Pricing counterparty default risks: Applications to FRNs and vulnerable options

This paper provides simple closed-form pricing models for floating-rate notes and vulnerable options under the counterparty risk framework of [Jarrow, R., Yu, F., 2001. Counterparty risk and the pricing of default risk. Journal of Finance 56, 1765-1799]. After deriving closed-form pricing models for them, this paper illustrates the impact of the default intensity of counterparty on the prices of floating-rate notes and vulnerable options. Numerical examples show that the default risk of counterparty is an important factor of the value of floating-rate notes and vulnerable options.     

Valuation of vulnerable American options with correlated credit risk

This article evaluates vulnerable American options based on the two-point Geske and Johnson method. In accordance with the Martingale approach, we provide analytical pricing formulas for European and multi-exercisable options under risk-neutral measures. Employing Richardson’s extrapolation gets the values of vulnerable American options. To demonstrate the accuracy of our proposed method, we use numerical examples to compare the values of vulnerable American options from our proposed method with the benchmark values from the least-square Monte Carlo simulation method. We also perform sensitivity analyses for vulnerable American options and show how the prices of vulnerable American options vary with the correlation between the underlying assets and the option writer’s assets.      

Fast estimation of true bounds on Bermudan option prices under jump-diffusion processes

Fast pricing of American-style options has been a difficult problem since it was first introduced to the financial markets in 1970s, especially when the underlying stocks’ prices follow some jump-diffusion processes. In this paper, we extend the ‘true martingale algorithm’ proposed by Belomestny et al. [Math. Finance, 2009, 19, 53–71] for the pure-diffusion models to the jump-diffusion models, to fast compute true tight upper bounds on the Bermudan option price in a non-nested simulation manner. By exploiting the martingale representation theorem on the optimal dual martingale driven by jump-diffusion processes, we are able to explore the unique structure of the optimal dual martingale and construct an approximation that preserves the martingale property. The resulting upper bound estimator avoids the nested Monte Carlo simulation suffered by the original primal–dual algorithm, therefore significantly improving the computational efficiency. Theoretical analysis is provided to guarantee the quality of the martingale approximation. Numerical experiments are conducted to verify the efficiency of our algorithm.     

Pricing credit-risky bonds and spread options modelling credit-spread term structures with two-dimensional Markov-modulated jump-diffusion

The relationship between company hazard rates and the business cycle becomes more apparent after a financial crisis. To address this relationship, a regime-switching process with an intensity function is adopted in this paper. In addition, the dynamics of both interest rates and asset values are modelled with a Markov-modulated jump-diffusion model, and a 2-factor hazard rate model is also considered. Based on this more suitable model setting, a closed-form model of pricing risky bonds is derived. The difference in yield between a risky bond and risk-free zero coupon bond is used to model a term structure of credit spreads (CSs) from which a closed-form pricing model of a call option on CSs is obtained. In addition, the degree to which the explicit regime shift affects CSs and credit-risky bond prices is numerically examined using three forward-rate functions under various business-cycle patterns.     

Pricing a class of exotic commodity options in a multi-factor jump-diffusion model

In a recent paper, Crosby introduced a multi-factor jump-diffusion model which would allow futures (or forward) commodity prices to be modelled in a way which captured empirically observed features of the commodity and commodity options markets. However, the model focused on modelling a single individual underlying commodity. In this paper, we investigate an extension of this model which would allow the prices of multiple commodities to be modelled simultaneously in a simple but realistic fashion. We then price a class of simple exotic options whose payoff depends on the difference (or ratio) between the prices of two different commodities (for example, spread options), or between the prices of two different (i.e. with different tenors) futures contracts on the same underlying commodity, or between the prices of a single futures contract as observed at two different calendar times (for example, forward start or cliquet options). We show that it is possible, using a Fourier transform-based algorithm, to derive a single unifying form for the prices of all these aforementioned exotic options and some of their generalizations. Although we focus on pricing options within the model of Crosby, most of our results would be applicable to other models where the relevant ‘extended’ characteristic function is available in analytical form.     

