Interest rate swap pricing with default risk under variance gamma process

Under the assumption that the dynamic assets price follows the variance gamma process, we establish a new bilateral pricing model of interest rate swap by integrating the reduced form model for swap pricing and the structural model for default risk measurement. Our pricing model preserves the simplicity of the reduced form model and also considers the dynamic evolution of the counterparty assets price by incorporating with the structural model for default risk measurement. We divide the swap pricing framework into two parts, simplifying the pricing model relatively. Simulation results show that, for a one year interest rate swap, a bond spread of one hundred basis points implies a swap credit spread about 0.1054 basis point.     

A multinomial tree model for pricing credit default swap options

Among the traded credit derivatives, the market interest in credit default swap options (CDSwaptions) is enormous. We propose a multinomial tree model to price Bermudan CDSwaptions. Our basic rationale is that we distribute the occurring probability for each node in a branch proportional to the probability density function of the assumed (normal) distribution. Through this approach, without the need of solving a large number of equations simultaneously, only the first four moments are required to build an arbitrarily large N-branches tree. We also demonstrate the detailed model implementation procedure including the valuation and the estimation of critical prices through an empirical example in Tucker and Wei (J Fixed Income 15(1):88–95, 2005). Numerical results show that, in the valuation, the proposed multinomial tree model is accurate and can significantly save pricing time under the same degree of accuracy as the binomial tree model. In the estimation of critical prices, the results are less accurate than those in the valuation, but the relative errors are acceptable.     

The pricing of credit risky securities under stochastic interest rate model with default correlation

In this paper, we study the pricing of credit risky securities under a three-firms contagion model. The interacting default intensities not only depend on the defaults of other firms in the system, but also depend on the default-free interest rate which follows jump diffusion stochastic differential equation, which extends the previous three-firms models (see R.A. Jarrow and F.Yu (2001), S.Y.Leung and Y.K.Kwok (2005), A.Wang and Z.Ye (2011)). By using the method of change of measure and the technology (H. S.Park (2008), R.Hao and Z.Ye (2011)) of dealing with jump diffusion processes, we obtain the analytic pricing formulas of defaultable zero-coupon bonds. Moreover, by the “total hazard construction”, we give the analytic pricing formulas of credit default swap (CDS).     

A tractable LIBOR model with default risk

We develop a model for the dynamic evolution of default-free and defaultable interest rates in a LIBOR framework. Utilizing the class of affine processes, this model produces positive LIBOR rates and spreads, while the dynamics are analytically tractable under defaultable forward measures. This leads to explicit formulas for CDS spreads, while semi-analytical formulas are derived for other credit derivatives. Finally, we give an application to counterparty risk.     

Pricing model of interest rate swap with a bilateral default risk

Under the foundation of Duffie & Huang (1996) [7], this paper integrates the reduced form model and the structure model for a default risk measure, giving rise to a new pricing model of interest rate swap with a bilateral default risk. This model avoids the shortcomings of ignoring the dynamic movements of the firm’s assets of the reduced form model but adds only a little complexity and simplifies the pricing formula significantly when compared with Li (1998) [10]. With the help of the Crank-Nicholson difference method, we give the numerical solutions of the new model to study the default risk effects on the swap rate. We find that for a one year interest rate swap with the coupon paid per quarter, the variance of the default fixed rate payer decreases from 0.1 to 0.01 only causing about a 1.35%’s increase in the swap rate. This is consistent with previous results.     

An analytical formula for pricing m-th to default swaps

The computational effort of pricing an m-th to default swap depends highly on the size n of the underlying basket. Usually, n different default times are modeled, but in fact the valuation only depends on the m-th smallest default time of this tuple. In this paper we attain an analytical formula for the distribution of this m-th default time. With the help of this distribution we simplify the valuation problem from an n-dimensional quadrature to a one-dimensional quadrature and break the curse of dimensionality. Applications of this modification are efficient pricing of m-th to default swaps, estimation of sensitivities and pricing of European max/min options.     

债券定价及利率风险的市场价格的重构方法

假设利率变化的模型是由随机微分方程给出,则可以用推导Black-Scholes方程的方法来推出债券价格满足的偏微分方程,得到一个抛物型的偏微分方程.但是,在债券定价的方程中隐含有一个参数λ称为利率风险的市场价格.所谓债券定价的反问题,就是由不同到期时间的债券的现在价格来得到利率风险的市场价格.对随机利率模型下债券定价的正问题先给予介绍和差分数值求解方法,并介绍了反问题,且对反问题给出了数值方法.     

