Distorted expectations can be expressed as weighted averages of quantiles. In this note, we show that this statement is essentially true, but that one has to be careful with the correct formulation of it. Furthermore, the proofs of the additivity property for distorted expectations of a comonotonic sum that appear in the literature often do not cover the case of a general distortion function. We present a straightforward proof for the general case, making use of the appropriate expressions for distorted expectations in terms of quantiles.
We introduce a new and easy-to-calculate measure for the expected degree of herd behavior or co-movement between stock prices. This forward looking measure is model-independent and based on observed option data. It is baptized the Herd Behavior Index (HIX).The degree of co-movement in a stock market can be determined by comparing the observed market situation with the extreme (theoretical) situation under which the whole system is driven by a single factor. The HIX is then de…ned as the ratio of an option-based estimate of the risk-neutral variance of the market index and an option-based estimate of the corresponding variance in case of the extreme single factor market situation.The HIX can be determined for any market index provided an appropriate series of vanilla options is traded on this index as well as on its components. As an illustration, we determine historical values of the 30-days HIX for the Dow Jones
We consider the pricing of American-type basket derivatives by numerically solving a partial differential equation (PDE). The curse of dimensionality inherent to basket derivative pricing is circumvented by using the theory of comonotonicity. We start with deriving a PDE for the Europeantype comonotonic basket derivative price, together with a unique self-financing hedging strategy. We show how to use the results for the comonotonic market to approximate American-type basket derivative prices for a basket with correlated stocks. Our methodology generates American basket option prices which are in line with the prices obtained via the standard Least-Square Monte Carlo approach. Moreover, the numerical tests illustrate the performance of the proposed method in terms of computation time, and highlight some deficiencies of the standard LSM method.
A basket option is an option whose underlying is a portfolio of individual stock prices. Due to the unknown dependence structure between stocks, basket option pricing relies in general on approximations or numerical methods like Monte Carlo simulation. We propose a methodology for pricing basket options in a multivariate Variance Gamma model. The stock prices composing the basket are then modeled by time changed geometric Brownian motions with a common Gamma subordinator. Using the additivity property of comonotonic stop-loss premiums together with Gauss-Laguerre polynomials, we derive a closed-form expression for the basket option price as a linear combination of Black & Scholes prices. This technique manages to approximate the real basket option price in an accurate way. Furthermore, our new basket option pricing formula enables us to calibrate the multivariate VG model in a fast way provided option quotes on the components and the basket itself are available. As an illustration, we show that the multivariate VG model can closely match the observed Dow Jones index options.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.