Survey sentiment and interest rate option smile

REVECO-01008; No of Pages 13 International Review of Economics and Finance xxx (2014) xxx–xxx

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International Review of Economics and Finance journal homepage: www.elsevier.com/locate/iref

Survey sentiment and interest rate option smile Cathy Yi-Hsuan Chen a,1, I-Doun Kuo b,⁎ a b

Department of Finance, Chung-Hua University, 707, Sec.2, WuFu Rd., Hsinchu 30012, Taiwan Department of Finance, Tunghai University, 181, Taichung-Kan Road, Taichung 407, Taiwan

a r t i c l e

i n f o

Article history: Received 14 October 2013 Received in revised form 11 November 2014 Accepted 18 November 2014 Available online xxxx JEL classification: G12 E43 Keywords: Survey sentiment Option smile Eurodollar options markets

a b s t r a c t This paper finds that the presence of interest rate smile can be fully explained neither by the model misspecification nor by the buying pressure. First, volatility smile obtained from alternative interest rate models is not flat and interest rate smile significantly relates to survey sentiment after controlling for fundamental and liquidity variables. Second, a dynamic relation between sentiment proxies and interest rate smiles meets the limit to arbitrage hypothesis, which is the focal point of market friction. Third, the relation between survey sentiment proxies and option smiles is more pronounced during the crisis period than the pre-crisis and post-crisis period. While investor sentiment drives the smile curve, interest rate models cannot fully capture the smile since these models are formulated in a frictionless environment. © 2014 Elsevier Inc. All rights reserved.

1. Introduction To provide alternative explanations for the smile effect and to extend our understanding of the nature of the smile, this paper studies the extent to which investor sentiment affects time series variation of the interest rate option smiles. Prior studies have explained the presence of smiles as due to model misspecification, wherein the implied distribution is inconsistent with the model assumptions. According to the literature, volatility smiles can be found in interest rate option markets. For example, the smiles backed out from Eurodollar options using a one-factor Heath, Jarrow, and Morton (1992) model generally appear to have an asymmetric pattern in which the out-of-the-money (OTM) volatility is generally greater than in-the-money (ITM) volatility, but at-the-money (ATM) volatility is the lowest (Amin & Morton, 1994). The smile pattern exists not only in one-factor models, but also is apparent in the multifactor models (Kuo & Paxson, 2006), the jump-diffusion model (Das, 1999; Zeto, 2002), and stochastic volatility and jump model (Jarrow, Li, & Zhao, 2007). Volatility smiles are present because of a deviation between option model prices and their corresponding market prices. BlackScholes models and all interest rate models are formulated in an environment of frictionless markets where it is possible to create a hedged position whose option value does not depend upon the underlying asset price and permits no-arbitrage opportunities. But in reality, option prices are determined in an environment which is not frictionless. Hence, a continuous hedged position behind the option model is not feasible in any option market and thus arbitrage opportunities exist. Therefore, backing out volatility from market option prices using a continuous model does not guarantee that the implied volatilities across strikes are equivalent. In addition, using alternative models, such as jump or/and stochastic volatility, or the model relaxing part of the assumption cannot fully fit into option prices across strikes, revealing that inconsistency between model and market prices. ⁎ Corresponding author. Tel.: +886 423590121 3586; fax: +886 4 23506835. E-mail addresses: [email protected] (C.Y.-H. Chen), [email protected] (I.-D. Kuo). 1 Tel.: +886 3 5186057; fax: +886 3 5186054.

http://dx.doi.org/10.1016/j.iref.2014.11.018 1059-0560/© 2014 Elsevier Inc. All rights reserved.

Please cite this article as: Chen, C.Y.-H., & Kuo, I.-D., Survey sentiment and interest rate option smile, International Review of Economics and Finance (2014), http://dx.doi.org/10.1016/j.iref.2014.11.018

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This study investigates the extent to which interest rate option prices across strikes are set beyond macroeconomic components and interest rate option models. To do so, all macroeconomic variables and the buying pressure (Bollen & Whaley, 2004) are controlled. Then, we examine whether investor sentiment affect interest rate option smile. By examining the time-series relationship between sentiment and smile pattern, we test two competing hypotheses, namely the limits to arbitrage and the positive feedback trading hypothesis, to capture the dynamic interplay between sentiment-driven traders and arbitrageurs. From this we identify the role of arbitrageurs in influencing Eurodollar option prices, and determine if there is a transitory or persistent sentiment effect. Our findings support the limits to arbitrage hypothesis that the sentiment effect is transitory and the time series variation of volatility smile are closely related to the level of impediments of arbitrage. Finally, we investigate whether time variation of investor sentiment affects volatility smiles during the states of high/low volatility and different economic periods. This study contributes to the literature in several aspects. First, the kurtosis as well as curvature of the smile are examined, which are important aspects of the smiles. Second, Han (2008) studys whether investor sentiment affects implied volatility smile on S&P500 options and found there is a significant relationship after control variables. In this study, we look at the role of sentiment in interest rate options markets rather than that in equity options market. In the equity options markets, the risk concentrates on the downside. In the interest rate options markets, however, the risk has been induced in the both rise and decline of interest rate. We find that, because of fear of rising borrowing costs, Eurodollar puts can be more influenced by investment sentiment than corresponding calls. Third, we extend Deuskar, Gupta, and Subrahmanyam (2008) who examine the relationship between economic determinants and interest rate caps smile. However, they do not consider the impact of sentiment on interest rate option smile. This paper also expends Chen and Kuo (2013) by looking at the relationship between Eurodollar option smile and investor sentiment with different economic states. More importantly, different survey sentiment measures are used to examine their impact on interest rate option smiles. Finally, this research provides implications for future development in the interest rate models, as well as pricing and hedging interest rate contingent claims in particular. The remainder of the paper is organized as follows. In Section 2, the theoretical background and various sentiment measures are introduced and discussed. Section 3 describes the data and measures the shape of smile. Empirical results are presented and analyzed in Section 4. 2. Data and shape of the volatility smile 2.1. Data description In this study, weekly observations of Eurodollar futures and options from January 1998 to December 2010 are used.2 They are selected in this study because Eurodollar options are probably one of the largest interest rate derivatives in the world traded in exchange-traded markets. Several filtering procedures are applied for the concern of illiquidity or microstructure effects. First, we exclude options with a maturity greater than 180 days or less than 6 days because the longer contracts are traded infrequently and the contracts of less than a week have an expiration effect. Second, we collect quarterly matured options since non-quarterly matured options are less frequently traded. Third, those options with moneyness greater than 0.3 or less than −0.3 are deleted because they are either deep ITM or deep OTM options. Following Amin and Morton (1994), and Kuo and Paxson (2006), moneyness is defined as the futures price less strike price for calls and strike price less futures price for puts.3 2.2. Definitions of the implied volatility smiles In this section, we describe how we obtain implied volatility from Heath et al. (1992). Since the objective of this study is to evaluate the impact of investor sentiment on volatility smile, we back out implied volatility from Eurodollar options using the one-factor HJM model. The implied volatility is obtained by minimizing the error between market and model price.4 To reduce the number of observations, moneyness with 0.25 and − 0.25 and maturity with 30 and 120 days are selected each day. Because there are no exact options matching these requirements for each day, a linear interpolation or extrapolation technique is used. Fig. 1 displays the Eurodollar call and put option volatility across different ranges of moneyness on 11 November 2009. Strikingly, the volatilities do not all lie on a horizontal line. This pattern is the volatility “smile” and constitutes evidence against the HJM constant model. The volatility smile indicates that OTM volatility is greater than ATMs and ITMs for calls, but it is opposite for puts. According to Fig. 1, the natural question to ask is how to measure the time series patterns of implied volatility smile in terms of their asymmetry and curvature. Following Deuskar et al. (2008), the measures of the asymmetry and curvature are used in this study, referred to respectively as “risk reversal” and “butterfly spread,” which are widely used by practitioners. These empirical 2 We select observations from each Wednesday since this day of the week is likely to be a trading day compared with others. If Wednesday is not a trading day, observations from the following day are selected. 3 In equity options, Bollen and Whaley (2004) suggest defining the moneyness using the option's delta, but this rule may not be perfectly applicable to fixed-income options. The level of change in interest rates is relatively smaller compared to stock prices within a certain period, and therefore the moneyness in fixed-income options may incur bias by adopting the measures from equity options. The moneyness we apply can be handled easily with the same sign of moneyness for calls and puts, and this feature could be attractive if a sign change is evident in the regression analysis. 4 The option price computed from one-factor model is obtained from the following procedure. First, we build a forward rate tree with 10 steps using the one-factor HJM model. Under the one-factor HJM model, the volatility function σ(t,T) is set to be σ0, where t is the starting time of forward rate and T is the terminating time. Second, an option price tree is built and calculated backward to the initial point. Then implied volatility for a given option is obtained when the difference between model and market prices are minimized.

