随机利率下带违约风险的利率互换定价
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摘要
利率互换,具有降低筹资成本、减少融资风险、简便易行等特点,自1981年产生后,以年平均增长率超过30%的速度增长。随着利率互换交易的发展,利率互换期权的产生,互换的一体化程度越来越高,互换在企业的资产与负债组合管理中的作用日益增大,已成为国际资本市场的最主要内容,并将继续成为国际资本市场的主流。中国的利率互换市场,从2006年开展交易以来,也进入了快速发展的阶段。利率互换在资本市场的重要性直接决定了利率互换定价的重要性。新兴的中国利率互换市场对于合理的利率互换定价模型的需求更加迫切。
本文正是基于这样一个契机,在前人的基础上,结合信用违约风险度量的理论发展,提出了逐级递进的三种随机利率下带违约风险的利率互换定价模型。
Duffie & Huang在1996年提出了把违约风险纳入折现因子的约化模型,从而使得利率互换可以像定价无风险资产一样进行定价。违约风险由外生给定的违约强度模型来进行刻画。
本文提出的第一、第二种模型便是在Duffie＆Huang的模型框架下,分别假设公司资产服从几何布朗运动过程以及EVG (Exponential Variance Gamma)过程,求出公司违约强度,再把这个违约强度加入到无风险贴现因子中,成为风险调整后的贴现因子,用来对互换未来现金流进行贴现从而得到互换价值。
EVG过程作为一个带跳的Levy过程,有着比几何布朗运动更良好的性质,它能够克服“波动率微笑”的困境,并能很好的刻画违约风险的厚尾分布。所以从理论上来说,第二个模型比第一个模型更能反应实际情况,是一个更高级的利率互换定价模型。
数值结果表明在前两个模型下,平均每100个基点为的信用价差(债券价差)都只能导致0.088个基点的互换价差。EVG过程在利率互换定价中并没能体现它在刻画资产价格过程中的优势来。这主要是由于互换自身特有的性质造成的,互换不交换本金,在付息时间也只交换净利息,这样的条款使得互换对信用风险并不敏感。而且,互换对于交易对手来说有可能是负债也有可能是资产,当公司破产时,只有互换对它来说是负债的情况下才会受公司破产的影响,因此这种性质也使得信用对于互换利率的影响比较小。另外,在我们的文章里假设的参数刚好都使得EVG过程下的对数收益率比几何布朗运动过程下的要高,这也部分抵消了EVG过程由于跳跃带来的风险。
由于前两个模型都是把利率互换定价分两步进行,第一步计算违约强度,第二步计算互换利率。在做数值模拟时,由于误差的存在,有可能会使最终的互换利率受双重数值模拟误差的影响。为了克服这种数值模拟上的双重误差,我们提出了第三种利率互换定价模型,即随机利率下的PIDE模型——在风险中性条件下,假设公司资产动态地服从EVG过程,利率服从Vasicek模型,把利率互换看成是公司资产、无风险利率以及时间的函数,运用结构化方法,求出利率互换所满足的PIDE方程。这样,我们就能一次性地把利率互换价值求出来。数值结果表明100个基点的信用价差也只能导致0.111个基点的互换价差。虽然在相对数量来看,这个结果比前面两个模型多了至少25%。但从绝对数量上来看,其结果变化并不明显。这说明前两个模型的双重误差在利率互换定价中并没有那么严重。可以说,三个模型都可以成为对利率互换定价的重要模型。
由于债券的定价比利率互换更加简单,把利率互换的三个模型稍加改动,便能对债券进行定价。数值结果表明,由于对信用风险的刻画不同,这三个模型的债券价格也就相差比较大,因此,对于像债券这种对信用风险敏感性较强的资产,需要慎重地选择更好的理论定价模型。第三种模型由于采用了更为理想的公司资产价值变去过程,而且不会有双重误差的缺陷,因此我们认为它是债券定价比较理想的选择。
Being one of the most popular and highly liquid instruments of the international cap-ital market, interest rate swap is in possession of functions such as price discovery, risk aversion and asset allocation. Since the first trade in the year of 1981, the year-on-year average growth of interest rate swap trading was over 30%. Interest rate swap has been popular all along and its influences on a corporation's capital management are becoming more and more significant. The Chinese interest rate swap market also goes into a fast growing stage from the opening in 2006. The purchase of appropriate pricing mode of the burgeoning interest rate swap market became more urgency.
Based on these reasons, this paper corporate the developing the credit default mea-surement theory, providing three stepwise interest rate swap pricing model with default risk. Duffie & Huang introduce a reduce model by merging the risk factors into risk free discount rate in 1996. With counterparties of different default risk, the promised cash flows of a swap are discounted by a switching discount rate that, at any given state and time, is equal to the discount rate of the counterparty of which the swap is currently out of the money (that is, a liability). They showed that defaultable securities can be priced in a way similar to the standard risk neutral pricing of default-free securities with the effective discount rate being used instead of the usual risk free short rate.
