FIX - The fear index. Measuring market fear.

In this paper, we propose a new fear index based on (equity) option surfaces of an index and its components. The quanti¯cation of the fear level will be solely based on option price data. The index takes into account market risk via the VIX volatility barometer, liquidity risk via the concept of implied liquidity, and systemic risk and herd-behavior via the concept of comonotonicity. It thus allows us to measure an overall level of fear (excluding credit risk) in the market as well as to identify precisely the individual importance of the distinct risk components (market, liquidity or systemic risk). As a side result we also derive an upperbound for the VIX.

Analysing market fear, liquidity and new capital instruments, before, during and after the financial crisis

The recent financial crisis was the unfortunate outcome of a widespread disruption in the global financial world. It showed, among other things, the need for both an adequate market stress level assessment and more stringent capital requirements for banks. In this thesis we focus on these two aspects. The thesis opens with a comprehensive introduction and summary of all mathematical and financial terms, models and methods employed in the rest of the work. In the second part we define different market fear indicators and propose a new overall stress index FIX. The third part covers the valuation and sensitivity analysis of Contingent Convertible bonds (CoCos), a new asset class that emerged due to the new regulatory setting following the financial crisis. Part II of this thesis focuses on measuring market fear and intraday liquidity. We argue that the "market fear" should not be recognized only by the level of the volatility, but should be measured by a composite indicator, which combines different factors into a single index. In particular, we have focused on three fear factors, which in our opinion; have a significant impact on the overall market stress level. More precisely, we propose to take into account market risk and nervousness using the well-known volatility index VIX. Further, we introduce two new components: LIQ - a market implied liquidity measure; CIX - an index of the degree of herd behavior in a market. Combination of these three leads to the new, overall fear index FIX. First we present the VIX index which is a key measure of market volatility delivered by Chicago Board Options Exchange (CBOE) and is originally based on the S&P 500 stock index option prices. The formula for VIX uses prices of all liquid out-of-the-money calls and puts for the front and second month expirations, leading to the 30-day forward looking volatility estimate. Secondly, we present the new liquidity measure LIQ. Although, several liquidity proxies already exist, these measures are typically defined for a given asset class only or are not unit free. To overcome these drawbacks, we define a new LIQ index, which isolates the liquidity risk in a unit free manner. It assesses the level of liquidity by employing the implied liquidity measure related to bid-ask spreads based on the conic finance theory of bid and ask prices. Conic pricing of vanilla options is performed using distorted expectations, instead of original expectations. For the implementation, concave distortion function from the minmaxvar family, characterized by a single parameter λ, are used. The value of λ for which the difference between the conic finance bid and ask prices and the market bid and ask prices is the smallest, is called the implied liquidity. Since the measure is purely based on option prices, we have a forward looking measure in contrast to many backward looking measures based on historical time series. A short term overall liquidity index LIQ is further constructed in the similar fashion to VIX out of the near- and next term index (or stock) implied liquidities. Subsequently, the herd behavior measure CIX is introduced as a third ingredient of the general panic or fear level. Although it is particularly hard to quantify phenomena which relies on human reactions in specific circumstances, the herd behavior in the financial world is related to the dependency relationship between traded assets. Thus, we make use of the concept of comonotonicity and comonotonic upper bounds for option prices. More precisely, we propose a new herd behavior index CIX as a ratio of VIX and its comonotonic counterpart, where option prices are replaced by their comonotonic upper bounds. This measure allows us to assess to which degree the whole market goes into one direction. The combination of these three leads to the new, overall fear index FIX presented in this thesis. VIX, LIQ and CIX are aggregated to one index, which in a "neutral" situation equals 100. A value of FIX>100 will reflect a market with a fear level above average, whereas a value of FIX<100 expresses less fear in the market than average. The theoretical part is followed by several numerical studies in which we apply the measures to real data in the time of the recent financial crisis, i.e. between January 2007 and October 2009, on Dow Jones index (DJX) option data. The results obtained in the numerical calculations clearly indicate the financial troubles within the considered period. VIX and LIQ triplicated its value in a timespan of a few weeks, indicating the culmination of the crisis. Both exhibit also a mean-reverting nature. CIX values furthermore demonstrated the credit crisis around October 2008 as well as smaller issues in the financial world. All three components represent different behavior. VIX gradually increase in response to market disturbances, whereas LIQ and CIX reacts relatively fast exhibiting sharp peaks. On the other hand, VIX and CIX seem to be more sensitive to smaller stress situations, showing peaks also e.g. around financial issues during the summer of 2007. Blending indices which exhibits different behavior should lead to a composite indicator carrying properties of its components. Indeed, results for FIX not only clearly reflects the financial problems round October 2008, but also demonstrates smaller peaks in summer 2007 which can be associated with stressful financial events and shows mean-reverting tendency. Also reaction to market distress seems to be rather immediate. All this confirms our motivation, that market fear should be recognized by a composite indicator, which combines various indices capturing different risks in different