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3

0.7

0.65

Volatility (%)

30-day call

0.6

120-day call 30-day put

0.55

120-day put

0.5

0.45 -0.4

-0.2

0

0.2

0.4

Moneyness Fig. 1. Implied volatility smile for the HJM model. Notes: The volatilities across moneyness are displayed for the date of 11 November, 2009. The procedure for how to obtain implied volatility can be found in the text. Interpolation and extrapolation are used to obtain the required moneyness and maturity of options, as described in the text.

measures explicitly capture the asymmetry and curvature of the smile curve so that they can be viewed as proxies for the skewness and kurtosis of the risk-neutral distribution of interest rates. The risk reversal (RR) and the butterfly spread (BS) are computed as, RR ¼ IV−0:25 −IV0:25

BS ¼

ð1Þ

IV0:25 þ IV−0:25 −IVATM 2

ð2Þ

where IV0.25 is the implied volatility for option moneyness being 0.25, and it is the same for IV−0.25. As in Fig. 2, the 30-day OTM call volatility is greater than the corresponding ITM volatility, but it is the opposite for puts. In this case, the value of RR will be positive for calls and negative for puts. We use the time variation of RR to evaluate the information revealed in the OTM options relative to ITM options. In line with Deuskar et al. (2008), the butterfly spread is built as a proxy for capturing the curvature of the smile. This spread is designed to measure implied volatility at the level of 0.25 away-from-the-money on either side of 0. We use the time variation of BS to evaluate the information revealed in the away-from-the-money options, compared with ATM options. Table 1 summarizes the statistics of Eurodollar smiles represented by RR and BS from the period between January 1998 and December 2010. The positive RR for calls, reflecting that OTM volatility appears to be greater than ITM volatility. This appears probably because there is a greater demand for OTM calls than ITM calls, but there is less supply for OTM calls than ITM calls. Market makers thus should sell higher OTM price and lower ITM price than normal, resulting in greater OTM call volatility than ITM volatility. We also report 25, 50, and 75 percentiles of RR and BS, showing an increase of the value for the top 75, 50, and 25 percentiles. The AR(1) indicates that the autocorrelation coefficient of the first lag for RR and BS. Although other lags are not reported, the coefficient values decline quickly as the lag increases, indicating that the current day's volatility is more related to the next day volatility. 0.015

1.0

Put

0.010

0.8

0.005 0.000

0.6

-0.005

0.4

-0.010 0.2

-0.015 -0.020 1998

1999

2000

2001

2002 RR

2003

2004 BS

2005

2006

2007

2008

2009

2010

0.0

Eurodollar Consensus

Fig. 2. Time variation of Put RR and BS of Eurodollar Smile and Investor sentiment. Notes: This figure shows the time variation of 30-day RR (Gray), BS (Black) and investor sentiment (thicker dark line based on the right horizontal axis) from 1998 to 2010. Eurodollar Consensus obtained from MarketVane is a proxy of investor sentiment, which is discussed in the text.

Please cite this article as: Chen, C.Y.-H., & Kuo, I.-D., Survey sentiment and interest rate option smile, International Review of Economics and Finance (2014), http://dx.doi.org/10.1016/j.iref.2014.11.018

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Table 1 Summary statistics of implied volatility smiles. TTM

Put

Call

30

120

Smile variables

RR

BS

RR

BS

Mean Std. 25%percentile 50%percentile 75%percentile AR(1) Mean Std. 25%percentile 50%percentile 75%percentile AR(1)

−0.2746 1.5204 −0.4745 −0.1211 0.1477 0.4943 0.5288 1.7756 0.0276 0.2564 0.7164 0.3826

0.2243 0.4215 −0.0003 0.0627 0.2574 0.6264 0.3687 2.2365 0.0038 0.0965 0.3426 0.0875

−0.0025 0.1623 −0.0672 0.0047 0.0784 0.9845 0.0427 0.2734 −0.0511 0.0223 0.1175 0.7986

0.0164 0.0555 −0.0083 0.0039 0.0201 0.8631 0.0185 0.0674 −0.0164 0.0047 0.0237 0.3588

Notes: This Table reports the summary statistics of Eurodollar option smiles during the sample period from January 1998 to December 2010. The shapes of smiles are characterized by asymmetry (RR), and curvature (BS) of the smiles. The RR represents the difference between the implied volatility of OTM and ITM options, defined in Eq. (1). The BS is defined in Eq. (2) to capture the curvature of the smiles. The 30- and 120-day smiles are calculated to provide further analysis for the term structure of the smiles. Moneyness is defined as futures price less strike price for calls and strike price less futures price for puts. In this Table, OTM and ITM are respectively with moneyness values of −0.25 and 0.25. The AR(1) indicates that the autocorrelation coefficient of the first lag for RR and BS.

