抛物型扩散模型参数识别的正则化方法

经济计量模型参数识别是对观察的经济数据进行分析而得到经济结果的方法。从数学上讲属于反问题,而这类问题大多是不适定的。本文针对一类典型的抛物型扩散模型采用处理不适定问题的Tikhonov正则化方法来求解。即采用先离散化后正则化的策略利用在误差水平已知的情况下具有三阶收敛速率的算法来处理数值微分,并基于一种快速选择正则参数的混合算法计算正则解。同时对待辨识的分布参数线性和非线性依赖于观测数据两种情况分别进行了数值试验。     

A PDE method for estimation of implied volatility

In this paper it is proved that the Black–Scholes implied volatility satisfies a second order non-linear partial differential equation. The obtained PDE is then used to construct an algorithm for fast and accurate polynomial approximation for Black–Scholes implied volatility that improves on the existing numerical schemes from literature, both in speed and parallelizability. We also show that the method is applicable to other problems, such as approximation of implied Bachelier volatility.     

Non-Linear Value-at-Risk

Value-at-risk methods which employ a linear ("delta only") approximationto the relation between instrument values and the underlyingrisk factors are unlikely to be robust when applied to portfolioscontaining non-linear contracts such as options. The most widelyused alternative to the delta-only approach involves revaluingeach contract for a large number of simulated values of theunderlying factors. In this paper we explore an alternativeapproach which uses a quadratic approximation to the relationbetween asset values and the risk factors. This method (i) islikely to be better adapted than the linear method to the problemof assessing risk in portfolios containing non-linear assets,(ii) is less computationally intensive than simulation usingfull-revaluation and (iii) in common with the delta-only method,operates at the level of portfolio characteristics (deltas andgammas) rather than individual instruments.     

Non-linear Dependence in Stock Returns: Does Trading Frequency Matter?

The UK has a quote-driven pure dealer market structure that is very different from order driven markets such as the NYSE and Japanese markets. This paper investigates non-linear dependence in stock returns for an exhaustive sample of UK stocks for a 21 year period. The results are analysed on the basis of trading frequency. It is found that non-linear dependence is highly significant in all cases for both individual stocks and stock portfolios formed on the basis of trading frequency. The non-linear dependence is primarily over a one day interval, although statistically significant non-linear dependence exists consistently even up to five trading days. Most of the non-linear dependence is in the form of ARCH-type conditional heteroskedasticity. However, statistically significant non-linearity in addition to an EGARCH(1,1) dependence also appears to be present. This additional non-linearity is greater for individual stocks than for portfolios and greater for smaller, less-liquid portfolios. Non-linear dependence does not appear to be caused by non-stationarity in underlying economic fundamentals or by non-linearity in the conditional mean. However, low dimensional chaos is not generally supported. The limited evidence on chaotic behaviour is stronger for portfolios with long price adjustment delays across component stocks. The main results are consistent with US studies on stock indices, suggesting that the process generating non-linear dependence is not dependent on market microstructure characteristics.     

An alternative approach to the valuation of American options and applications

In this paper we examine the structure of American option valuation problems and derive the analytic valuation formulas under general underlying security price processes by an alternative but intuitive method. For alternative diffusion processes, we derive closed-form analytic valuation formulas and analyze the implications of asset price dynamics on the early exercise premiums of American options. In this regard, we introduce useful and interesting diffusion processes into American option-pricing literature, thus providing a wide range of choices of pricing models for various American-type derivative assets. This work offers a useful analytic framework for empirical testing and practical applications such as the valuation of corporate securities and examining the impact of options trading on market micro-structure.     

Follow the money: The monetary roots of bubbles and crashes

A reduced form model for the join dynamics of liquidity and asset prices is proposed. The self-reinforcing feedback between credit creation and the market value of the financial assets employed as collateral in the bank loans (the so called financial accelerator) is modeled by a coupled non-linear stochastic process. We show that such non-linear interaction produces explosive dynamics in the financial variables announcing a regime change in finite time in the form of a market crash which can also be modeled by the same coupled non-linear stochastic process with inverted signs. Casting the financial accelerator dynamics into a highly stylized macroeconomic model, we study its macro-dynamics implications for real economy and for monetary policy interventions. Finally, by exploiting the implications of the proposed model on the dynamics of financial asset returns, we introduce an extension of the GARCH process, that can provide an early warning identification of bubbles.     

