中美微积分教学改革的比较研究

本文首先介绍了中美两国微积分教学改革的主要背景,特别是美国微积分教学改革的背景,并详细地说明了中美两国微积分教学改革的主要做法.通过对两国微积分教学改革的比较,找出我国在微积分教学中的差距,起到对我国高校微积分教学改革的启示和借鉴作用.     

A General Approach to Hedging Options: Applications to Barrier and Partial Barrier Options

In this paper we consider a Black and Scholes economy and show how the Malliavin calculus approach can be extended to cover hedging of any square integrable contingent claim. As an application we derive the replicating portfolios of some barrier and partial barrier options.     

A return to Bentham's felicific calculus: From moral welfarism to technical non-welfarism

Abstract

This paper is a study of Bentham's felicific calculus. Challenging a view that contrasts ‘cardinal’ to ‘ordinal’ calculus, we show that these two forms of calculus constitute instead different phases of a single approach. Bentham sometimes has to rely upon proxy variables because of operational constraints, and consequently upon factors other than utility. As a utilitarian, Bentham is de facto welfarist from an ethical point of view. Surprisingly however, this study shows that Bentham resorts to non-welfare information in the actual application of his calculus. His approach reconciles non-welfarism from the technical standpoint with welfarism as a fundamental moral principle.     

Additional utility of insiders with imperfect dynamical information

In this paper we consider a market driven by a Wiener process where there is an insider and a regular trader. The insider has privileged information which has been deformed by an independent noise vanishing as the revelation time approaches. At this time, the information of every trader is the same. We obtain the semimartingale decomposition of the original Wiener process under dynamical enlargement of the filtration, and we prove that if the rate at which the additional noise in the insiders information vanishes is slow enough then there is no arbitrage and the additional utility of the insider is finite.Received: 1 October 2003, Mathematics Subject Classification: 60G48, 90A09, 60H07, 90A60JEL Classification: D82, G11, G14     

A generalization of the Hull and White formula with applications to option pricing approximation

By means of Malliavin calculus we see that the classical Hull and White formula for option pricing can be extended to the case where the volatility and the noise driving the stock prices are correlated. This extension will allow us to describe the effect of correlation on option prices and to derive approximate option pricing formulas.A previous version of this paper has benefited from helpful comments by two anonymous referees.     

Binary funding impacts in derivative valuation

We discuss the binary nature of funding impact in derivative valuation. Under some conditions, funding is either a cost or a benefit, that is, one of the lending/borrowing rates does not play a role in pricing derivatives. When derivatives are priced, considering different lending/borrowing rates leads to semilinear backward stochastic differential equations (BSDEs) and partial differential equation (PDEs), and thus it is necessary to solve the equations numerically. However, once it can be guaranteed that only one of the rates affects pricing, linear equations can be recovered, and analytical formulae can be derived. Moreover, as a by‐product, our results explain how debt value adjustment (DVA) and funding benefits are dissimilar. It is often believed that considering both DVA and funding benefits results in a double‐counting issue but it will be shown that the two components are affected by different mathematical structures of derivative transactions. We find that funding benefit is related to the decreasing property of the payoff function, but this relationship decreases as the funding choices of underlying assets are transferred to repo markets.     

Model-free CPPI

We consider Constant Proportion Portfolio Insurance (CPPI) and its dynamic extension, which may be called Dynamic Proportion Portfolio Insurance (DPPI). It is shown that these investment strategies work within the setting of Föllmer's pathwise Itô calculus, which makes no probabilistic assumptions whatsoever. This shows, on one hand, that CPPI and DPPI are completely independent of any choice of a particular model for the dynamics of asset prices. They even make sense beyond the class of semimartingale sample paths and can be successfully defined for models admitting arbitrage, including some models based on fractional Brownian motion. On the other hand, the result can be seen as a case study for the general issue of robustness in the face of model uncertainty in finance.     

OPTIMAL LIQUIDATION OF DERIVATIVE PORTFOLIOS

We consider the problem facing a risk averse agent who seeks to liquidate or exercise a portfolio of (infinitely divisible) perpetual American style options on a single underlying asset. The optimal liquidation strategy is of threshold form and can be characterized explicitly as the solution of a calculus of variations problem. Apart from a possible initial exercise of a tranche of options, the optimal behavior involves liquidating the portfolio in infinitesimal amounts, but at times which are singular with respect to calendar time. We consider a number of illustrative examples involving CRRA and CARA utility, stocks, and portfolios of options with different strikes, and a model where the act of exercising has an impact on the underlying asset price.