A fractional credit model with long range dependent default rate

Motivated by empirical evidence of long range dependence in macroeconomic variables like interest rates we propose a fractional Brownian motion driven model to describe the dynamics of the short and the default rate in a bond market. Aiming at results analogous to those for affine models we start with a bivariate fractional Vasicek model for short and default rate, which allows for fairly explicit calculations. We calculate the prices of corresponding defaultable zero-coupon bonds by invoking Wick calculus. Applying a Girsanov theorem we derive today’s prices of European calls and compare our results to the classical Brownian model.     

一般均衡下基于产品市场和资本市场的单名CDS定价

次贷危机呼吁新的信用衍生品定价模型, 因此为存在产品市场和资本市场的经济结构建立一般均衡的单名CDS定价模型, 使用最优化求解一般均衡下的商品价格和CDS价格. 可以发现一般均衡的CDS定价具有资本市场和产品市场的因素, 这表示CDS的价格不再是由单纯的资本市场因素决定的, 而是由无风险利率、资本产出弹性、违约率、回收率同时决定的. 通过数量约束用模拟的方式研究多个均衡的动态变化, 发现违约风险的增加使得价格剧烈波动且市场交易萎缩. 在为以中国工商银行为参考资产的CDS定价过程中, 发现各种因素在不同的时期都可能成为定价的主要影响因素. 可以发现, 次贷危机的定价体系存在着信用调整问题和定价与实体经济脱节的问题. 可以认为, 一般均衡下基于产品市场和资本市场的单名CDS定价可以囊括多个市场的交叉影响, 为衍生品定价提供一个新的方向.     

A non‐default rate regression model for credit scoring

In this paper, we propose a new non‐default rate survival model. Our approach enables different underlying activation mechanisms which lead to the event of interest. The number of competing causes, which may be responsible for the occurrence of the event of interest, is assumed to follow a geometric distribution, while the time to event is assumed to follow an inverse Weibull distribution. An advantage of our approach is to accommodate all activation mechanisms based on order statistics. We explore the use of maximum likelihood estimation procedure. Simulation studies are performed and experimental results are illustrated based on a real Brazilian bank personal loan portfolio data. Copyright © 2015 John Wiley & Sons, Ltd.     

基于指数效用无差异价值过程的不完备市场下的可违约期权的定价模型

本文研究不完备市场情况下的可违约期权的动态指数效用无差异定价。不同于大多数的可违约期权定价文献,本文没有假定鞅的不变性,即通常的H 假设,而是通过信息流的扩张和测度的变换,将信用风险敏感的资产转换为一个G 局部鞅,其后引入一个具体的倒向随机微分方程（BSDE）,并证明该方程解的存在性与唯一性;然后利用无差异价值过程C_t（B,α）在最小熵鞅测度下对一般的投资策略为上鞅,而在最优投资策略下为鞅的事实,证明无差异价值过程C_t（B,α）就是BSDE 的解,从而给出可违约期权的定价。     

A Bayesian approach to pricing longevity risk based on risk-neutral predictive distributions

We present a Bayesian approach to pricing longevity risk under the framework of the Lee-Carter methodology. Specifically, we propose a Bayesian method for pricing the survivor bond and the related survivor swap designed by Denuit et al. (2007). Our method is based on the risk neutralization of the predictive distribution of future survival rates using the entropy maximization principle discussed by Stutzer (1996). The method is illustrated by applying it to Japanese mortality rates.     

A mathematical model for asset pricing

Asset price dynamics is studied by using a system of ordinary differential equations which is derived by utilizing a new excess demand function introduced by Caginalp [4] for a market involving more information on demand and supply for a stock rather than their values at a particular price. Derivation is based on the finiteness of assets (rather than assuming unbounded arbitrage) in addition to investment strategies that are based on not only price momentum (trend) but also valuation considerations. For this new model and the older models which were extracted using the classical excess demand function by Caginalp and Balenovich [2] and [3], time evolutions of asset price are compared through numerical simulations.     

