A Simple Novel Approach to Valuing Risky Zero Coupon Bond in a Markov Regime Switching Economy

We have addressed the problem of pricing risky zero coupon bond in the framework of Longstaff and Schwartz structural type model by pricing it as a Down-and-Out European Barrier Call option on the company’s asset-debt ratio assuming Markov regime switching economy. The growth rate and the volatility of the stochastic asset debt ratio is driven by a continuous time Markov chain which signifies state of the economy. Regime Switching renders market incomplete and selection of a Equivalent martingale measure (EMM) becomes a subtle issue. We price the zero coupon risky bond utilizing the powerful technique of Risk Minimizing hedging of the underlying Barrier option under the so called “Risk Minimal” martingale measure via computing the bond default probability.     

最小方差准则下一个未定权益的近似对冲

本文研究了不完备的离散时间股票市场下未定权益的定价的对冲问题.利用在最小方差准则下选择概率测度Q或权重函数LN来求最优投资组合的方法,给出了离散时间情况下的鞅表示定理,在最小方差准则下提供一个简单的方法来近似对冲一个未定权益或一个欧氏期权.     

Hedging Large Portfolios of Options in Discrete Time*

The problem studied is that of hedging a portfolio of options in discrete time where underlying security prices are driven by a combination of idiosyncratic and systematic risk factors. It is shown that despite the market incompleteness introduced by the discrete time assumption, large portfolios of options have a unique price and can be hedged without risk. The nature of the hedge portfolio in the limit of large portfolio size is substantially different from its continuous time counterpart. Instead of linearly hedging the total risk of each option separately, the correct portfolio hedge in discrete time eliminates linear as well as second and higher order exposures to the systematic risk factors only. The idiosyncratic risks need not be hedged, but disappear through diversification. Hedging portfolios of options in discrete time thus entails a trade‐off between dynamic and cross‐sectional hedging errors. Some computations are provided on the outcome of this trade‐off in a discrete‐time Black–Scholes world.     

GENERALIZED STOCHASTIC DURATION INMARKOVIAN HEATH-JARROW-MORTONFRAMEWORK

1 IntroductionIll moderll financial industry, risk managen1ent is a major task tliat fiuancial institutionsnlust deal witli in every tradiug day alld it is becouilng urore and more important for mailltaining tl1eir access to clieap capital and meeting risk-based caPital requirements. Meanwhile,sonle fiuaucial iustitutions (FI) sucl1 as co1nuercial ba1lks, iusurallce companies and securitiescolllpanies. etc., hold a large proportioll of fixed income security which price is sensitive totlle mar…     

Hedging guarantees in variable annuities under both equity and interest rate risks

Effective hedging strategies for variable annuities are crucial for insurance companies in preventing potentially large losses. We consider discrete hedging of options embedded in guarantees with ratchet features, under both equity (including jump) risk and interest rate risk. Since discrete hedging and the underlying model considered lead to an incomplete market, we compute hedging strategies using local risk minimization. Our results suggest that risk minimization hedging, under a joint model for the underlying and interest rate, leads to effective risk reduction. Moreover, hedging with standard options is superior to hedging with the underlying when both equity and interest rate risks are appropriately modeled.     

离散条件下标的资产带跳的德尔塔对冲误差研究

讨论了离散条件下的德尔塔对冲以及含泊松跳跃的布莱克—休斯模型下期权的定价问题.在布莱克—休斯模型中对冲被假设为连续发生的,当应用于离散的交易时,对冲误差就产生了.考虑到对冲误差,得出一种离散条件下标的资产带泊松跳跃的修正的布莱克—休斯方程和依赖再对冲区间长度的更精确的德尔塔值.     

基于债券久期的国债期货套期保值模型分析

使用久期的方法在中国国债期货市场上进行套期保值是否有效?使用久期的方法研究国债期货套期保值的效率问题在国外已经很多,然而这种方法是否适合于目前中国的国债市场,相关研究还不多见,还有待进一步的证实.为此借鉴国外相关理论,采用比较研究的方法,以国债期货上市后2013年9月到2014年5月初,国债现货和国债期货的数据为样本,以基于久期的最优套期保值比率模型为主,其他模型为辅,比较出最优套期保值效率.研究结果表明,基于久期的套期保值方法在目前中国的国债市场效果一般.     

On discrete time hedging errors in a fractional Black-Scholes model

In this paper we investigate asymptotic behavior of error of a discrete time hedging strategy in a fractional Black-Scholes model in the sense of Wick-Ito-Skorohod integration.The rate of convergence of the hedging error due to discrete-time trading when the true strategy is known for the trader,is investigated.The result provides new statistical tools to study and detect the effect of the long-memory and the Hurst parameter for the error of discrete time hedging.     

Optimal design of profit sharing rates by FFT

This paper addresses the calculation of a fair profit sharing rate for participating policies with a minimum interest rate guaranteed. The bonus credited to policies depends on the performance of a basket of two assets: a stock and a zero coupon bond and on the guarantee. The dynamics of the instantaneous short rates are driven by a Hull and White model, whereas the stocks follow a double exponential jump-diffusion model. The participation level is determined such that the return retained by the insurer is sufficient to hedge the interest rate guaranteed. Given that the return of the total asset is not lognormal, we rely on a Fast Fourier Transform to compute the fair value of bonus and guarantee options.     