2.3. Fundamental concepts Fig. 2 depicts the time variation of skewness and curvature backed out from Eurodollar put option prices using the one-factor HJM model. The flat smile will produce zero skewness and curvature. The RR over time is significantly positive during the early 2000 and turns out to be negative before the global financial crisis in July 2007, and then it increases dramatically during the crisis. Time variation of curvature is also up and down across zero, and particularly unstable in the crisis period. How can the fluctuation of skewness and curvature computed from Eurodollar option smile be explained? The non-zero skewness and curvature might be inconsistent to non-normal distribution, differing from what the model assumes.5 In addition, virtually all models, including interest rate models, are built up in an environment where the market is frictionless and prices are determined under a no-arbitrage condition. Although some of these assumptions can be removed in an extension of the model, complete removals are not possible. Hence, no-arbitrage pricing cannot be achieved in reality and market option price in fact is determined by risky arbitrage conditions. The other important economic concept in determining asset price is supply and demand. The macroeconomic variables may explain the change in skewness and curvature of the interest rate smile since the changes in these variables affect the relative demand and supply for the series of Eurodollar options. Subsequently, the option prices and their volatilities will be affected. Next, the market structure of Eurodollar option markets can be another source of volatility smile. The function of market makers is to facilitate market liquidity, but this arrangement can influence the option prices and their volatilities. Particularly, during a crisis period, the market participants demand more OTM Eurodollar calls to protect against the expected large fall in Libor rates. When market makers face net buying pressure from these options, they will offer higher price for these options to cover costs for hedging unbalanced inventories, the net buying pressure of these options will lead to a rise in volatility. Green and Figlewski (1999) indicate that this hedge strategy will be prohibitively expensive, which leads to some exposure of price risk for market makers. Under this circumstance, they have to charge greater prices than fundamentals, and so OTM volatilities will be greater than normal. Then positive RR and BS will be expected. In addition to the macroeconomic factors mentioned above that affect skewness and kurtosis, bullishness or bearishness of investors, also affects the particular series of option prices. With the presence of limits to arbitrage (Shleifer & Vishny, 1997) in options markets, option prices and their volatilities are influenced not only by macroeconomic variables but also by investor sentiment. In Fig. 2, during the US subprime crisis, with the greater limit to arbitrage and higher volatility, the smiles seem to have a closer association with investor sentiment.

3. Sentiment and interest rate option smile One of the objectives of this paper is to examine the determinants of the volatility smiles in interest rate option markets. A clear understanding of the determinants of these smile patterns can help in developing models that eventually explain the entire smile. To this end, we explore the extent to which interest rate volatility smile is affected by investor sentiment. Appropriate measure of investor investment is critical since any proxy may contain some components unrelated to sentiment. Apart from investor sentiment, some fundamental variables may relate to the smile, and their dynamic relations are discussed in the following sections. First, we provide a description of sentiment proxies and then we discuss related issues. 5

The HJM model assumes that the forward rate movements satisfy the geometric Brownian motion.

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3.1. Sentiment measures Sentiment is a measure of the bearishness or bullishness of market participants. Although Brown and Cliff (2004) show a list of direct and indirect sentiment proxies, in this paper, we mainly use direct sentiment proxies obtained from a survey because they are cleaner sources to measure the bullishness or bearishness of investors. Indirect sources obtained from market trading activities may contain macroeconomic components, which are not directly linked to sentiment. Several survey sentiment proxies are selected to study the relations with interest rate option smile. First, sentiment measure is the bullish consensus for Eurodollar prices (EurodollarConsensus) obtained from MarketVane.6 The bullish consensus, sometimes referred to as market sentiment, is the degree of bullish sentiment for a particular asset, such as gold, commodities, or the S&P 500 index. MarketVane tracks the buy/sell recommendations of leading advisers7 in the futures market for that specific asset. The index price of the Eurodollar futures is equal to 100 less the futures rate, i.e. Futures prices = 100-futures rates, where the futures rate is expressed in percentage points. The expectation for EurodollarConsensus is opposite to that for Eurodollar futures rate. A rising EurodollarConsensus implies a rising Eurodollar futures expectation and equivalently a declining Eurodollar futures rate expectation. If the consensus for Eurodollar futures price becomes bearish, the borrowers will hold more puts and are willing to pay more for them to lock in the borrowing cost. We then expect a steeper put smile and a flatter call smile as the consensus for Eurodollars futures price becomes bearish, and vice versa. We also use the bullish consensus for S&P500 futures (S&P500Consensus) as the other sentiment proxy collected from MarketVane. A bullish consensus of 60% for S&P500 futures imsaplies that 60% of the traders are bullish and expect S&P500 futures prices to rise. If the consensus for S&P500 futures becomes bullish, the borrowers will hold more puts and are willing to pay more for them to lock in borrowing cost. We then expect a steeper put smile and a flatter call smile as the consensus for S&P500 futures price becomes bullish, and vice versa. The third sentiment is the monthly index of Consumer Sentiment (CS) collected by the University of Michigan based on a survey of households’ perceptions about current and future financial conditions. The main objective of creating this index is to judge the consumer's level of optimism/pessimism through assessing near-time consumer attitudes on the business climate or gauging the economic expectation of consumer saving and spending behavior. Since throughout this paper all estimations are used on a weekly basis, the value of CS each week is applied using the value for the nearest month. Another sentiment proxy is the monthly Baker and Wurgler Sentiment Index (Baker). Baker and Wurgler (2006) constructed first principal component from six sentiment proxies and their lags. This index is commonly used because it removes idiosyncratic and non-sentiment-related components and it can determine the lead-lag relationships of those variables. Likewise, the weekly observations are replicated by using the most recently available monthly value. A further appealing property is that it levels out the extreme observations that may be biased and affecting the inference. The last sentiment measure is American Association of Individual Investors (AAII). The AAII surveys individual investors weekly in respect to their outlook, asking them to state whether they believe that over the next six months the stock market will be bullish, bearish, or neutral. The AAII sentiment index is defined as the ratio of bullish responses to bearish responses. Table 2 reports the summary statistics of all sentiment proxies. Panel A displays the statistics of all sentiment proxies showing that the Michigan consumer sentiment index on average is higher than others, whereas Baker sentiment index is the most unstable. Panel B of Table 2 shows the correlation coefficients between sentiment proxies among all sentiment proxies. The correlations between sentiment proxies are within the range of 0.3 and −0.5, but most of them are lower than the absolute value of 0.4. The wide range of sentiment values reflects the different perspectives, different participants of survey, conducting approaches, and estimating methods of sentiment measure. In particular, the EurodollarConsensus is negatively correlated with all measures, except the Baker sentiment index because the Eurodollar futures price is based on the setting of the index rather than rates. Since index and rates are negatively related, when we expect the Eurodollar futures market to be bullish, in turn the Eurodollar rate will decline, leading the EurodollarConsensus to contrary to other sentiment indices. To study the relations of sentiment and volatility smile, it is important to illustrate how RR (proxy of skewness) relates to the investor sentiment proxies because the direction of the skewness conveys important information. In Table 3, Panel A shows the likely graph of volatility smile where graph (a) has the OTM volatility being greater than the ITM volatility and graph (b) has the ITM volatility being greater than the OTM volatility. Panel B gives details regarding how the variables relate to Graph (a) and Graph (b) in Panel A.

3.2. Illustration of complex relations Since the Eurodollar futures price conversely relates its corresponding rate, it is important to clarify the direction for how they relate to sentiment proxies. Note that moneyness for Eurodollar options is defined as the difference between futures price and strike price rather than the difference between Eurodollar futures rate and strike rate, so RR is the proxy of skewness for Eurodollar futures 6 The bullish consensus of MarketVane is available daily or weekly back in 1964, covering major futures product in US. Details can be found is their website (http:// www.marketvane.net/). 7 These recommendations are collected from the following sources: 1. Reading current market letters from these advisers. 2. Calling hotlines provided by advisers. 3. Contacting major brokerage houses to learn what the house analysts are recommending for different markets. 4. Reading faxes and e-mails sent from advisers. The buy/ sell recommendations from each adviser are tracked during the day to verify the entry and exit of each trading position. The bullish consensus is compiled at the end of the day to reflect the open positions of the advisers.

