ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi‐ or single‐asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi‐asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single‐asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi‐asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi‐asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi‐asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.     

A MULTINOMIAL APPROXIMATION FOR AMERICAN OPTION PRICES IN LÉVY PROCESS MODELS

This paper gives a tree-based method for pricing American options in models where the stock price follows a general exponential Lévy process. A multinomial model for approximating the stock price process, which can be viewed as generalizing the binomial model of Cox, Ross, and Rubinstein (1979) for geometric Brownian motion, is developed. Under mild conditions, it is proved that the stock price process and the prices of American-type options on the stock, calculated from the multinomial model, converge to the corresponding prices under the continuous time Lévy process model. Explicit illustrations are given for the variance gamma model and the normal inverse Gaussian process when the option is an American put, but the procedure is applicable to a much wider class of derivatives including some path-dependent options. Our approach overcomes some practical difficulties that have previously been encountered when the Lévy process has infinite activity.     

ESTIMATION OF VALUE AT RISK AND RUIN PROBABILITY FOR DIFFUSION PROCESSES WITH JUMPS

In this paper we give upper bounds for both the Value at Risk   VaR _α,  0 < α < 1  , and for ruin probabilities associated with the supremum of a process driven by a Brownian motion and a compound Poisson process. We obtain lower bounds for the same Value at Risk, and for different cases we discuss the behavior of the bounds for small α. We prove our bounds are asymptotically optimal, as α tends to zero. The ruin probabilities obtained are related to other bounds found in recent literature.     

解读我国的对外贸易依存度

外贸依存度反映的是一国经济对他国经济或对世界经济相互依赖的程度。本文在准确把握外贸依存度内涵的基础上，指出我国外贸依存度的现状和特征，进而分析了外贸依存度被高估的因素，最后提出我国今后为降低外贸依存度应采取的经济发展战略。     

DIFFUSION MODELS FOR EXCHANGE RATES IN A TARGET ZONE

We present two analytically tractable diffusion models for an exchange rate in a target zone. One model generalizes a model proposed by De Jong, Drost, and Werker (2001) to allow asymmetry between the currencies which is often an important feature of data. Estimation of the model parameters by the method of Kessler and Sørensen (1999) using eigenfunctions of the generator is investigated and shown to give well-behaved estimators that are easy to calculate. The method is well suited to the models because the eigenfunctions are known so that explicit estimating functions are obtained, and because the state space is a finite interval, for which it is known that the method can be made arbitrarily efficient by including sufficiently many eigenfunctions. The model fits data on exchange rates in the European Monetary System well. In particular, the asymmetry parameter is significantly different from zero for three out of four currencies. An alternative diffusion model is presented with similarly nice properties, but with different dynamics that allow constant volatility near the boundaries of the target zone. No-arbitrage pricing of derivative assets is considered, and the effect of realignments is briefly discussed.     

中国未来不同生育水平下的经济增长后果比较研究

对中国未来90年不同生育水平下的经济增长后果进行了人口-经济动态模拟。在生育水平过低导致劳动力减少过快、人口老龄化过重、劳动负担加重的情况下,将使经济增长大大放缓;而较高的生育水平下,虽然经济增长速度略快,但是人均GDP增长速度慢于中方案生育水平下的经济增长,并且人均GDP水平也具有较大差异。完善当前生育政策,使生育水平稳定在1.9-2.0之间,如此人口在本世纪缓慢地减少也将有利于我国的经济增长和人均生活水平的提高。同时,在低生育水平下,依靠劳动增加和资本积累的粗放型经济增长将不复存在,经济发展方式转变是必然选择,技术创新、技术进步将是未来经济增长首要源泉。     

Hedging and Portfolio Optimization in Financial Markets with a Large Trader

We introduce a general continuous-time model for an illiquid financial market where the trades of a single large investor can move market prices. The model is specified in terms of parameter-dependent semimartingales, and its mathematical analysis relies on the nonlinear integration theory of such semimartingale families. The Itô–Wentzell formula is used to prove absence of arbitrage for the large investor, and, using approximation results for stochastic integrals, we characterize the set of approximately attainable claims. We furthermore show how to compute superreplication prices and discuss the large investor's utility maximization problem.     

Error Calculus and Path Sensitivity in Financial Models

In the framework of risk management, for the study of the sensitivity of pricing and hedging in stochastic financial models to changes of parameters and to perturbations of the stock prices, we propose an error calculus that is an extension of the Malliavin calculus based on Dirichlet forms. Although useful also in physics, this error calculus is well adapted to stochastic analysis and seems to be the best practicable in finance. This tool is explained here intuitively and with some simple examples.     

TIME‐CHANGED MARKOV PROCESSES IN UNIFIED CREDIT‐EQUITY MODELING

This paper develops a novel class of hybrid credit‐equity models with state‐dependent jumps, local‐stochastic volatility, and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time‐changed Markov diffusion process with state‐dependent local volatility and killing rate (default intensity). When the time change is a Lévy subordinator, the stock price process exhibits jumps with state‐dependent Lévy measure. When the time change is a time integral of an activity rate process, the stock price process has local‐stochastic volatility and default intensity. When the time change process is a Lévy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state‐dependent jumps, local‐stochastic volatility, and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the time‐changed process is expressed in closed form as a single integral in the complex plane. If the payoff is square integrable, the complex integral is further reduced to a spectral expansion. To illustrate our general framework, we time change the jump‐to‐default extended constant elasticity of variance model of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local‐stochastic volatility, and default intensity. These models can be used to jointly price equity and credit derivatives.