Continuous-time perpetuities and time reversal of diffusions

We consider the problem of estimating the joint distribution of a continuous-time perpetuity and the underlying factors which govern the cash flow rate, in an ergodic Markovian model. Two approaches are used to obtain the distribution. The first identifies a partial differential equation for the conditional cumulative distribution function of the perpetuity given the initial factor value, which under certain conditions ensures the existence of a density for the perpetuity. The second (and more general) approach, using techniques of time reversal, identifies the joint law as the stationary distribution of an ergodic multidimensional diffusion. This latter approach allows efficient use of Monte Carlo simulation, as the distribution is obtained by sampling a single path of the reversed process.     

Solvable local and stochastic volatility models: supersymmetric methods in option pricing

In this paper we provide an extensive classification of one- and two-dimensional diffusion processes which admit an exact solution to the Kolmogorov (and hence Black–Scholes) equation (in terms of hypergeometric functions). By identifying the one-dimensional solvable processes with the class of integrable superpotentials introduced recently in supersymmetric quantum mechanics, we obtain new analytical solutions. In particular, by applying supersymmetric transformations on a known solvable diffusion process (such as the Natanzon process for which the solution is given by a hypergeometric function), we obtain a hierarchy of new solutions. These solutions are given by a sum of hypergeometric functions, generalizing the results obtained in a paper by Albanese et al. (Albanese, C., Campolieti, G., Carr, P. and Lipton, A., Black–Scholes goes hypergeometric. Risk Mag., 2001, 14, 99–103). For two-dimensional processes, more precisely stochastic volatility models, the classification is achieved for a specific class called gauge-free models including the Heston model, the 3?/?2-model and the geometric Brownian model. We then present a new exact stochastic volatility model belonging to this class.     

The Expected Discounted Penalty at Ruin for a Markov-Modulated Risk Process Perturbed by Diffusion

Abstract

A Markov-modulated risk process perturbed by diffusion is considered in this paper. In the model the frequencies and distributions of the claims and the variances of the Wiener process are influenced by an external Markovian environment process with a finite number of states. This model is motivated by the flexibility in modeling the claim arrival process, allowing that periods with very frequent arrivals and ones with very few arrivals may alternate. Given the initial surplus and the initial environment state, systems of integro-differential equations for the expected discounted penalty functions at ruin caused by a claim and oscillation are established, respectively; a generalized Lundberg’s equation is also obtained. In the two-state model, the expected discounted penalty functions at ruin due to a claim and oscillation are derived when both claim amount distributions are from the rational family. As an illustration, the explicit results are obtained for the ruin probability when claim sizes are exponentially distributed. A numerical example also is given for the case that two classes of claims are Erlang(2) distributed and of a mixture of two exponentials.     

A hyperbolic diffusion model for stock prices

In the present paper we consider a model for stock prices which is a generalization of the model behind the Black–Scholes formula for pricing European call options. We model the log-price as a deterministic linear trend plus a diffusion process with drift zero and with a diffusion coefficient (volatility) which depends in a particular way on the instantaneous stock price. It is shown that the model possesses a number of properties encountered in empirical studies of stock prices. In particular the distribution of the adjusted log-price is hyperbolic rather than normal. The model is rather successfully fitted to two different stock price data sets. Finally, the question of option pricing based on our model is discussed and comparison to the Black–Scholes formula is made. The paper also introduces a simple general way of constructing a zero-drift diffusion with a given marginal distribution, by which other models that are potentially useful in mathematical finance can be developed.     

我国证券投资基金、股票市场和债券市场的溢出风险测度——来自上海证券市场的证据

本文基于VAR-DCC-MGARCH模型分析了沪市基金指数与股票指数和国债指数的波动相关性和溢出效应,估计了三个市场的VaR,并通过失败检验法进行了验证,研究发现:基金市场与股票市场的条件相关系数一直呈现正向相关关系;股票市场与国债市场以及基金市场与国债市场的动态条件相关系数具有很强的时变特征,而且走势呈现相似性,但是统计检验显示基金市场与国债市场的相关性不明显。基金市场对自身和股票市场存在显著波动溢出效应,股票市场对国债市场和基金市场存在一定显著的波动溢出效应。在给定期望损失概率下,发现基金指数收益率的VaR波动最为剧烈;股票指数收益率的VaR变动风险与均值的比值是最高的;结合剔除其他市场波动影响的VaR发现,三个市场的风险承受度更高,可以接受更大的损失收益率。     

Pricing Exotic Options and American Options: A Multidimensional Asymptotic Expansion Approach

This paper introduces a new method for pricing exotic options whose payoff functions depend on several stochastic indices and American options in multidimensional models. This method is based on two ideas. One is an application of the asymptotic expansion method for the law of a multidimensional diffusion process. The other is the combination of the asymptotic expansion method and the method called backward induction. The author applies the method to the problems of pricing call options on the maximum of two assets in the CEV model, average strike options in the Black–Scholes model and American options in the Heston model. Numerical examples show practical effectiveness of the proposed method.     