Please cite this article as: Chen, C.Y.-H., & Kuo, I.-D., Survey sentiment and interest rate option smile, International Review of Economics and Finance (2014), http://dx.doi.org/10.1016/j.iref.2014.11.018

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Table 2 Summary statistics of sentiment measures. Eurodollar Consensus Panel A: Summary statistics Mean Std. 25% percentile 75% percentile Panel B: Correlation coefficients Eurodollar Consensus S&P 500 Consensus CS Baker AAII

S&P 500 Consensus

CS

Baker

AAII

0.495 0.215 0.320 0.680

0.494 0.155 0.350 0.630

0.918 0.108 0.861 0.974

0.288 0.674 −0.055 0.439

0.426 0.108 0.350 0.507

1 −0.427 −0.447 0.231 −0.192

1 −0.141 −0.425 0.217

1 0.101 0.223

1 −0.066

1

Notes: Panel A reports the summary statistics of sentiment measures collected from survey. Panel B shows the correlation coefficients among sentiment variables. The sentiment proxies obtained from MarketVane are the bullish consensus for Eurodollar prices (EurodollarConsensus) and bullish consensus for S&P500 futures (S&P500Consensus). The ratio of bullish responses to bearish responses is collected by American Association of Individual Investors (AAII). These three measures are bounded between zero and one since they are ratio data. The index of Consumer Sentiment (CS) collected by the University of Michigan is started in 1967 and is benchmarked to year 1985 as 100, we divide the original values by 100 to be comparable with other sentiment measures. The Baker and Wurgler sentiment index (Baker) is constructed by principle component analysis, and has been further rescaled so that the index has unit variance.

price. Table 3 shows the relations for RR and sentiment proxies. Panel A shows possible shapes of volatility smile, while Panel B states how the sentiment proxies relate to skewness of Eurodollar futures price and futures rate. For Graph (a) in Panel A, OTM volatility is greater than ITM volatility and thus positive RR will be obtained. Under this circumstance, positive (negative) skewness of the Eurodollar futures price observed in the call (put) smile, or positive skewness in terms of Eurodollar futures rate will be observed, as stated in Panel B. Conversely, the analogous interpretation can be repeated for Graph (b). The relation between RR and sentiment proxies should be interpreted cautiously. Panel B in Table 3 shows that two sentiment proxies, the EurodollarConsensus and the S&P500Consensus, have opposite relations with RR. Note that the EurodollarConsensus is the survey for Eurodollar futures price rather its corresponding rate. If the consensus of Eurodollar futures price is more bullish, OTM calls will be more in demand than ITM calls, and subsequently the former options will be priced higher than the latter. Under this circumstance, OTM call volatility will be expected to be greater than ITM calls, and a positive RR in calls (or Graph A) will be expected. The situation is the reverse for puts. A negative RR (or Graph B) is expected since ITM volatility will be higher than OTM volatility. An identical interpretation can be used for less bullishness of the EurodollarConsensus. However, the relation between the S&P500Consensus and RR is different from that between the EurodollarConsensus and RR. If the S&P500Consensus is more bullish, it indicates an expected rise in Eurodollar futures rate or expected decline in corresponding Eurodollar futures price, so a negative call RR and a positive put RR may be expected. Conversely, with a less bullish consensus of

Table 3 Illustration for Relations for RR and sentiment proxies. Panel A volatility smile

Panel B Variable

RR skewness of Eurodollar futures price skewness of Eurodollar futures rate EurodollarConsensus is more bullish EurodollarConsensus is less bullish S&P500Consensus is more bullish S&P500Consensus is less bullish

Graph (a)

Graph (b)

call

put

call

put

positive negative positive ✓

positive negative positive

negative positive negative

negative positive negative ✓

✓ ✓

✓ ✓

✓

✓ ✓

Note that Panel A shows the graphs of volatility smile, where vol, in, and out indicate volatility, options with in-the-money, and out-of-the-money. Moneyness is defined as the futures price less strike price for calls and strike price less futures price for puts. In Panel B, RR is defined as the difference between OTM volatility and ITM volatility. For example, OTM option is the option with moneyness being −0.25, whereas ITM is the moneyness with 0.25.

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S&P 500 futures price, we may expect a positive call RR and a negative put RR. Analogously, bullishness of Consumer Sentiment and AAII may be identical to the relation between RR and S&P500Consensus, which is confirmed by their correlation in Table 2. To test our conjecture for the relation between RR and sentiment proxies mentioned above, regressions are conducted to test their relations and robustness by adding other controlling variables. 3.3. Sentiment in Eurodollar options smiles Table 4 reports the regression results for the relation between the shape of interest rate smiles and several investor sentiment proxies. The results of calls and puts are shown separately in Panels A and B. The regressions employ weekly time-series observations where the dependent variable of the regression is RR and BS with 30 and 120 days, as obtained from contemporaneous Eurodollar options prices. Meanwhile, a lagged-dependent variable is included as a regressor to control for its positive autocorrelation and to capture the persistence in the smiles. The t-statistics of parameter estimates are obtained from standard errors that have been adjusted for heteroskedasticity and serial correlation according to Newey and West (1987). Table 4 shows that the RR for 30-day and 120-day puts are negatively related to the EurodollarConsensus and the Baker sentiment index, but are positively related to other sentiment proxies. These findings are consistent with our conjecture discussed in Section 3.2. The negative coefficient estimate, −0.931, obtained by regressing RR on EurodollarConsensus for 30-day puts, implies that a more bullish consensus for Eurodollar futures price is related to a negative RR (positive skewness of Eurodollar futures price or negative skewness of Eurodollar futures rate). In contrast, the positive coefficient estimates in S&P500Consensus, CS and AAII indicate that a bullish shift in market sentiment is related to a positive RR (negative skewness of Eurodollar futures price or positive skewness of Eurodollar futures rate). With respect to calls and puts, the relation between RR and all sentiment measures is stronger for the options with the 30-day maturity than those with 120-day maturity. For example, the coefficient of the Michigan consumer index for 30-day puts is 1.15 and it is 0.039 for 120-day puts. In addition, the results from 30-day puts have stronger statistical significance. Similar patterns can be found for other sentiment proxies, suggesting that short-term options are more strongly affected by investor sentiment than long-term options. Typically, short-term options are convenient ways for speculating on interest rate movements because they are more sensitive to delta and gamma risk than long-term options. Short-term OTM options are better targets of speculation than long-term OTM options, leading to greater skewness for short-term options than long-term options. Table 4 Investor sentiment and the shape of interest rate smiles. Option Type

Put

TTM

120

Sentiment measures

RR

BS

RR

BS

EurodollarConsensus

−0.931* (−5.08) 0.559* (2.32) 1.1522* (4.57) −0.110* (−2.36) 0.601* (2.21) 0.449* (24.39) 0.261 0.902* (4.20) −0.884* (−3.10) −1.025* (−2.63) 0.101 (1.84) −0.573 (−1.78) 0.355* (18.50) 0.551

−0.183* (−2.81) 0.190* (3.07) −0.014 (−0.17) −0.003 (−0.21) 0.005 (0.072) 0.600* (36.50) 0.410 −0.034 (−0.11) 0.668 (0.41) 0.445 (0.78) −0.001 (−0.01) 0.546 (1.16) 0.080* (3.91) 0.236

−0.023* (−2.67) 0.027* (2.36) 0.039* (2.46) −0.008* (−3.84) 0.035* (2.75) 0.902* (106.96) 0.869 0.083* (4.25) −0.122* (−4.62) −0.104* (−2.95) 0.032* (6.15) −0.128* (−4.32) 0.709* (49.59) 0.608