ways. In a supplementary study, we focus only on the LIQ measure in a high frequency data framework. The goal is to investigate the potential time-of-the-day and day-of-the-week effect of this new liquidity index. For that reason, the DJX index LIQ has been calculated every 5 minutes in period between August 6, 2012 and August 8, 2013. Results show, that Mondays and Fridays are, on average, the least liquid days of the week. Furthermore, considering the intra-day evolution of the LIQ, we observed that the beginning and the end of the session is characterized with a decrease of the liquidity. The study confirms the U-shaped daily pattern, which is in line with previous findings reported in the literature. Despite many contributions, there is still room for further improvement in the research in the framework of fear indices. We recognize potential improvements in the way of aggregating the fear factors, study on behavior of the historical time series of all the risk measures and discussion on the most liquid moments. The above mentioned financial crisis led to the advent of a new financial instrument - contingent convertible bonds, CoCos in short. CoCos are characterized with a loss absorption capacity with the aim of preventing or limiting the use of taxpayer's money in future bank rescue operations. During the crisis many financial institutions have been bailed-out by governments with the tax payers' money. To avoid in the future again the situation that the public finance would have to rescue banks, more regulatory capital has been demanded. This led to more stringent capital requirements for banks proposed in September 2010 by the Basel committee and finally resulted in the Basel III framework. Revised in July 2011 Basel III requirements allowed CoCo bonds to be part of the Tier 2 and Additional Tier 1 (AT1) in the capital. Since then banks have been embracing this new asset class. One can model CoCos was in a traditional (one-parameter) Black-Scholes framework assuming a constant volatility. However, increasing interest in the CoCos opens the quest for more appropriate and sophisticated pricing models. Thus, in this thesis we employ the stochastic volatility, five parameter Heston model as a more sophisticated method to describe the stock market dynamics. In particular, we investigate the impact of the volatility skew on the pricing of CoCo bonds. The third part of the thesis starts with explaining the anatomy of a CoCo. A CoCo can be understood as a bond issued by a financial institution that is converted into a predetermined number of shares or (fully/partially) written down as soon as the issuer gets into a less viable state, i.e. upon a trigger event. There exist different kinds of triggers: market based trigger, accounting trigger and regulatory trigger. The valuation of CoCos boils down to the quantification of the trigger probability and the expected loss suffered by the investors in case of conversion or write down. Apart from taking into account this loss absorption risk, coupon cancellation and extension risk are other risk factors in the CoCo design and are becoming also important in the pricing. There exist at least two ways of CoCos valuation - structural and market implied models. In this thesis we follow two market implied models: a credit derivatives approach and an equity derivatives approach. They are characterized by the fact that the derivation is based on market data such as share prices, credit default swaps (CDS) and implied volatilities. The credit derivatives approach model, is a rather simple model: it does not take into account neither the risk associated with the coupons nor coupons cancellation. For this reason we refer to this model as a "rule of thumb" model. The equity derivatives approach goes one step further and uses the coupon structure of the CoCo when calculating its theoretical price. The thesis section related to the research on CoCos contains several numerical studies, based on the Tier 2 (T2) 10NC 7 5/8 % CoCo issued by Barclays in 2012 and maturing on 21.11.2022 (ISIN: US06740L8C27). Both, option and CDS data are used for the Black-Scholes and the Heston model calibration. Here, the Heston model demonstrates its preponderance, yielding a better overall model fit to the market data. Using the calibrated parameters, we can price the Barclays CoCo under two mentioned valuation methods. In the thesis, we further calculate also the CoCo price for a range of trigger levels and identify the particular trigger level that corresponds to the observed market CoCo price. This level is called the implied trigger level. For the credit derivatives approach we obtain an implied trigger of around 17% of the current stock price in case of the Black-Scholes model and 25.8% for the Heston model; both models resulted in almost the same probability that the Barclays CoCo will be written down. We note that in the Heston model case the computational time was significant due to involved exotic option pricing with a Monte Carlo method. Subsequently, we tested the sensitivity of the CoCo price to changes in the Heston parameters values in the current market situation. In that way, one can assess the impact of the volatility skew on the pricing of the CoCo bonds. Multiple relationships have been unveiled, demonstrating significant variation of the price for different parameter sets. For instance, shifting the speed of mean reversion κ from 0.2 to 0.6 results in decrease of the CoCo price by around 25%. Further, the same impact was investigated in artificial distress and non-distressed cases. The correlation coefficient ρ is the parameter which has the most impact. Furthermore, the direction of impact is situation dependent. When the stock is close to the barrier, the change of ρ has a different impact on the CoCo price, compared to the situation further away from the barrier. Note that an increase in ρ (less negative) result in a flatter volatility curve (less skewed). A numerical estimation of the turning point, i.e. an implied barrier H for which the change of ρ has no impact on the CoCos' price is proposed in a following study. Using an estimate of the CoCo price slope related to different values of parameter ρ, we calculated that the turning point is around the level of 39% of the initial stock price in our example. In the very last study we make use of the observation that the Heston model parameters are also interlinked with so-called volatility smiles. In this