Reduced-form Models with Regime Switching: An Empirical Analysis for Corporate Bonds

Empirical evidence shows that there is a close link between regime shifts and business cycle fluctuations. A standard term structure of interest rates, such as the Cox et al. (1985 Econometrica, 53, 385–407; CIR) model, is sharply rejected in the Treasury bond data. Only Markov regime-switching models on the entire yield curve of the Treasury bond data can account for the observed behavior of the yield curve. In this paper, we examine the impact of regime shifts on AAA-rated and BBB-rated corporate bonds through the use of a reduced-form model. The model is estimated by the Efficient Method of Moments (EMM). Our empirical results suggest that regime-switching risk has significant implications for corporate bond prices and hence has a material impact on the entire corporate bond yield curve, providing evidence for the approach of rating through the cycle employed by rating agencies.     

Numerical solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market

We study the numerical solutions for an integro-differential parabolic problem modeling a process with jumps and stochastic volatility in financial mathematics. We present two general algorithms to calculate numerical solutions. The algorithms are implemented in PDE2D, a general-purpose, partial differential equation solver.     

An approximate distribution of delta-hedging errors in a jump-diffusion model with discrete trading and transaction costs

We introduce a jump-diffusion model for asset returns with jumps drawn from a mixture of normal distributions and show that this model adequately fits the historical data of the S&P500 index. We consider a delta-hedging strategy (DHS) for vanilla options under the diffusion model (DM) and the proposed jump-diffusion model (JDM), assuming discrete trading intervals and transaction costs, and derive an approximation for the probability density function (PDF) of the profit-and-loss (P&L) of the DHS under both models. We find that, under the log-normal model of Black–Scholes–Merton, the actual PDF of the P&L can be well approximated by the chi-squared distribution with specific parameters. We derive an approximation for the P&L volatility in the DM and JDM. We show that, under both DM and JDM, the expected loss due to transaction costs is inversely proportional to the square root of the hedging frequency. We apply mean–variance analysis to find the optimal hedging frequency given the hedger's risk tolerance. Since under the JDM it is impossible to reduce the P&L volatility by increasing the hedging frequency, we consider an alternative hedging strategy, following which the P&L volatility can be reduced by increasing the hedging frequency.     

基于多层次模糊综合评价的中小企业信用风险评估

引入多层次模糊评价方法,设计包括行业状况、上下游状况、产品状况、管理水平、财务状况和资信状况的指标体系,构建多层次模糊综合评价的中小企业信用风险评估模型。算例分析表明该方法综合了定性因素和定量因素,能有效地评价中小企业的信用风险。     

On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility

In this paper we use Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be a diffusion or a Markov process, as the examples in Sect. 7 show. This expression depends on the derivative of the volatility in the sense of Malliavin calculus. E. Alòs’ research is supported by grants MEC FEDER MTM 2006 06427 and SEJ2006-13537. J.A. León’s research is partially supported by the CONACyT grant 45684-F. J. Vives’ research is supported by grant MEC FEDER MTM 2006 06427.     

Growth options,macroeconomic conditions,and the cross section of credit risk

This paper develops a structural equilibrium model with intertemporal macroeconomic risk, incorporating the fact that firms are heterogeneous in their asset composition. Compared with firms that are mainly composed of invested assets, firms with growth options have higher costs of debt because they are more volatile and have a greater tendency to default during recession when marginal utility is high and recovery rates are low. Our model matches empirical facts regarding credit spreads, default probabilities, leverage ratios, equity premiums, and investment clustering. Importantly, it also makes predictions about the cross section of all these features.     