−0.010* (−2.20) 0.002 (0.32) −0.010 (−1.18) −0.001 (−0.09) 0.001 (0.01) 0.743* (62.97) 0.641 −0.004 (−0.46) 0.010 (0.89) −0.046* (−3.00) −0.003 (−1.29) 0.014 (1.11) 0.338* (18.34) 0.146

S&P500Consensus CS Baker AAII Lagged dependent

Call

30

R-square EurodollarConsensus S&P500Consensus CS Baker AAII Lagged dependent R-square

Notes: This Table reports the regression results for the relation between the shape of interest rate smiles and the various investor sentiment measures. The results for puts and calls are separately shown. The smiles are calculated based on 30-day and 120-day maturities to examine whether the relation depends on the maturity of options. A lagged-dependent variable is included as a regressor to control for positive autocorrelation and to capture persistence in the smiles. The numbers in parentheses are t-statistics, obtained from standard errors that have been adjusted for heteroskedasticity and serial correlation according to Newey and West (1987). The symbol * denotes significance at the 5% level. On each Wednesday, the one-factor HJM model is calibrated to best fit the price of each Eurodollar option over the period between January 1998 and December 2010. RR and BS are then computed and defined in Table 1. Moneyness is defined as futures price less strike price for calls and strike price less futures price for puts. In this Table, OTM and ITM options have moneyness values of −0.25 and 0.25, respectively. TTM denotes the day to maturity for options.

Please cite this article as: Chen, C.Y.-H., & Kuo, I.-D., Survey sentiment and interest rate option smile, International Review of Economics and Finance (2014), http://dx.doi.org/10.1016/j.iref.2014.11.018

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The relation between investor sentiment and BS is also reported in Table 4. For puts, a significant estimated coefficient in S&P500Concensus, 0.190, means that a higher bullish sentiment links to an expectation of rising interest rate. A rising curvature of put smile caused by bullish sentiment indicates that the additional information has been revealed in the away-from-the-money options (especially for OTM options), compared with ATM options. From the perspective of investors, to benefit from an expected decrease in Eurodollar futures or an expected increase in futures rate, the OTM put is a convenient and less expensive instrument compared to ATM puts. However, the estimated coefficient in EurodollarConsences is significantly negative, which means that bearish sentiment leads to a reverse curvature due to lack of demand for OTM puts. The impact of sentiment on smiles is also economically significant. Based on the estimated coefficients in Table 4, when the EurodollarConsensus drops by one sample standard deviation, the RR(BS) from puts becomes more positive by about a 0.14 (0.10) sample standard deviation and RR from calls becomes more negative by about 0.12 sample standard deviation. Furthermore, based on the summary statistics of smiles in Table 1, we find that one standard-deviation moves in the EurodollarConsensus, S&P500Consensus and CS lead to the changes in RR from puts by about 20%, 9% and 12%, which are about 76%, 33% and 47% of the average magnitude of RR from puts. We also found two patterns in Table 4. First, the association for survey sentiment proxies with BS is weaker than those with RR. One possible reason is that the curvature measured by BS, which is proxy of kurtosis, relates to severe changes (or jumps) in Eurodollar futures. Asymmetry of the smile measured by RR, which is a proxy of skewness, relates to a small change in Eurodollar futures. Since the large change in Eurodollar prices occurs infrequently and is not much affected by investor sentiment, the relation for investor sentiment with BS is not as strong as that with RR. Second, the sensitivity of put smiles on sentiment measures is higher than that of call smiles, which may be due to higher values on absolute coefficients and t-statistics. The put smiles are more vulnerable than call smiles because puts are important vehicles for hedging against rises in interest rate (or declines in interest rate futures) for borrowers. In particular, OTM puts are less expensive than ATM puts for the protection again the risk of interest rates and hence these contracts can be more popular and is likely to be affected by investor sentiment. 3.4. Robustness to control variables The economic and liquidity variables should determine the shape of interest rate smiles. This section checks the significance of sentiment effect after including these fundamental variables. The selected control variables are (1) the spot 1-month U.S. interbank rate (Interbank), (2) the slope of term structure captured by the difference between 5-year T-bond rate and T-bill 1-month rate (YieldCurve), (3) the default spread defined as the difference between the 1-month U.S. interbank rate and the 1-month Treasury bill rate (DefaultSpread), and (4) the model-free implied volatility extracted from Eurodollar options markets (Model-free IV), developed by Kuo and Chen (2011). Considering the 1-month Interbank rate and the slope of the yield curve is necessary because they are indicators of general economic conditions and the direction of expected interest rate changes in the future. For example, if interest rates are meanreverting, very high interest rates are likely to be followed by rate decreases. Similarly, a downward-sloping yield curve is also indicative of future rate decreases. This would manifest in a lower demand for OTM Eurodollar futures in the market, thus affecting the prices of these options, and possibly the shape of the implied volatility smile itself. Typically, the default spread is regarded as a measure of aggregate liquidity as well as the default risk of the constituent banks. A wider spread indicates a higher default risk for the constituent banks, and possibly also higher risk of default for interest rate option dealers. It could affect the prices of away-from-the-money options more than the prices of ATM options, thus affecting the shape of the smile. The model-free volatility is included to examine whether the patterns of the smile vary significantly with the level of uncertainty in the market. During more uncertain times, as reflected by higher volatility, market makers may charge higher than normal asking prices for away-from-the-money options, since they may be more averse to taking short position at these strike rates, leading to a steeper smile. Also, during times of greater uncertainty, a risk-averse market maker may demand higher compensation for providing liquidity to the market, which would affect the shape of the smile. Table 5 shows that there is contemporaneous relationship between put smiles, and sentiment, economic, and liquidity variables.8 As expected, interest rate level, yield curve and default spread are positively and significantly related to the RR. However, they are insignificantly associated with the BS, except for model-free volatility. During more uncertain times, a higher curvature can be found because the away-from-the-money options are more valuable than ITM options. After considering economic and liquidity variables, investor sentiment proxies remain significantly relating to the smiles of Eurodollar options. 3.5. Considering alternative models and order imbalance So far, RR and BS are used as proxies of skewness and kurtosis. They can be incorrectly estimated because of model misspecification. To test the robustness of the association between survey sentiment and skewness and kurtosis, this section examines whether the relationship between sentiment and interest rate option smile can be explained by alternative option pricing models or the net buying pressure hypothesis. Recently, two major groups of studies, option pricing models and net buying pressure hypothesis, have been considered to explain the smile. The first group of studies derives modified versions of options pricing models using different volatility assumptions such as jump-diffusion assumptions (Das, 1999), or deterministic volatility assumptions (such as multi8

For brevity, the results of the relationship between sentiment and call smile after including control variable is not reported here, though they can be provided on request.