part of the research we have checked the CoCo price and skewness for different sets of the Heston model parameters. The sets were chosen so that they preserved the same implied volatility level for at-the-money options (i.e. for moneyness M=100%) in one case and for moneyness M=20% in the second case. We observed price impacts as high as 10%. All above findings lead to the conclusion, that CoCos are significantly skew sensitive and argue for advanced models to accurately capture related risks in the assessment of CoCos. Despite significant insight in the CoCo pricing under the Heston model and the sensitivity of the CoCo price to different parameter sets given in this thesis, there is still room for further research, such as the improvement of the Monte-Carlo method, application of other stochastic volatility models, research on broader set of CoCo examples or investigation of optimum parameters sensitivity to the initial guess during the calibration. Although the recent financial crisis is regarded by many as the worst time for financial markets since the Great Depression of the 1930s, it allowed many underestimated risks to materialize. On the other hand it triggered the introduction of multiple new regulations. This gave a boost to the academic research on the roots that may have caused the crisis and gave motivation for further interesting research in the world of finance.I INTRODUCTION financial instruments and mathematical foundations 1 1 Financial assets and derivatives 3 1.1 Financial assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Derivative assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Credit Default Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Mathematical foundations 13 2.1 No-arbitrage and equivalent martingale measures . . . . . . . . . . . . . . . . . 13 2.1.1 Non-arbitrage principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 Probability space, filtration, equivalent martingale measure . . . . . . . 14 2.2 Mathematical models of financial markets . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Black-Scholes model - constant volatility model . . . . . . . . . . . . . . 17 2.2.2 Heston model - stochastic volatility model . . . . . . . . . . . . . . . . . 22 II FIX & LIQ market fear and the optimal trading time for derivatives markets 27 1 Introduction 29 2 Measuring market risk via the implied volatility index VIX 33 2.1 The model-free estimator for volatility: the VIX . . . . . . . . . . . . . . . . . 35 2.2 VIX step by step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 Measuring liquidity risk via the implied liquidity index LIQ 41 3.1 Coherent risk measure, distortion functions and distorted expectations . . . . . 41 3.2 Bid and ask pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 LIQ step by step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Measuring herd-behavior via the comonotonicity index CIX 51 4.1 Theory of comonotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Comonotonic upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2.1 The Comonotonicity ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3 COM step by step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5 Numerical results 63 5.1 VIX, LIQ and CIX - results for 2007-2009 . . . . . . . . . . . . . . . . . . . . . 63 5.2 FIX - The Fear Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 LIQ - determining the optimal trading time for derivative markets . . . . . . . 68 6 Summary and future work 71 III COCOs new asset class - structure, pricing, sensitivity 73 1 Introduction 75 2 Definition, structure and foundations 79 2.1 What is a CoCo? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.2 Risk profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.3 CoCo like products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.4 CoCo dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.5 CoCo investors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.6 Basel III and CoCos in the market . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.7 Pro and cons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3 CoCo anatomy 87 3.1 Trigger event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.2 Loss absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.3 Conversion types and price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.4 Host instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4 Pricing techniques 93 4.1 Credit derivatives approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Equity derivatives approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5 Barclays case study and implied barrier 101 5.1 The data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2 Weighting scheme and calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.2.1 Weighting scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2.2 Calibration of the Black-Scholes model . . . . . . . . . . . . . . . . . . . 106 5.2.3 Calibration of the Heston Model . . . . . . . . . . . . . . . . . . . . . . 108 5.2.4 Calibration on CDS data . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.3 Implied barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.3.1 Credit derivatives approach . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.3.2 Equity derivatives approach . . . . . . . . . . . . . . . . . . . . . . . . . 113 6 Sensitivity analysis of CoCo price 115 6.1 The current Barclays’ CoCo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Distressed versus non-distressed situation . . . . . . . . . . . . . . . . . . . . . 118 6.3 Turning point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.4 Skew . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7 Summary and future work 125nrpages: 155status: publishe

FIX - The fear index. Measuring market fear

In this paper, we propose a new fear index based on (equity) option surfaces of an index and its components. The quanti¯cation of the fear level will be solely based on option price data. The index takes into account market risk via the VIX volatility barometer, liquidity risk via the concept of implied liquidity, and systemic risk and herd-behavior via the concept of comonotonicity. It thus allows us to measure an overall level of fear (excluding credit risk) in the market as well as to identify precisely the individual importance of the distinct risk components (market, liquidity or systemic risk). As a side result we also derive an upperbound for the VIX.status: publishe