Non‐linear Predictability of Value and Growth Stocks and Economic Activity

Recent empirical evidence suggests that stock market index returns are predictable from a variety of financial and macroeconomic variables. We extend this research by examining value and growth portfolios constructed by book-to-market ratio, and consider whether such predictability is evident here. Further, we assess whether such predictability is better characterised by a non-linear form and whether such non-linear predictability can be exploited to provide superior forecasts to those obtained from a linear model. General non-linearities are examined using non-parametric techniques, which suggest possible threshold behaviour. This leads to estimation of a smooth-transition threshold model, with the results indicating an improved in-sample performance and marginally superior out-of-sample forecast results.     

Pricing American Put Options on Defaultable Bonds

In the last two decades, the market of credit derivativeshas expanded rapidly, and the importance of pricing problemsfor credit derivatives has been recognized especially in the last decade.Among these securities, the pricing problems of credit derivativeswith an early exercise, such as American put options,have not received enough attention. In view of this need, this paper develops a continuous stochastic modelof American put options on defaultable bonds.The method of obtaining a solution is based on a new result of the optimalstopping problem for a diffusion process with a jump.Some characterizations of American put options are providedusing partial differential equations.     

A direct solution method for pricing options involving the maximum process

One often encounters options involving not only the stock price, but also its running maximum. We provide, in a fairly general setting, explicit solutions for optimal stopping problems concerned with a diffusion process and its running maximum. Our approach is to use excursion theory for Markov processes and rewrite the original two-dimensional problem as an infinite number of one-dimensional ones. Our method is rather direct without presupposing the existence of an optimal threshold or imposing a smooth-fit condition. We present a systematic solution method by illustrating it through classical and new examples.     

Non-Linear Value-at-Risk

Value-at-risk methods which employ a linear (delta only) approximation to the relation between instrument values and the underlying risk factors are unlikely to be robust when applied to portfolios containing non-linear contracts such as options. The most widely used alternative to the delta-only approach involves revaluing each contract for a large number of simulated values of the underlying factors. In this paper we explore an alternative approach which uses a quadratic approximation to the relation between asset values and the risk factors. This method (i) is likely to be better adapted than the linear method to the problem of assessing risk in portfolios containing non-linear assets, (ii) is less computationally intensive than simulation using full-revaluation and (iii) in common with the delta-only method, operates at the level of portfolio characteristics (deltas and gammas) rather than individual instruments.     

Recovery of volatility coefficient by linearization

Abstract

We study the problem of reconstruction of the asset price dependent local volatility from market prices of options with different strikes. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an explicit functional which is linear in perturbation of volatility. We obtain an integral equation for this functional and we show that under some natural conditions it can be inverted for volatility. We demonstrate the stability of the linearized problem, and we propose a numerical algorithm which is accurate for volatility functions with different properties.     

A quantum derivation of a reputational risk premium

Using quantum modeling, we propose a novel approach to reputational risk management arguing that taking care of corporate reputation can be considered a coalitional strategy framed into a quantum game theory schema. Following a stochastic mechanics approach, and assuming that the revenues of a firm can be modeled as an Ornstein–Uhlenbeck process represented by the well-known Langevin equation for the diffusion of a particle with unit mass under non-linear friction, we offer a mathematical derivation for the discount rate of a firm that does not manage its reputational risk and for a company that optimally manages its reputational risk. By comparison, we derive an analytical expression for the reputational risk premium. Our approach provides useful managerial and financial implications, suggesting that the use of the quantum ideology may result useful in enlarging the body of knowledge of corporate management.     

Estimation of hyperbolic diffusion using the Markov chain Monte Carlo method

In this paper we propose a Bayesian method to estimate the hyperbolic diffusion model. The approach is based on the Markov chain Monte Carlo (MCMC) method with the likelihood of the discretized process as the approximate posterior likelihood. We demonstrate that the MCMC method Provides a useful tool in analysing hyperbolic diffusions. In particular, quantities of posterior distributions obtained from the MCMC outputs can be used for statistical inference. The MCMC method based on the Milstein scheme is unsatisfactory. Our simulation study shows that the hyperbolic diffusion exhibits many of the stylized facts about asset returns documented in the discrete-time financial econometrics literature, such as the Taylor effect, a slowly declining autocorrelation function of the squared returns, and thick tails.     