Please cite this article as: Chen, C.Y.-H., & Kuo, I.-D., Survey sentiment and interest rate option smile, International Review of Economics and Finance (2014), http://dx.doi.org/10.1016/j.iref.2014.11.018

C.Y.-H. Chen, I.-D. Kuo / International Review of Economics and Finance xxx (2014) xxx–xxx

9

Table 5 Investor sentiment and the shape of interest rate put smiles after control economic and liquidity variables. TTM

30

Variables

RR

BS

RR

BS

EurodollarConsensus

−0.934* (−3.33) 0.293 (1.07) 1.573* (4.02) −0.080* (−1.97) 0.409 (1.47) 5.637* (3.65) 0.488* (5.78) 0.462* (2.67) 0.621 (1.19) 0.398* (21.12) 0.287

−0.238* (−3.33) 0.124 (1.78) 0.306* (3.09) −0.043* (2.56) 0.064 (0.90) 3.029 (0.69) 0.018 (0.87) −0.030 (−0.68) 0.918* (6.72) 0.547* (31.77) 0.421

0.004 (0.31) −0.011 (−0.93) 0.061* (3.28) −0.004 (−1.28) 0.023 (1.79) 3.191* (2.71) 0.016* (4.08) 0.022* (2.71) −0.002 (−0.08) 0.861* (87.57) 0.873

0.011 (1.57) −0.007 (−1.05) 0.031* (3.17) −0.002 (−1.34) 0.003 (0.42) 0.323 (0.75) 0.010* (2.68) −0.003 (−0.74) 0.076* (5.66) 0.711* (57.07) 0.650

S&P500Consensus CS Baker AAII Interbank YieldCurve DefaultSpread model-freeIV Lagged dependent R-square

120

Notes: This table checks the robustness of the relation between the shape of put smiles and the sentiment measures by including several control variables, mainly from interest rate markets. The chosen control variables include: (1) the spot 1-month U.S. interbank rate (Interbank), (2) the slope of term structure obtained by the difference between 5-year T-bond rate and T-bill 1-month rate (YieldCurve), (3) the default spread defined as the difference between 1-month U.S. interbank rate and 1-month Treasury bill rate (DefaultSpread), and (4) the model-free implied volatility extracted from Eurodollar options markets (model-free IV). The numbers in parentheses are t-statistics, obtained from standard errors that have been adjusted for heteroskedasticity and serial correlation according to Newey and West (1987). The symbol * denotes significance at the 5% level. On each Wednesday, the one-factor HJM model is calibrated to best fit the price of each Eurodollar option over the period between January 1998 and December 2010. RR and BS are then computed and defined in Table 1. Moneyness is defined as futures price less strike price for calls and strike price less futures price for puts. In this table, OTM and ITM have moneyness values of −0.25 and 0.25, respectively.

factor versions of the HJM model). In addition to model-dependent skewness, model-free skewness can also be obtained to check the robustness of a model-free framework (Bakshi, Kapadia, & Madan, 2003; Jiang & Tian, 2005; Taylor, Yadav, & Zhang, 2009). The modeldependent volatilities are subject to model misspecification errors (Jiang & Tian, 2005), whereas the model-free skewness is calculated from observed option prices without using any specific options-pricing model. Bakshi et al. (2003) formalize a mechanism developed by Bakshi and Madan (2000) to extract the skewness of the risk-neutral return density from a contemporaneous collection of OTM calls and puts. However, the accuracy of model-free skewness may be limited if there are insufficient strikes for OTM options.9 This section evaluates robustness of the relation between investor sentiment and skewness derived by alternative approaches. To check robustness from model-dependent skewness, we include one-, two-, and three-factor HJM models to reduce model specification error.10 In this paper, the volatility smile is computed from a one-factor HJM model and from interpolation to obtain other volatilities for different strikes. We check robustness using the approach of Deuskar et al. (2008) by fitting into a volatility function and obtain the volatility smile from the function.11 This approach has the advantage of obtaining smooth volatility smiles, but has the risk of fitting into inappropriate volatility functions. The second group of literature interprets the smile as a phenomenon from the options market microstructure. The net buying pressure hypothesis proposed by Bollen and Whaley (2004) points out that market makers markup the volatility for a particular series of options, particularly for OTM puts, to compensate for the risk they take and the liquidity they provide. They found that buying pressure on the index put option appears to drive permanently downward sloping shape of S&P 500 index option. This variable may also affect the skewness and kurtosis for the Eurodollar options and hence should be used as a control variable. In sum, a significant relation between the interest rate option smile and sentiment may be attributed to the possibility that the sentiment proxies are correlated with risk factors in the models or the sentiment proxies are associated with net buying pressure. Table 6 examines whether the relationship between sentiment and interest rate option smile can be explained by either option pricing models or net buying pressure hypothesis. As shown in this table, the relation between four sentiment proxies and the model-free skewness is significant. In the third column, the skewness is derived from the skewness variable in the jump-diffusion model (Das, 1999).12 The results show that even considering jump-diffusion models or the model-free method, the smile is present and is significantly related to several sentiment proxies, suggesting a robust relation between volatility smile and investor sentiment.

9 The mathematical expressions of model-free skewness and their computation procedure can be found in Bakshi et al. (2003) and Han (2008) so they are not shown here. 10 Recent literature shows that multifactor models outperform single-factor models (See Driessen, Klaassen, & Melenberg, 2003; Gupta and Subrahmanyam, 2005; Kuo and Wang, 2009; Zeto, 2002. 11 For the specification of volatility function, we use models 1, 5, and 7 from Kuo and Wang (2009) to represent one-, two-, and three-factor models, respectively. 12 Across the sample period, the mean, median, and standard deviation of μ is −0.00016, −0.00042, and 0.00152.

Please cite this article as: Chen, C.Y.-H., & Kuo, I.-D., Survey sentiment and interest rate option smile, International Review of Economics and Finance (2014), http://dx.doi.org/10.1016/j.iref.2014.11.018

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C.Y.-H. Chen, I.-D. Kuo / International Review of Economics and Finance xxx (2014) xxx–xxx

Table 6 Considering alternative interest rate models, model-free method, and net buying pressure. Variables

model-free skewness

Jump model

HJM one factor

HJM two factor

HJM three factor

EurodollarConsensus

−0.158 (−0.97) 0.407* (2.56) 0.576* (2.54) −0.081* (−2.11) 0.418* (2.58) 50.338* (5.03) −0.327* (−6.72) −0.508* (−5.04) 0.448 (1.48) 0.094 (0.13) 0.414

−0.745* (−3.86) 0.572* (3.04) 1.025* (3.82) −0.021 (−0.45) 0.774* (4.04) 35.244* (2.97) 0.321* (5.56) −0.355* (−2.97) 3.261* (9.09) 0.085 (0.12) 0.496

−0.015* (−4.52) 0.013* (4.06) 0.032* (6.92) −0.005* (−6.27) 0.013* (3.82) −0.040 (−0.19) 0.010* (9.40) 0.001 (0.21) −0.001 (−0.16) 0.0672 (0.09) 0.220

−0.025* (−4.71) 0.069* (13.45) 0.101* (13.77) 0.001 (0.99) 0.005 (1.01) −2.553* (−7.88) 0.012* (7.34) −0.025* (−7.87) −0.013 (−1.38) 0.074 (0.09) 0.434

−0.069* (−9.33) 0.057* (7.91) 0.030* (2.85) −0.008* (−4.26) 0.037* (5.10) 1.713* (3.74) 0.014* (6.25) −0.017* (−3.78) −0.017 (−1.25) 0.078 (0.10) 0.487

S&P500Consensus CS Baker AAII Interbank YieldCurve DefaultSpread model-freeIV BuyingPressure R-square

Notes: The dependent variables are the 30-day skewness derived from the model-free method of Bakshi et al. (2003), the jump-diffusion model of Das (1999), and 1-, 2-, and 3-factor HJM models. For the jump-diffusion model, the first factor of the jump model is specified in exponential form. The specification of volatility function for one-, two-, and three-factor HJM models is obtained from Models 1, 5, and 7 of Kuo and Wang (2009). The procedure to compute volatility parameters for these three models and HJM jump model is identical to that for a one-factor HJM model with constant volatility function. RR in this table for those models is obtained from a put smile. Following Han (2008), the net buying pressure is measured as the ratio of the open interest of OTM put options to the open interest of the near and ATM call and put options. The OTM put options correspond to those with moneyness between −0.35 and −0.25, while the ATM options correspond to those with moneyness between −0.05 and 0.05. The numbers in parentheses are t-statistics, obtained from standard errors that have been adjusted for heteroskedasticity and serial correlation according to Newey and West (1987).