The first two models presented in this paper is based by the reduced model of Duffie & Huang (1996), assuming the dynamics of firm assets following the Geometrical Brown motion process and the Exponential Variance Gamma process respectively. The risk free discount rate is adjusted by the default density after it is calculated by structure model. Then the interest rate swap is discounted by the risk adjustment discount rate to get the final rates.
As a process with jump item, EVG process possesses some advantages, such as it can solve the problem of volatility smile and the fat-tail phenomena. The second model in this paper is an advanced model than the first one theoretically, because the EVG model is a more proper process in describing dynamics assets price.
Both of the two models'numerical results show that, a one hundred basis point of credit spread (bond spread) cause only 0.088 basis point of swap spread averagely. It seems that the advance of assets dynamics price model don't behave better in interest rate swap pricing. It's because of the special swap clauses of without exchange the principle and the netting, causing the little sensitivity of swap to default risk. And by another side, the swap is not always assets or liability to a counterparty, and swap is suffered from the counter-party's default risk only when the swap is a liability to this counterparty. Additionally, our paramenters in this paper are all let the log return under the EVG model is larger than under the Geometrical Brown Motion, and this positive difference of log return counteracts the risk caused by the jump under EVG model.
The first two models both calculate the swap rate by two steps, first step calculate the default density and the second step solve the PDE, which will cause double error. In order to solve this problem, we present another interest rate swap pricing model, which we consider the interest rate swap as a contingent claim of three stochastic variables:assets price, risk free interest rate, and time t, under the risk neutral measurement. We suppose the firm assets price follows the EVG process, and then deduce the PIDE which the value of the interest rate swap satisfies under the stochastic interest rate. The numerical results show that, a one hundred basis point of bond spread results in a 0.111 basis point of swap spread. Although the result seems increasing in more than 25% corresponding to the former two models, but the absolute difference is still very small too. It seems that the double error caused by the first two models is not so obviously in swap pricing, and all the three models are suitable pricing model of swap.
We apply the three models into bond pricing with small modification since bond pric-ing is relative to swap pricing. The numerical results show that the behave of the three models are very different in bond pricing. Under the same parameters, the difference of bond rate can be more than 1000 basis point which is very significant. As a result, as- sets as bond, which are sensitive to default risk, the choice of a proper pricing model is very important. And theoretically, the third model use the advanced assets dynamics price model-EVG process and solve the problem of double error, we consider the third model is the relative proper model in bond pricing.
本文正是基于这样一个契机,在前人的基础上,结合信用违约风险度量的理论发展,提出了逐级递进的三种随机利率下带违约风险的利率互换定价模型。
Duffie & Huang在1996年提出了把违约风险纳入折现因子的约化模型,从而使得利率互换可以像定价无风险资产一样进行定价。违约风险由外生给定的违约强度模型来进行刻画。
本文提出的第一、第二种模型便是在Duffie＆Huang的模型框架下,分别假设公司资产服从几何布朗运动过程以及EVG (Exponential Variance Gamma)过程,求出公司违约强度,再把这个违约强度加入到无风险贴现因子中,成为风险调整后的贴现因子,用来对互换未来现金流进行贴现从而得到互换价值。
EVG过程作为一个带跳的Levy过程,有着比几何布朗运动更良好的性质,它能够克服“波动率微笑”的困境,并能很好的刻画违约风险的厚尾分布。所以从理论上来说,第二个模型比第一个模型更能反应实际情况,是一个更高级的利率互换定价模型。
数值结果表明在前两个模型下,平均每100个基点为的信用价差(债券价差)都只能导致0.088个基点的互换价差。EVG过程在利率互换定价中并没能体现它在刻画资产价格过程中的优势来。这主要是由于互换自身特有的性质造成的,互换不交换本金,在付息时间也只交换净利息,这样的条款使得互换对信用风险并不敏感。而且,互换对于交易对手来说有可能是负债也有可能是资产,当公司破产时,只有互换对它来说是负债的情况下才会受公司破产的影响,因此这种性质也使得信用对于互换利率的影响比较小。另外,在我们的文章里假设的参数刚好都使得EVG过程下的对数收益率比几何布朗运动过程下的要高,这也部分抵消了EVG过程由于跳跃带来的风险。
由于前两个模型都是把利率互换定价分两步进行,第一步计算违约强度,第二步计算互换利率。在做数值模拟时,由于误差的存在,有可能会使最终的互换利率受双重数值模拟误差的影响。为了克服这种数值模拟上的双重误差,我们提出了第三种利率互换定价模型,即随机利率下的PIDE模型——在风险中性条件下,假设公司资产动态地服从EVG过程,利率服从Vasicek模型,把利率互换看成是公司资产、无风险利率以及时间的函数,运用结构化方法,求出利率互换所满足的PIDE方程。这样,我们就能一次性地把利率互换价值求出来。数值结果表明100个基点的信用价差也只能导致0.111个基点的互换价差。虽然在相对数量来看,这个结果比前面两个模型多了至少25%。但从绝对数量上来看,其结果变化并不明显。这说明前两个模型的双重误差在利率互换定价中并没有那么严重。可以说,三个模型都可以成为对利率互换定价的重要模型。
由于债券的定价比利率互换更加简单,把利率互换的三个模型稍加改动,便能对债券进行定价。数值结果表明,由于对信用风险的刻画不同,这三个模型的债券价格也就相差比较大,因此,对于像债券这种对信用风险敏感性较强的资产,需要慎重地选择更好的理论定价模型。第三种模型由于采用了更为理想的公司资产价值变去过程,而且不会有双重误差的缺陷,因此我们认为它是债券定价比较理想的选择。