A fully consistent,minimal model for non-linear market impact

We propose a minimal theory of non-linear price impact based on the fact that the (latent) order book is locally linear, as suggested by reaction–diffusion models and general arguments. Our framework allows one to compute the average price trajectory in the presence of a meta-order that consistently generalizes previously proposed propagator models. We account for the universally observed square-root impact law, and predict non-trivial trajectories when trading is interrupted or reversed. We prove that our framework is free of price manipulation and that prices can be made diffusive (albeit with a generic short-term mean-reverting contribution). Our model suggests that prices can be decomposed into a transient ‘mechanical’ impact component and a permanent ‘informational’ component.     

A variance reduction technique based on integral representations

Abstract

Standard Monte Carlo methods can often be significantly improved with the addition of appropriate variance reduction techniques. In this paper a new and powerful variance reduction technique is presented. The method is based directly on the Itô calculus and is used to find unbiased variance-reduced estimators for the expectation of functionals of Itô diffusion processes. The approach considered has wide applicability: for instance, it can be used as a means of approximating solutions of parabolic partial differential equations or applied to valuation problems that arise in mathematical finance. We illustrate how the method can be applied by considering the pricing of European-style derivative securities for a class of stochastic volatility models, including the Heston model.     

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 .     

Limits to growth rates in an ethereal economy

It has been argued that economic growth can continue despite the finite nature of the Earth and its ecological systems if growth is concentrated in an ethereal economy where ideas and information dominate over physical inputs. In this paper, we agree that in a sustainable society continued growth must eventually be concentrated in the ethereal economy; however, we argue that such growth cannot occur at the ongoing exponential rate that currently underpins the constant rate of returns relied upon within our economies. As there is a limit to how fast a population can adopt new ideas, and as such adoption and innovation itself occurs in unpredictable bursts, growth in an ethereal economy will follow a model of punctuated equilibrium composed of exponential bursts, logistic growth, and stable/stagnating periods in a manner similar to ecological evolutionary processes. Although such an economic environment is likely far in the future, lessons in not overtaxing ecological capital and encouraging information dissemination and knowledge diffusion are applicable to problems we face today.     

Planes,Trains, and Automobiles: The Impact of Traffic Noise on House Prices

Because of large planned infrastructural projects like expansion of the main airport and construction of high-speed railways, noise nuisance has become a national social topic in the Netherlands. Moreover, according to EU-guidelines, determination and enforcement of differentiated noise limits will be delegated from national to local governments in the near future. The value of noise has never been this important. In this paper, we estimate the non-linear impact of traffic noise on property prices. The used data set is very extensive; over 100,000 sales transactions are studied, with many individual property characteristics, combined with noise levels for 2 million small 100 by 100 m areas. We use spatial autocorrelation techniques to overcome the regular problems of traditional NIMBY-analysis performed by hedonic regression. In a rising market, we find that the impact of traffic noise ranges to 12 percent, with an average of about 5 percent. The discount varies across sub-markets, and is a non-linear function of the noise level.     

Optimal portfolio liquidation in target zone models and catalytic superprocesses

We study optimal buying and selling strategies in target zone models. In these models, the price is modelled by a diffusion process which is reflected at one or more barriers. Such models arise, for example, when a currency exchange rate is kept above a certain threshold due to central bank interventions. We consider the optimal portfolio liquidation problem for an investor for whom prices are optimal at the barrier and who creates temporary price impact. This problem is formulated as the minimization of a cost–risk functional over strategies that only trade when the price process is located at the barrier. We solve the corresponding singular stochastic control problem by means of a scaling limit of critical branching particle systems, which is known as a catalytic superprocess. In this setting, the catalyst is given by the barriers of the price process. For the cases in which the unaffected price process is a reflected arithmetic or geometric Brownian motion with drift, we moreover give a detailed financial justification of our cost functional by means of an approximation with discrete-time models.     

A behavioural model of investor sentiment in limit order markets

By incorporating behavioural sentiment in a model of a limit order market, we show that behavioural sentiment not only helps to replicate most of the stylized facts in limit order markets simultaneously, but it also plays a unique role in explaining those stylized facts that cannot be explained by noise trading, such as fat tails in the return distribution, long memory in the trading volume, an increasing and non-linear relationship between trade imbalance and mid-price returns, as well as the diagonal effect, or event clustering, in order submission types. The results show that behavioural sentiment is an important driving force behind many of the well-documented stylized facts in limit order markets.