The remaining columns show the results based on one-, two-, and three-factor HJM models. The procedure to obtain volatility parameters and volatility smile is identical to that in the one-factor model with constant volatility function. The results in Table 6 show that the relation between RR obtained from put smile and sentiment proxies is significant to various models, especially for multifactor models. The relations for multi-factor models are stronger, indicating that the association of sentiment and smile is robust to different model specifications and volatility smile smoothing approach. If the relation between sentiment and smile only reflects rational updating of belief on future volatility or the net buying pressure from supply and demand imbalance, sentiment would have no incremental importance to determine interest rate option smile. Conversely, if this relation reflects the impact of aggregate belief errors on option prices and if this relation is beyond order imbalance, then sentiment remains significant. As shown in Table 6, after considering a number of option pricing models and the net buying pressure effect, the magnitude of sentiment coefficients barely changes in comparison with those in Table 4 and still remains significant.

3.6. Movement of sentiment and option smile A smile study based on the "change" rather than the "level" enables us to determine whether the sentiment impact is persistent or transitory. On the other hand, we are able to test two competing hypotheses that address a dynamic association between sentiment and smile. In theory, there are two hypotheses that state this dynamic relationship. The limits to arbitrage hypothesis claim that arbitrageurs bring the prices to their fundamentals although they may entail risk and cost, and it predicts that the coefficient of lagged change in smile is significantly negative. Bollen and Whaley (2004) apply this hypothesis to explain the change of implied volatility of S&P 500 index options. If sentiment has a price impact in the presence of limits to arbitrage, this impact is likely to be transitory. As arbitrageurs gradually hedge their positions, prices should be reversed at least partly to their previous levels. The positive feedback hypothesis predicts a positive coefficient on the lagged change in smile.13 When arbitrageurs receive excessively bullish sentiment, they recognize that increasing OTM Eurodollar puts will stimulate buying by positive feedback traders the next day. In anticipation of this purchasing caused by the sentiment effect, arbitrageurs buy more that day, and thus drive that day’s OTM puts prices higher than fundamental values. On the next day, positive feedback traders become more bullish and buy in response to the previous day's price increase, so they keep prices above fundamentals. As a result, the positive feedback trading hypothesis implies a positive autocorrelation of change in smiles, indicating that the sentiment effect is persistent in the presence of positive feedback trading reinforced by arbitrageurs who have destabilized prices. 13 Aspects of the variations of the positive feedback trading hypothesis have been widely discussed by Simon and Wiggins (2001), Wang (2003), Kurov (2008) and Schwarz (2001).

Please cite this article as: Chen, C.Y.-H., & Kuo, I.-D., Survey sentiment and interest rate option smile, International Review of Economics and Finance (2014), http://dx.doi.org/10.1016/j.iref.2014.11.018

C.Y.-H. Chen, I.-D. Kuo / International Review of Economics and Finance xxx (2014) xxx–xxx

11

Table 7 Testing two competing hypotheses under low and high market volatility. Volatility of Interest rate

RR

BS

market

Low

High

Low

High

EurodollarConsensus

−1.187 (−0.08) −5.314 (−0.93) 3.953 (0.39) −1.969 (−0.27) 0.387 (0.13) −0.518* (−14.4) 0.268

−2.798* (−4.03) 1.104* (1.97) 2.461* (2.23) −0.155 (−0.53) 0.062 (0.21) −0.357* (−9.67) 0.181

−4.249 (−1.30) 0.003 (0.00) 2.014 (0.86) 0.404 (0.24) −0.082 (−0.11) −0.445* (−12.07) 0.211

−0.187 (−0.86) 0.393* (2.26) 1.041* (2.12) −0.012 (−0.13) 0.047 (0.51) −0.423* (−14.55) 0.284

S&P500Consensus CS Baker AAII Lagged dependent R-square

Notes: This table shows the response of the changes in the shape of put smiles to the changes of sentiment under a high and low volatility of interest rate market, respectively. High volatility is defined as interest rate volatility measured by the model-free method being above its 75th sample quantile, while low volatility is defined as a variable whose value is lower than its 25th sample quantile. For brevity, the regression coefficients for these additional control variables are not reported. The numbers in parentheses are t-statistics, obtained from standard errors that have been adjusted for heteroskedasticity and serial correlation, according to Newey and West (1987). The symbol * denotes significance at the 5% level. On each Wednesday, the one-factor HJM model is calibrated to best fit the price of each Eurodollar option from the period January 1998 and December 2010. 30-day RR and BS are computed and defined in Table 1.

Table 7 reports the results of testing two competing hypotheses under a high and low volatility period. Since high volatility increases the capital required per unit of investment, impediments of arbitrage are greater during a highly volatile period (Brunnermeier & Pedersen, 2009) than during a calm period. The level of model-free implied volatility for Eurodollar options (Kuo & Chen, 2011) is used to represent general volatility in interest rate markets. Testing two competing hypotheses together with consideration of arbitrage cost provides a comprehensive understanding of the interaction between sentiment-driven investors and arbitrageurs. As seen in Table 7, the coefficients of lagged change of the put smile are significantly negative for all smile measures, indicating that the current day's Eurodollar put prices are driven up by bullish sentiment on Eurodollar rate (or bearish sentiment on Eurodollar futures price), thus leading to a higher RR and BS. On the next day, arbitrageurs take the opportunity to sell overvalued puts and simultaneously hedge delta risk. The arbitrage strategy leads to partial price reversions and a smoother pattern of smile, supporting the limits to the arbitrage hypothesis and implying that the sentiment effect is transitory. However, there is less reversion under a high volatility period than under a low volatility period, corresponding to greater impediments of arbitrage in a volatile period. The price reversal is relatively restrictive in a volatile period so that the sentiment effect on smile is salient and may be more persistent.