A note on utility based pricing and asymptotic risk diversification

In principle, liabilities combining both insurancial risks (e.g. mortality/longevity, crop yield,...) and pure financial risks cannot be priced neither by applying the usual actuarial principles of diversification, nor by arbitrage-free replication arguments. Still, it has been often proposed in the literature to combine these two approaches by suggesting to hedge a pure financial payoff computed by taking the mean under the historical/objective probability measure on the part of the risk that can be diversified. Not surprisingly, simple examples show that this approach is typically inconsistent for risk adverse agents. We show that it can nevertheless be recovered asymptotically if we consider a sequence of agents whose absolute risk aversions go to zero and if the number of sold claims goes to infinity simultaneously. This follows from a general convergence result on utility indifference prices which is valid for both complete and incomplete financial markets. In particular, if the underlying financial market is complete, the limit price corresponds to the hedging cost of the mean payoff. If the financial market is incomplete but the agents behave asymptotically as exponential utility maximizers with vanishing risk aversion, we show that the utility indifference price converges to the expectation of the discounted payoff under the minimal entropy martingale measure.     

On the total operating costs up to default in a renewal risk model

The paper proposes a new approach to study a general class of ruin-related quantities in the context of a renewal risk model. While the classical approaches in Sparre Andersen models have their own merits, the approach presented in this paper has its advantages from the following perspectives. (1) The underlying surplus process has the flexibility to reflect a broad range of scenarios for surplus growth including dividend policies and interest returns. (2) The solution method provides a general framework to unify a great variety of existing ruin-related quantities such as Gerber–Shiu functions and the expected present value of dividends paid up to ruin, and facilitates derivations of new ruin-related quantities such as the expected present value of total claim costs up to ruin, etc. In the end, many specific examples are explored to demonstrate its application in renewal risk models.     

No-arbitrage theorem for multi-factor uncertain stock model with floating interest rate

In the stock models, the prices of the stocks are usually described via some differential equations. So far, uncertain stock model with constant interest rate has been proposed, and a sufficient and necessary condition for it being no-arbitrage has also been derived. This paper considers the multiple risks in the interest rate market and stock market, and proposes a multi-factor uncertain stock model with floating interest rate. A no-arbitrage theorem is derived in the form of determinants, presenting a sufficient and necessary condition for the new stock model being no-arbitrage. In addition, a strategy for the arbitrage is provided when the condition is not satisfied.     

A simple plant-location model for quantity-setting firms subject to price uncertainty

The uncapacitated plant location problem under uncertainty is formulated in a mean-variance framework with prices in various markets correlated via their response to a common random factor. This formulation results in a mixed-integer quadratic programming problem. However, for a given integer solution, the resulting quadratic programming problem is amenable to a very simple solution procedure. The simplicity of this algorithm means that reasonably large problems should be solvable using existing branch-and-bound techniques.     

A pricing model for clearing end-of-season retail inventory

The problem of setting prices for clearing retail inventories of fashion goods is a difficult task that is further exacerbated by the fact that markdowns enacted near the end of the selling season have a smaller impact on demand. In this article, we present discrete-time models for setting clearance prices in such an environment. When demand is deterministic, we compute optimal prices and show that decreasing reservation prices lead to declining optimal prices. When demand is stochastic and arbitrarily correlated across planning periods, we obtain bounds on the optimal expected revenue and on optimal prices. We also develop a heuristic procedure for finding near-optimal prices and test its accuracy through numerical experiments. These experiments reveal new insights for practitioners. For example, the penalty for choosing clearance price once and keeping it unchanged for the remainder of the selling season is found to be small when either the mean reservation prices do not change appreciably over time or when they drop sharply after the first period.     

Forecasting model with asymmetric market response and its application to pricing of consumer package goods

This paper presents a dynamic forecasting model that accommodates asymmetric market responses to marketing mix variable—price promotion—by threshold models. As a threshold variable to generate a mechanism for different market responses, we use the counterpart to the concept of a price threshold applied to a representative consumer in a store. A Bayesian approach is taken for statistical modelling because of advantages that it offers over estimation and forecasting. The proposed model incorporates the lagged effects of a price variable. Thereby, myriad pricing strategies can be implemented in the time horizon. Their effectiveness can be evaluated using the predictive density. We intend to improve the forecasting performance over conventional linear time series models. Furthermore, we discuss efficient dynamic pricing in a store using strategic simulations under some scenarios suggested by an estimated structure of the models. Empirical studies illustrate the superior forecasting performance of our model against conventional linear models in terms of the root mean square error of the forecasts. Useful information for dynamic pricing is derived from its structural parameter estimates. This paper develops a dynamic forecasting model that accommodates asymmetric market responses to marketing mix variable—price promotion—by the threshold models. Copyright © 2005 John Wiley & Sons, Ltd.