Being one of the most popular and highly liquid instruments of the international cap-ital market, interest rate swap is in possession of functions such as price discovery, risk aversion and asset allocation. Since the first trade in the year of 1981, the year-on-year average growth of interest rate swap trading was over 30%. Interest rate swap has been popular all along and its influences on a corporation's capital management are becoming more and more significant. The Chinese interest rate swap market also goes into a fast growing stage from the opening in 2006. The purchase of appropriate pricing mode of the burgeoning interest rate swap market became more urgency.
Based on these reasons, this paper corporate the developing the credit default mea-surement theory, providing three stepwise interest rate swap pricing model with default risk. Duffie & Huang introduce a reduce model by merging the risk factors into risk free discount rate in 1996. With counterparties of different default risk, the promised cash flows of a swap are discounted by a switching discount rate that, at any given state and time, is equal to the discount rate of the counterparty of which the swap is currently out of the money (that is, a liability). They showed that defaultable securities can be priced in a way similar to the standard risk neutral pricing of default-free securities with the effective discount rate being used instead of the usual risk free short rate.
The first two models presented in this paper is based by the reduced model of Duffie & Huang (1996), assuming the dynamics of firm assets following the Geometrical Brown motion process and the Exponential Variance Gamma process respectively. The risk free discount rate is adjusted by the default density after it is calculated by structure model. Then the interest rate swap is discounted by the risk adjustment discount rate to get the final rates.
As a process with jump item, EVG process possesses some advantages, such as it can solve the problem of volatility smile and the fat-tail phenomena. The second model in this paper is an advanced model than the first one theoretically, because the EVG model is a more proper process in describing dynamics assets price.
Both of the two models'numerical results show that, a one hundred basis point of credit spread (bond spread) cause only 0.088 basis point of swap spread averagely. It seems that the advance of assets dynamics price model don't behave better in interest rate swap pricing. It's because of the special swap clauses of without exchange the principle and the netting, causing the little sensitivity of swap to default risk. And by another side, the swap is not always assets or liability to a counterparty, and swap is suffered from the counter-party's default risk only when the swap is a liability to this counterparty. Additionally, our paramenters in this paper are all let the log return under the EVG model is larger than under the Geometrical Brown Motion, and this positive difference of log return counteracts the risk caused by the jump under EVG model.
The first two models both calculate the swap rate by two steps, first step calculate the default density and the second step solve the PDE, which will cause double error. In order to solve this problem, we present another interest rate swap pricing model, which we consider the interest rate swap as a contingent claim of three stochastic variables:assets price, risk free interest rate, and time t, under the risk neutral measurement. We suppose the firm assets price follows the EVG process, and then deduce the PIDE which the value of the interest rate swap satisfies under the stochastic interest rate. The numerical results show that, a one hundred basis point of bond spread results in a 0.111 basis point of swap spread. Although the result seems increasing in more than 25% corresponding to the former two models, but the absolute difference is still very small too. It seems that the double error caused by the first two models is not so obviously in swap pricing, and all the three models are suitable pricing model of swap.
We apply the three models into bond pricing with small modification since bond pric-ing is relative to swap pricing. The numerical results show that the behave of the three models are very different in bond pricing. Under the same parameters, the difference of bond rate can be more than 1000 basis point which is very significant. As a result, as- sets as bond, which are sensitive to default risk, the choice of a proper pricing model is very important. And theoretically, the third model use the advanced assets dynamics price model-EVG process and solve the problem of double error, we consider the third model is the relative proper model in bond pricing.
引文
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