Portfolio risk management in a data-rich environment

We study risk assessment using an optimal portfolio in which the weights are functions of latent factors and firm-specific characteristics (hereafter, diffusion index portfolio). The factors are used to summarize the information contained in a large set of economic data and thus reflect the state of the economy. First, we evaluate the performance of the diffusion index portfolio and compare it to both that of a portfolio in which the weights depend only on firm-specific characteristics and an equally weighted portfolio. We then use value-at-risk, expected shortfall, and downside probability to investigate whether the weights-modeling approach, which is based on factor analysis, helps reduce market risk. Our empirical results clearly indicate that using economic factors together with firm-specific characteristics helps protect investors against market?risk.     

An alternative approach to evaluating the agreement between financial markets

This research investigates that the price relationship between a stock index and its associated nearby futures markets can be explained by the cost-of-carry model using the concordance correlation (CC) coefficient in the US financial markets. The main purpose of this research is to confirm that the CC coefficient is an appropriate methodology to determine ex post arbitrage opportunities and to maximize ex ante arbitrage profits through the analysis of the price relationship derived from the cost-of-carry model. To increase the robustness of the results and to enable us to generalize our conclusions, this analysis is carried out in consideration of external uncertainty, including the marking-to-market procedure of futures contracts and the transaction cost on the stock index and its futures markets, under several assumptions related to the conditions of transactions. Examining transaction price data on the S&P 500 stock index and its futures markets shows that the CC coefficient gives a good result for ex ante arbitrage profits and is appropriate for analyzing the relationship between the observed stock index futures market price and its theoretical price derived from the cost-of-carry model.     

Bond pricing in a hidden Markov model of the short rate

Abstract. We consider a diffusion type model for the short rate, where the drift and diffusion parameters are modulated by an underlying Markov process. The underlying Markov process is assumed to have a stochastic differential driven by Wiener processes and a marked point process. The model for the short rate thus falls within the category of hidden Markov models.     

Equity-linked annuities with multiscale hybrid stochastic and local volatility

In recent times, hybrid underlying models have become an industry standard for the pricing of derivatives and other problems in finance. This paper chooses a hybrid stochastic and local volatility model to evaluate an equity-linked annuity (ELA), which is a sort of tax-deferred annuity whose credited interest is linked to an equity index. The stochastic volatility component of the hybrid model is driven by a fast mean-reverting diffusion process while the local volatility component is given by the constant elasticity of variance (CEV) model. Since contracts of the ELA usually have long maturities over 10 years, a slowly moving factor in the stochastic volatility of stock index is expected to play a significant role in the valuation of the ELA, and thus, it is added to the aforementioned model. Based on this multiscale hybrid model, an analytic approximate formula is obtained for the price of a European option in terms of the CEV probability density function and then the result is applied to the value of the point-to-point ELA. The formula leads to the dependence structure of the ELA price on the fast and slow scale stochastic volatility and the elasticity of variance.     

Asymptotics of the probability of minimizing ‘down-side’ risk under partial information

We consider minimizing the probability of falling below a target growth rate of the wealth process up to a time horizon T in an incomplete market model under partial information and then study the asymptotic behavior of the minimizing probability as T → ∞. This problem is closely related to an ergodic risk-sensitive stochastic control problem under partial information in the risk-averse case. Indeed, in our main theorem we relate the former problem to the latter as its dual. As a result we obtain an explicit expression for the limit value of the former problem in the case of linear Gaussian models.     

宏微观分析相结合的信贷风险预测模型研究

我国现有的信贷风险评估方法存在宏微观分析结合不紧密以及风险评估不全面的问题.本文在基于财务分析的企业风险评估模型基础上,构建了宏微观分析相结合的信贷风险预测模型.建模的主要工作包括:选择反映行业信贷风险的指标与反映宏观经济变化的指标;确定反映宏观经济变化的指标与反映行业信贷风险指标之间的函数关系;依据以上的关联函数和宏观经济指标预测值,计算行业信贷风险调整系数;据此对属于该行业的企业即期信贷风险指标进行调整,对企业的信贷风险进行前瞻性预测.作者还利用证券市场数据检验了上述模型的准确性,结果表明模型可以有效预测企业未来的信贷风险.     