3.7. Sentiment and financial crisis In 1995, Alan Greenspan, the US Federal Reserve Chair used the strong words “irrational exuberance” to warn on overpriced stocks in US, and after five years the stock bubble burst. Irrational exuberance can be one aspect of sentiment since over-exuberant investors naively extrapolate the market upward trend into the future (Shefrin, 2008, Ch8). The outbreak of a financial crisis may be due to a burst of speculative bubbles and systemic risk across countries.14 The stock price speculative bubbles affect option markets in which OTM calls are traded much more than OTM puts, leading to call volatility skew or positive skewness. After the bubble has burst, investors fear a further drop in price, and so OTM puts become traded more frequently than OTM calls. Under this circumstance, OTM puts will be priced with higher volatility than corresponding ITM and ATM puts, leading to positive put skewness. To examine this conjecture, we investigate the relationship between sentiment and skewness (and curvature) of Eurodollar option smile before, during, and after the recent subprime financial crisis. Table 8 shows the regression analysis for three subperiods, the pre-crisis (May 2005 and June 2007), the crisis (July 2007 and December 2008) and the post-crisis (January 2009 and December 2010). The crisis period begins from July 2007 since it began from bankcruptcy of Lebmam Brothers and accelerated with the announcement of bailouts for key financial firms by the US government in July 2008. It is assumed that the crisis ended in December 2008 because the S&P 500 stock index starts to rise in the early 2009. There are two major findings. First, according to R-square, the number of significant sentiment proxies and t-statistics of coefficients, the relation between sentiment and put smiles is more pronounced during the crisis than other periods. Even after the crisis, the impact of the EurodollarConsensus on put smiles is still prominent. The lag coefficients of RR and BS are 0.64 and 0.345 in the crisis period, which is much greater than that in the other periods, indicating a pronounced persistence in the smile. Hence, during the crisis, the smile is more vulnerable to sentiment effect. The limit to arbitrage is severer in the crisis period, since the market price, which has been influenced by sentiment, deviates from its model price.

14

Su, Lee, and Chiu (2014) measure skewness and kurtosis from equity option prices to test whether they affect volatility and return during global financial crisis.

Please cite this article as: Chen, C.Y.-H., & Kuo, I.-D., Survey sentiment and interest rate option smile, International Review of Economics and Finance (2014), http://dx.doi.org/10.1016/j.iref.2014.11.018

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Table 8 Financial crises and investor sentiment. Pre-Crisis

EurodollarConsensus S&P500Consensus CS Baker AAII Interbank YieldCurve DefaultSpread model-freeIV M2 Infaltion Lagged dependent R-square

During Crisis

Post-Crisis

RR

BS

RR

BS

RR

BS

−2.747* (3.03) 0.450 (−1.49) 0.003 (−0.69) −0.795* (−4.40) 0.031 (0.14) 4.496* (3.80) 0.441* (2.26) −0.079 (−0.21) 0.103 (0.16) −3.676 (−0.80) −1.897 (−0.43) 0.321* (7.53) 0.293

−0.314 (1.08) 0.064 (−0.65) 0.004 (−1.16) −0.155* (−2.71) 0.045 (0.63) 2.220 (1.94) 0.138* (2.18) −0.078 (−0.66) −0.294 (−1.45) 1.820 (1.21) −5.464* (−3.72) 0.270* (6.61) 0.282

−0.662* (4.68) 0.225* (−2.24) 0.001 (0.78) −0.366* (−4.13) 0.045 (0.46) 2.023 (0.74) 0.083 (1.83) −0.002 (−0.05) −0.035 (−0.55) −0.035 (−0.02) 4.603* (4.35) 0.640* (18.05) 0.832

−0.158* (2.05) 0.120* (−2.21) 0.001 (1.09) 0.025 (0.57) 0.057 (1.12) 2.135 (1.47) −0.009 (−0.39) 0.026 (1.43) −0.208* (−5.86) 0.921 (1.20) −0.138 (−0.26) 0.345* (8.46) 0.494

−1.522* (5.30) 0.452 (−1.69) 0.002 (−0.38) −0.014 (−0.18) 0.163 (1.24) 3.962 (0.48) −0.025 (−0.77) −0.377 (−1.24) 0.417* (2.31) −0.870 (−0.70) 0.414 (0.26) 0.310* (6.92) 0.403

−0.414* (3.71) 0.248* (−2.38) 0.001 (−0.26) −0.006 (−0.17) 0.009 (0.17) 1.984 (0.19) 0.015 (1.20) 0.024 (0.20) 0.037 (0.53) −0.510 (−1.05) −0.896 (−1.44) 0.246* (6.24) 0.331

Note: This Table shows the regression results for 30-day RR and BS on several variables, defined and explained in Table 4, for the pre-crisis period (May 2007-June 2007), the crisis period (July 2007-December 2008), and the post-crisis period (January 2009–December 2010).

4. Conclusions The previous explanation of interest rate volatility smile with model misspecification (Jarrow et al., 2007, Kuo & Wang, 2009) was unsatisfactory since stochastic volatility or/and jumps models cannot fully capture interest rate options smiles. Using weekly observations of Eurodollar futures and options from 1998 to 2010, our results show that investor sentiment is an important determinant of Eurodollar put smiles, and this indicates that the market is more sensitive to rises rather than declines of interest rates. The shape of smiles from short-maturity options is more vulnerable to sentiment than corresponding long-term ones. In addition, during a period of high volatility in the interest rate markets, a significant impediment to arbitrage results in a pronounced sentiment effect. We thus find that the impact of sentiment on the smile shape is temporary and the results support the limits to arbitrage hypothesis against the positive feedback trading hypothesis. For further subperiod analysis, the appealing impacts of sentiment on option prices can be evident during the financial crisis. During the crisis, the smile is more sensitive to sentiment effect because arbitrage is more restrictive. This relationship is robust even after controlling for fundamental variables, considering other options pricing models and the net buying pressure effect. The noise trader plays a part in the determinations of the put smiles, which is beyond the fundamental, liquidity variables and the buying pressure. Our results provide two implications for option model development, in general, and pricing and hedging interest rate contingent claims, in particular. Since equilibrium interest rate models usually start with assumptions of economic variables and are developed under no-arbitrage conditions (Hull, 2012, chap. 8). Investor sentiment, nevertheless, has no impact on interest rates and option prices. Our findings show that investor sentiment play a certain role in interest rate option prices so that future works may consider a model including sentiment variables and permit option price determined with arbitrage opportunities. However, the risk-neutral valuation, which governs the determination of option prices, cannot be applied and no-arbitrage conditions will fail. This posts challenges for future works. References Amin, K.I., & Morton, A.J. (1994). Implied volatility functions in arbitrage-free term structure models. Journal of Financial Economics, 35, 141–180. Baker, M., & Wurgler, J. (2006). Investor sentiment and the cross-section of stock returns. Journal of Finance, 61, 1645–1680. Bakshi, G., Kapadia, N., & Madan, D. (2003). Stock return characteristics, skew laws, and the differential pricing of individual equity options. Review of Financial Studies, 16, 101–143. Bakshi, G., & Madan, D. (2000). Spanning and derivative security valuation. Journal of Financial Economics, 55, 205–238. Bollen, N.P., & Whaley, R.E. (2004). Does net buying pressure affect the shape of implied volatility functions? Journal of Finance, 59, 711–753. Brown, G.W., & Cliff, M.T. (2004). Investor sentiment and the near-term stock market. Journal of Empirical Finance, 11, 1–27. Brunnermeier, M.K., & Pedersen, L.H. (2009). Market liquidity and funding liquidity. Review of Financial Studies, 22, 2201–2238. Chen, C.Y.H., & Kuo, I.D. (2013). Investor sentiment and interest rate volatility smile: evidence from Eurodollar options markets. Review of Quantitative Finance and Accounting, 40(4), 1–27.

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Please cite this article as: Chen, C.Y.-H., & Kuo, I.-D., Survey sentiment and interest rate option smile, International Review of Economics and Finance (2014), http://dx.doi.org/10.1016/j.iref.2014.11.018