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Measuring the degree of herd behavior in financial markets.
The main goal of this dissertation is to measure the extent to which stock prices will move together in the future. In Part III we propose two different ways to measure herd behavior between stock prices: a model-free and a model-based approach. Measuring co-movement in a model-free waycan be considered as an additional market stress indicator besides the well-known volatility estimates like the VIX. The model-based approach can be considered as an improvement of the well-known implied correlation.
Part I:Comparing risks, equality in distribution and </>comonotonicity</>
In Part I of this dissertation, we investigate how the random sum of the components of a random vector can be used toreveal information about the multivariate nature of the random vector. In Part III, these results help us to construct herd behavior measures based on the random sum. Chapter 3 introduces some important integral stochastic orders. For example, we consider the convex and supermodular orders, which will play an important role throughout this dissertation. An interesting result in this chapter states that the expected utility of arandom variable can be expressed as a mixture of upper and lower tail transforms of this random variable. In Chapter 6 we derive a similar result for the distorted expectation of a random variable: the distorted expectation of a random variable can be expressed as a mixture of Tail Values-at-Risk of this random variable. These representations for the expected utility and the distorted expectation will be used in Chapter 7 to show that under the appropriate conditions, one can prove an equality in distribution by comparing expected utilities or distorted expectations.
The concept of comonotonicity is considered in Chapter4. Loosely speaking, comonotonicity refers to a situation where all random variables are non-decreasing functions of only one random source, i.e. they will move in the same direction. The distribution function andthe stop-loss premiums of a sum of comonotonic random variables can be determined in an analytical form, which makes the comonotonic sum an attractive random variable; see Chapter 5. We finish the first part of thisdissertation with a set of characterizations of comonotonicity based onthe distribution of the sum.
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Part II: Pricing index options</>
An index option is an option which has as underlying a basketof stocks. The dependence structure between the different stocks makes it a hard task to derive closed form solutions for the price of such an exotic option. In this dissertation we consider two different approaches.
Chapter 9 derives a static super-replicating strategy for thepay-off of a basket option. This strategy uses only traded vanilla options and is model-free. We consider the finite market case</>, where the prices of the options on the stocks of which the index is composed are available for a finite number of strikes. We introduce a framework where the super-replicating strategy for an index call and an index put is derived at the same time. We prove that such an integrated approachgives rise to an efficient algorithm which is fast to calculate.
Even if the individual stocks composing the basket can be described by the celebrated Black & Scholes model, the price of an index option is still hard to determine. In Chapter 11 we derive an upper and lowerbound for this price, based on the theory of comonotonicity. A linear
combination of these two bounds results in a close approximation for the price of the index option.
</>
Part III:Measuring herd behavior in stock markets</>
The last part of this dissertation deals with the problem of measuring dependence between stock prices. Chapter 12and Chapter 13 construct a family of dependence measures, based on the theory of comonotonicity and the information content of the random sum. Measuring dependence can become cumbersome when the number of r.v.s involved becomes large. Using the information contained in the random sum results in a tractable dependence measure for the average level of
co-movement. Chapter 12 uses swap rates on the index to determine the degree of herd behavior whereas Chapter 13 is based on distorted expectationsof the index. It will be shown that both the swap rate and the distorted expectation can be determined in a model-free way using the available option data. As result, the corresponding estimate for the degree of herd behavior is forward looking </>and model-free</>.
Chapter 14 considers a model-based approach to measure the dependence between asset prices. More precisely, we assume that the stocks can be described by a multivariate Black & Scholes model. Invertingthe pricing formula for index options results in an implied estimate for the mean level of correlation.
Part I:Comparing risks, equality in distribution and </>comonotonicity</>
In Part I of this dissertation, we investigate how the random sum of the components of a random vector can be used toreveal information about the multivariate nature of the random vector. In Part III, these results help us to construct herd behavior measures based on the random sum. Chapter 3 introduces some important integral stochastic orders. For example, we consider the convex and supermodular orders, which will play an important role throughout this dissertation. An interesting result in this chapter states that the expected utility of arandom variable can be expressed as a mixture of upper and lower tail transforms of this random variable. In Chapter 6 we derive a similar result for the distorted expectation of a random variable: the distorted expectation of a random variable can be expressed as a mixture of Tail Values-at-Risk of this random variable. These representations for the expected utility and the distorted expectation will be used in Chapter 7 to show that under the appropriate conditions, one can prove an equality in distribution by comparing expected utilities or distorted expectations.
The concept of comonotonicity is considered in Chapter4. Loosely speaking, comonotonicity refers to a situation where all random variables are non-decreasing functions of only one random source, i.e. they will move in the same direction. The distribution function andthe stop-loss premiums of a sum of comonotonic random variables can be determined in an analytical form, which makes the comonotonic sum an attractive random variable; see Chapter 5. We finish the first part of thisdissertation with a set of characterizations of comonotonicity based onthe distribution of the sum.
</>
Part II: Pricing index options</>
An index option is an option which has as underlying a basketof stocks. The dependence structure between the different stocks makes it a hard task to derive closed form solutions for the price of such an exotic option. In this dissertation we consider two different approaches.
Chapter 9 derives a static super-replicating strategy for thepay-off of a basket option. This strategy uses only traded vanilla options and is model-free. We consider the finite market case</>, where the prices of the options on the stocks of which the index is composed are available for a finite number of strikes. We introduce a framework where the super-replicating strategy for an index call and an index put is derived at the same time. We prove that such an integrated approachgives rise to an efficient algorithm which is fast to calculate.
Even if the individual stocks composing the basket can be described by the celebrated Black & Scholes model, the price of an index option is still hard to determine. In Chapter 11 we derive an upper and lowerbound for this price, based on the theory of comonotonicity. A linear
combination of these two bounds results in a close approximation for the price of the index option.
</>
Part III:Measuring herd behavior in stock markets</>
The last part of this dissertation deals with the problem of measuring dependence between stock prices. Chapter 12and Chapter 13 construct a family of dependence measures, based on the theory of comonotonicity and the information content of the random sum. Measuring dependence can become cumbersome when the number of r.v.s involved becomes large. Using the information contained in the random sum results in a tractable dependence measure for the average level of
co-movement. Chapter 12 uses swap rates on the index to determine the degree of herd behavior whereas Chapter 13 is based on distorted expectationsof the index. It will be shown that both the swap rate and the distorted expectation can be determined in a model-free way using the available option data. As result, the corresponding estimate for the degree of herd behavior is forward looking </>and model-free</>.
Chapter 14 considers a model-based approach to measure the dependence between asset prices. More precisely, we assume that the stocks can be described by a multivariate Black & Scholes model. Invertingthe pricing formula for index options results in an implied estimate for the mean level of correlation.
Date:1 Oct 2009 → 27 Jun 2013
Keywords:index option prices, comonotonicity, model-free index, co-movement, supermodular order, dependence, Herd behavior, VIX index, HIX index, market fear
Disciplines:Applied economics
Project type:PhD project