Mean-variance hedging for continuous processes: New proofs and examples

Let be a special semimartingale of the form and denote by the mean-variance tradeoff process of . Let be the space of predictable processes for which the stochastic integral is a square-integrable semimartingale. For a given constant and a given square-integrable random variable , the mean-variance optimal hedging strategy by definition minimizes the distance in between and the space . In financial terms, provides an approximation of the contingent claim by means of a self-financing trading strategy with minimal global risk. Assuming that is bounded and continuous, we first give a simple new proof of the closedness of in and of the existence of the F?llmer-Schweizer decomposition. If moreover is continuous and satisfies an additional condition, we can describe the mean-variance optimal strategy in feedback form, and we provide several examples where it can be computed explicitly. The additional condition states that the minimal and the variance-optimal martingale measures for should coincide. We provide examples where this assumption is satisfied, but we also show that it will typically fail if is not deterministic and includes exogenous randomness which is not induced by .     

Conditions on option prices for absence of arbitrage and exact calibration

Under the assumption of absence of arbitrage, European option quotes on a given asset must satisfy well-known inequalities, which have been described in the landmark paper of Merton [Merton, R., 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4 (1), 141–183]. If we further assume that there is no interest rate volatility and that the underlying asset continuously pays deterministic dividends, cross-maturity inequalities must also be satisfied by the bid and ask option prices.     

On a general class of one-factor models for the term structure of interest rates

We propose a general one-factor model for the term structure of interest rates which based upon a model for the short rate. The dynamics of the short rate is described by an appropriate function of a time-changed Wiener process. The model allows for perfect fitting of given term structure of interest rates and volatilities, as well as for mean reversion. Moreover, every type of distribution of the short rate can be achieved, in particular, the distribution can be concentrated on an interval. The model includes several popular models such as the generalized Vasicek (or Hull-White) model, the Black-Derman-Toy, Black-Karasinski model, and others. There is a unified numerical approach to the general model based on a simple lattice approximation which, in particular, can be chosen as a binomial or -nomial lattice with branching probabilities .     

A note on the existence of unique equivalent martingale measures in a Markovian setting

Simple sufficient conditions for the existence of a unique equivalent martingale measure are provided. Furthermore, these conditions give us a handle on situations where an equivalent martingale measure cannot exist. The existence of a unique equivalent martingale measure is of relevance to problems in mathematical finance. Two examples of models for which the question of existence was unresolved are studied. By means of our results existence of a unique equivalent measure up to an explosion time is proved.     

Continuous-time term structure models: Forward measure approach

The problem of term structure of interest rates modelling is considered in a continuous-time framework. The emphasis is on the bond prices, forward bond prices and so-called LIBOR rates, rather than on the instantaneous continuously compounded rates as in most traditional models. Forward and spot probability measures are introduced in this general set-up. Two conditions of no-arbitrage between bonds and cash are examined. A process of savings account implied by an arbitrage-free family of bond prices is identified by means of a multiplicative decomposition of semimartingales. The uniqueness of an implied savings account is established under fairly general conditions. The notion of a family of forward processes is introduced, and the existence of an associated arbitrage-free family of bond prices is examined. A straightforward construction of a lognormal model of forward LIBOR rates, based on the backward induction, is presented.     

Optional decomposition and Lagrange multipliers

Let be the set of equivalent martingale measures for a given process , and let be a process which is a local supermartingale with respect to any measure in . The optional decomposition theorem for states that there exists a predictable integrand such that the difference is a decreasing process. In this paper we give a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption.     

Bad beta good beta,state‐space news decomposition and the cross‐section of stock returns

This study employs an innovative market‐based approach, where return on equity (ROE) is employed as a proxy for cash‐flow news and a state‐space model is used for market news decomposition. We document that the bad beta good beta (BBGB) model of Campbell and Vuolteenaho (2004) explains about 30 per cent of the cross‐sectional variations in US stock returns. We also find that the BBGB model adequately explains the size effect leading to its superior performance in this area. Our method controls for the news decomposition method and market news proxies’ bias. We contribute to the literature by providing an alternative easy‐to‐implement and consistent market‐based method for news decomposition.     