基于信用利差的中国城投债券信用风险分析

在"四万亿"投资的推动下,中国城投债券的发行规模大幅增加。与此同时,城投公司的整体债务规模也急剧攀升,其债务总量超过了地方政府的财政收入水平,城投公司的整体信用水平不容乐观。因此,城投债券的信用风险需引起足够重视。本文从量化角度研究了影响中国城投债券信用风险的四个因素,结果显示,中国城投债券的信用利差与发债企业的资产规模以及发债企业所在地区的人均GDP水平负相关,与企业债券收益率以及担保正相关。     

Relevance of the disposition effect on the options market: New evidence

A moneyness‐based propensity to sell (MPS) measure, at the aggregate level, determines the propensity of option holders to exercise their winning relative to losing positions. Using data on individual stock and S&P 500 Index options, we find that the MPS measure has significant predictive power over the cross section of delta‐hedged option returns. We test the disposition effect in the options market based on a long–short strategy that exploits price distortions induced by the disposition bias. More pronounced evidence of the disposition bias is found for individual at‐the‐money call options than put options where the significance of abnormal returns remains robust across different subsamples even after we control for the portfolio option greeks and market‐based risk factors. The profitability of the long–short strategy is related to limit‐to‐arbitrage proxies suggesting that behavioral explanations help explain the positive relation between the MPS measure and delta‐hedged option returns.     

Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results

Stochastic volatility (SV) and local stochastic volatility (LSV) processes can be used to model the evolution of various financial variables such as FX rates, stock prices and so on. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes. Many issues remain, though, including the efficacy of the standard alternating direction implicit (ADI) numerical methods for solving SV and LSV pricing problems. In general, the amount of required computations for these methods is very substantial. In this paper, we address some of these issues and propose a viable alternative to the standard ADI methods based on Galerkin-Ritz ideas. We also discuss various approaches to solving the corresponding pricing problems in a semi-analytical fashion. We use the fact that in the zero correlation case some of the pricing problems can be solved analytically, and develop a closed-form series expansion in powers of correlation. We perform a thorough benchmarking of various numerical solutions by using analytical and semi-analytical solutions derived in the paper.     

散户投资者如何影响债券价格？——基于交易所同一只信用债的价格差分析

交易所债券市场存在竞价交易和大宗交易两种方式,由于交易门槛限制,散户投资者只能参与竞价交易。本文考察同一只信用债、同一天在两种交易方式下的价格差,以识别散户投资者的影响。实证结果发现:竞价交易下的债券价格显著更高,并且高收益率债券的交易更为频繁,说明竞价交易下的投资者可能存在片面追逐收益率而忽视基本面的特点。进一步分析发现,在散户需求更大(如票面利率更高、认知度更高)的债券上,价格差也更大,说明散户投资者对高收益率债券的需求是竞价交易价格更高的主因之一。本文还提供证据说明,价格差的持续存在主要是由于套利限制而不是由流动性溢价所导致。最后,本文考察了“云南城投债危机”这一外生冲击对价格差的影响,得到了与上述一致的结论。     

债务展期金融契约视角下信用担保风险定价研究

通过把存款保险定价思路引入到信用担保风险定价领域,从信贷债务展期金融契约视角构建的信用担保风险定价模型及其数值求解方法,并进行的实证分析证明了基于债务展期金融契约视角下信用担保风险定价模型的科学性与应用前景.     

A Note on Option Pricing for the Constant Elasticity of Variance Model

We study the arbitrage free optionpricing problem for the constant elasticity of variance (CEV) model. To treatthestochastic aspect of the CEV model, we direct attention to the relationship between the CEV modeland squared Bessel processes. Then we show the existence of a unique equivalentmartingale measure and derive the Cox's arbitrage free option pricing formulathrough the properties of squared Bessel processes. Finally we show that the CEVmodel admits arbitrage opportunities when it is conditioned to be strictlypositive.     

Options on the minimum or the maximum of two average prices

This paper studies options on the minimum/maximum of two average prices. We provide a closed-form pricing formula for the option with geometric averaging starting at any time before maturity. We show overwhelming numerical evidence that the variance reduction technique with the help of the above closed-form solution dramatically improves the accuracy of the simulated price of an option with arithmetic averaging. The proposed options are found widely applicable in risk management and in the design of incentive contracts. The paper also discusses some parity relationships within the family of average-rate options and provides the upper and lower bounds for the proposed options with arithmetic averaging. This revised version was published online in June 2006 with corrections to the Cover Date.