A note on convergence of an approximate hedging portfolio with liquidity risk

When the underlying asset price depends on activities of traders, hedging errors include costs due to the illiquidity of the underlying asset and the size of this cost can be substantial. Cetin et al. (2004), Liquidity risk and arbitrage pricing theory, Finance and Stochastics, 8(3), 311-341, proposed a hedging strategy that approximates the classical Black–Scholes hedging strategy and produces zero liquidity costs. Here, we compute the rate of convergence of the final value of this hedging portfolio to the option payoff in case of a European call option; i.e. we see how fast its hedging error converges to zero. The hedging strategy studied here is meaningful due to its simple liquidity cost structure and its smoothness relative to the classical Black–Scholes delta.     

A systematic approach to pricing and hedging international derivatives with interest rate risk: analysis of international derivatives under stochastic interest rates

This paper deals with the valuation and the hedging of non-path-dependent European options on one or several underlying assets in a model of an international economy allowing for both, interest rate risk and exchange rate risk. Using martingale theory and, in particular, the change of numeraire technique we provide a unified and easily applicable approach to pricing and hedging exchange options on stocks, bonds, futures, interest rates and exchange rates. We also cover the pricing and hedging of compound exchange options.     

我国股指期货的套期保值比率研究

本文主要运用OLS、VAR、ECM、diagonal-BEKK、full-BEKK、scalar-BEKK对沪深300股指期货仿真交易的套期保值比率进行研究,比较了静态套期模型和动态套期保值模型的效果,并研究不同参数化形式对动态套期保值模型的影响。结果表明尽管动态套期保值在样本内要优于静态套期保值模型,但样本外效果却不是很好;另外,动态模型的不同的参数化形式对结果的影响也比较大。     

Option pricing under residual risk and imperfect hedging

This paper is concerned in the option pricing in a discrete time incomplete market. We emphasize the interplay between option pricing and residual risk as well as imperfect hedging. It has been shown that the value of a European option satisfies a hyperbolic, rather than parabolic, partial differential equation. The closed-form solution for this hyperbolic equation has been obtained, which will collapse to the Black–Scholes formula as the time scaling converges to zero.     

Discrete–time delta hedging and the Black–Scholes model with transaction costs

The paper deals with the problem of discrete–time delta hedging and discrete-time option valuation by the Black–Scholes model. Since in the Black–Scholes model the hedging is continuous, hedging errors appear when applied to discrete trading. The hedging error is considered and a discrete-time adjusted Black–Scholes–Merton equation is derived. By anticipating the time sensitivity of delta in many cases the discrete-time delta hedging can be improved and more accurate delta values dependent on the length of the rebalancing intervals can be obtained. As an application the discrete-time trading with transaction costs is considered. Explicit solution of the option valuation problem is given and a closed form delta value for a European call option with transaction costs is obtained.     

Progressive hedging and tabu search applied to mixed integer (0,1) multistage stochastic programming

Many problems faced by decision makers are characterized by a multistage decision process with uncertainty about the future and some decisions constrained to take on values of either zero or one (for example, either open a facility at a location or do not open it). Although some mathematical theory exists concerning such problems, no general-purpose algorithms have been available to address them. In this article, we introduce the first implementation of general purpose methods for finding good solutions to multistage, stochastic mixed-integer (0, 1) programming problems. The solution method makes use of Rockafellar and Wets' progressive hedging algorithm that averages solutions rather than data. Solutions to the induced quadratic (0,1) mixed-integer subproblems are obtained using a tabu search algorithm. We introduce the notion of integer convergence for progressive hedging. Computational experiments verify that the method is effective. The software that we have developed reads standard (SMPS) data files.     

Black-Scholes期权定价的风险(英文)

Black-Scholes期权定价的推导假定对冲是连续的以达到无风险. 但事实上, 股市收市后将不再有交易, 所以投资者不能连续的调整其投资组合, 故期权定价的风险是存在的. 本文讨论了这种不连续对冲带来的期权定价的风险, 并以美国股市的几种指标股为例, 给出其比率. 比率多在5%以上, 有的可以达到38%, 可见传统期权定价的风险不容小觑.     

Option Pricing and Hedging under a Markov Switching Lévy Process Model

In this paper,we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging.In this model,the market interest rate,the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process.We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure.The option price using this model is obtained by the Fourier transform method.We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.     

Bachelier model with stopping time and its insurance application

A modification of a classical Bachelier model by letting a stock price absorb at zero is revisited. Alternative proofs are given to derive option pricing formulas under the modified Bachelier model and numerical comparison with the Black–Scholes formula is provided. Quantile hedging methodology is developed for both classical and modified Bachelier models and application to pricing the pure endowment with fixed guarantee life insurance contracts is demonstrated, both theoretically and by means of a numerical example.     

带有风险价值的最优期货套期保值策略

本文研究了带有风险价值约束的期货套期保值优化问题.用最优化方法获得了套期保值策略的存在性、求解模型的增广拉格朗日算法及其收敛性.文中的结果推广了期货收益率服从正态分布的单变量套期保值策略的研究,表现为用服从椭圆分布的随机变量刻画市场风险因子的厚尾特征、用风险价值控制套期保值的风险、构建了均值-VaR组合套期保值理论模型并给出了求解算法.     

利率风险与或有权益定价

应用无差异方法研究不完全市场中或有权益的保值和定价问题,并证明了或有权益的价格不仅依赖于或有权益的不可复制部分,而且受利率风险的影响．在最优保值意义下利率风险分解为可控风险和不可控风险．利率的可控风险与资本市场波动有关,可通过套期保值方法避免,可能产生正、零或负的期望收益．利率的不可控风险与资本市场波动无关,无法对冲,而且带来正的期望收益．利率风险的分解有助于更准确地解释或有权益的价格-它受利率的不可控风险影响,而与可控风险无关．当利率的不可控收益与或有权益的不可复制部分正(负)相关时,或有权益的不可复制部分的风险越大导致或有权益的价格越高(低)．