Towards a general theory of bond markets

The main purpose of the paper is to provide a mathematical background for the theory of bond markets similar to that available for stock markets. We suggest two constructions of stochastic integrals with respect to processes taking values in a space of continuous functions. Such integrals are used to define the evolution of the value of a portfolio of bonds corresponding to a trading strategy which is a measure-valued predictable process. The existence of an equivalent martingale measure is discussed and HJM-type conditions are derived for a jump-diffusion model. The question of market completeness is considered as a problem of the range of a certain integral operator. We introduce a concept of approximate market completeness and show that a market is approximately complete iff an equivalent martingale measure is unique.     

On the range of options prices

In this paper we consider the valuation of an option with time to expiration and pay-off function which is a convex function (as is a European call option), and constant interest rate , in the case where the underlying model for stock prices is a purely discontinuous process (hence typically the model is incomplete). The main result is that, for “most” such models, the range of the values of the option, using all possible equivalent martingale measures for the valuation, is the interval , this interval being the biggest interval in which the values must lie, whatever model is used.     

On a Generalization of certain inequalities by Tchebychef and Jensen

Abstract

The inequalities dealt with in this paper were published by me in Danish in ?Matematisk Tidsskrift? B. 1925 p. 49¹. I was not at the time aware, that an inequality of similar form had already been given by JENSEN; but as the conditions under which his theorem is valid differ from mine and are, in fact, included in them, it appears that my investigation is not superfluous. I am reverting to the subject partly in order to clear up the relationship between JENSEN'S and my own results, partly to show, by means of a few examples, how these inequalities may with advantage be applied to actuarial problems.     

Volatility and covariation of financial assets: A high-frequency analysis

Using high frequency data for the price dynamics of equities we measure the impact that market microstructure noise has on estimates of the: (i) volatility of returns; and (ii) variance–covariance matrix of n assets. We propose a Kalman-filter-based methodology that allows us to deconstruct price series into the true efficient price and the microstructure noise. This approach allows us to employ volatility estimators that achieve very low Root Mean Squared Errors (RMSEs) compared to other estimators that have been proposed to deal with market microstructure noise at high frequencies. Furthermore, this price series decomposition allows us to estimate the variance covariance matrix of n assets in a more efficient way than the methods so far proposed in the literature. We illustrate our results by calculating how microstructure noise affects portfolio decisions and calculations of the equity beta in a CAPM setting.     

LIBOR and swap market models and measures

A self-contained theory is presented for pricing and hedging LIBOR and swap derivatives by arbitrage. Appropriate payoff homogeneity and measurability conditions are identified which guarantee that a given payoff can be attained by a self-financing trading strategy. LIBOR and swap derivatives satisfy this condition, implying they can be priced and hedged with a finite number of zero-coupon bonds, even when there is no instantaneous saving bond. Notion of locally arbitrage-free price system is introduced and equivalent criteria established. Stochastic differential equations are derived for term structures of forward libor and swap rates, and shown to have a unique positive solution when the percentage volatility function is bounded, implying existence of an arbitrage-free model with such volatility specification. The construction is explicit for the lognormal LIBOR and swap “market models”, the former following Musiela and Rutkowski (1995). Primary examples of LIBOR and swap derivatives are discussed and appropriate practical models suggested for each.     

经济增长论的发展与动向——关于均衡的安定性与非决定性

为什么相同的经济基础,相同的地理自然环境的条件下,有的国家经济能够突飞猛进地发展,有的国家却停滞不前甚至越来越差?到底是什么因素促进了经济发展,到底有哪些因素造成了发展的不均衡?很多经济学者正在用数理经济的方法试图说明这一问题。关于这方面的研究可以应用到关于我国的经济分析中,去说明经济发展不均衡的问题,并促进我们去探讨怎样才能促进发展缓慢地区的经济发展。本文介绍了关于经济增长论的非决定性分析的最新成果和尚待解决的问题。