jump diffusion model to price Amer- ican options when jump risk is systematic. He examines the stock market crash of 1987 and seeks to ascertain whether the distributions and parameters implicit in option prices can explain and rationalise its occurrence. He ultimately finds that the crash was an- ticipated in the market on the basis of information implicit in these options prices. In the year before the crash implicit distributions were negativity skewed demonstrating a significant perception of downside risk.

In recent years the Merton jump diffusion option pricing model has been extended in various ways some of which include stochastic volatility (See Heston (1993), Bates (1996) and Scott (1997) for examples), jumps in volatility (Eraker et al. (2003) and Eraker (2004)) and double exponential jumps (Kou (2002) and Kou and Wang (2004)). The jump diffusion model has also been extended to many other asset classes outside of equities including commodities, exchange rate markets, interest rates and credit mar- kets. In an application to exchange rate markets, Bates (1996) combines the stochastic volatility jump model of Heston (1993) with Bates (1991) model for American op- tions with systematic jump and volatility risk. Bates estimates parameters implicit in Deutsche mark options over the time period 1984-1991 using various option pricing models. He finds that the stochastic volatility process is unable to fully explain the implicit leptokurtic distribution which results in the volatility smile that is observed. Attributing the excess kurtosis to the fear of jumps in the market he finds that this can better explain the shape of the smile and the presence of outliers. In commodity markets, Murphy and Ronn (2014) examine jumps in the crude-oil market. Owing to the non-normal distribution of commodity returns the existing literature suggests that jump diffusion models are well suited to describe commodity price movements. They

find that the implied jump diffusion parameters bear a close relationship to economic and geopolitical events. This chapter follows a similar methodology and uses a Ball and Torous (1983) to estimate jump parameters in the underlying future contract and then proceeds to estimate option implied jump parameters using a mixed Gaussian Poisson process.

There is a substantial body of literature in the area of interest rate jump models. The three most common type of models are jump augmented HJM models (for examples see Jarrow and Madan (1995) and Das (1998a)), jump augmented factor models such as augmented Vasicek or CIG models (Das (1998b), Ahn and Thompson (1988) and Baz and Das (1996)) and finally jump augmented price kernel models (Attari (1999) and Das and Foresi (1996)). Various papers have empirically tested the relationship between jumps in interest rates and the central bank or macroeconomic announcements. Das (2002) develops a Poisson-Gaussian model of the Federal Funds rate to capture jump effects and finds that this model provides a good statistical description of the nature of the short rate. Several applications of the model are demonstrated and show that the jump model can successfully be used to find day of week effects, the effect of the