A Two-Factor Model for Low Interest Rate Regimes

This paper derives a two-factor model for the term structure of interest rates that segments the yield curve in a natural way. The first factor involves modelling a non-negative short rate process that primarily determines the early part of the yield curve and is obtained as a truncated Gaussian short rate. The second factor mainly influences the later part of the yield curve via the market index. The market index proxies the growth optimal portfolio (GOP) and is modelled as a squared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, this study focuses on the difficult but important case where the short rate stays close to zero for a prolonged period of time. For the proposed model, an equivalent risk neutral martingale measure is neither possible nor required. Hence we use the benchmark approach where the GOP is chosen as numeraire. Fair derivative prices are then calculated via conditional expectations under the real world probability measure. Using this methodology we derive pricing functions for zero coupon bonds and options on zero coupon bonds. The proposed model naturally generates yield curve shapes commonly observed in the market. More importantly, the model replicates the key features of the interest rate cap market for economies with low interest rate regimes. In particular, the implied volatility term structure displays a consistent downward slope from extremely high levels of volatility together with a distinct negative skew. 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: G10, G13     

A model of discontinuous interest rate behavior,yield curves,and volatility

This paper develops an equilibrium model in which interest rates follow a discontinuous (generalized) gamma process. The gamma process has finite variation, takes an infinite number of “small” jumps in every interval, and includes the Wiener process as a limiting case. The gamma interest rate model produces yield curves that closely resemble those of diffusion models. But in contrast to diffusion models, the curvature of the yield curve does not directly depend on the true volatility of the interest rate process, but instead depends on a different risk-neutral volatility. The gamma model appears to fit the distribution of interest rates changes and the jump characteristics of interest rate paths. Empirical tests reject a diffusion model of interest rates in favor of the more general gamma model because daily interest rate innovations are highly leptokurtic. The author appreciates comments from George Constantinides, Jon Ingersoll, Herbert Johnson, Ray Rishel, and an anonymous referee, computational assistance from Kerry Back and Saikat Nandi, and support from Atlantic Asset Management. Any errors are the responsibility of the author.     

A risk-sensitive stochastic control approach to an optimal investment problem with partial information

We consider an infinite time horizon optimal investment problem where an investor tries to maximize the probability of beating a given index. From a mathematical viewpoint, this is a large deviation probability control problem. As shown by Pham (in Syst. Control Lett. 49: 295–309, 2003; Financ. Stoch. 7: 169–195, 2003), its dual problem can be regarded as an ergodic risk-sensitive stochastic control problem. We discuss the partial information counterpart of Pham (in Syst. Control Lett. 49: 295–309, 2003; Financ. Stoch. 7: 169–195, 2003). The optimal strategy and the value function for the dual problem are constructed by using the solution of an algebraic Riccati equation. This equation is the limit equation of a time inhomogeneous Riccati equation derived from a finite time horizon problem with partial information. As a result, we obtain explicit representations of the value function and the optimal strategy for the problem. Furthermore we compare the optimal strategies and the value functions in both full and partial information cases.

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On Viable Diffusion Price Processes of the Market Portfolio

The assumption that the market portfolio follows a specified diffusion process implies, in a simple equilibrium framework, that the representative individual must have a certain utility function which is identified in the paper. Not every diffusion process is viable, i.e., can be “endogenized” to be the market portfolio's price process in such an equilibrium model. The paper provides necessary and sufficient conditions for viability which imply that viable diffusion processes constitute a rather restricted family.     

A Semiparametric Two-Factor Term Structure Model

This article proposes a semiparametric two-factor term structuremodel based on a consol rate and the spread between a shortrate and the consol rate. The diffusion functions in both theconsol rate and spread processes are nonparametrically specifiedso that the model allows for maximal flexibility of diffusionfunctions in fitting into data. The drift function of the spreadprocess is specified as a mean-reverting function, while thedrift function of the consol rate process is left unrestricted.A nonparametric procedure is developed for estimating the diffusionfunctions. The asymptotic biases of the nonparametric estimatorsare quantified when the step of discretization is fixed, whilethe asymptotic distributions of the nonparametric estimatorsare derived when the step of discretization tends to zero. Thepricing and hedging performances of the model are evaluatedin a simulated economic environment. Results show that the modelperforms quite well in the simulated economy.