我们对每个债券类型都进行了严谨的利率建模,确保了各种债券类型之间的一致性,同时我们针对各种情况都建立了零息收益率曲线。此外,考虑到数据馈送的不规则性和稀疏性,我们决定使用半参数法,而不使用完全非参数表示法(如自助抽样法),
因为后者将无法反映出市场各个期限结构之间的相互影响。这样一来,
即使定价偶尔会与报价存在细微偏差,SuperCurve®的曲线也能在每个时间点都确保“拟合的一致性”。
 
The consistency between the various instruments is ensured by rigorous interest rate modeling for each 
class of instruments. A “zero-coupon” yield curve is built in each case. Given the irregularity and sparsity of the data feed, a semi-parametric representation is used instead of a fully non-parametric representation, 
such as bootstrapping, which would be unable to reflect the mutual influence of various areas of the market
onto one another. A “best consistent fit” is provided at each point in time, 
which may occasionally slightly differ from existing quotes.
 
 

核心算法 / Core algorithm

 

零息收益率曲线的半参数表

Semi-Parametric Representation of the Zero-Coupon Yield Curve

 

为了适应中国固定收益市场的复杂情况,SuperCurve®选择使用半参数法来拟合零息收益率曲线。
我们审慎选择样条曲线表示方法、样条数量和拟合算法,使得系统能够提供符合一致性的稳定定价,并持续反映当前市场状况。
To handle the complexities of the Chinese fixed income market, SuperCurve® has chosen a semi-parametric approach to model the zero-coupon curve. 
The spline representation, the number of splines and the fitting algorithm have been carefully 
chosen to provide consistent and stable prices that permanently reflect the current state of the market.
 
 

剩余期限和利率的对数标度

Logarithmic Scale for the Maturity and the Rates

 

当通货膨胀和利率都很高时,利率的波动或多或少都会与利率水平成正比;但在通胀率和利率都很低而又不至于为负数时,利率波动又会独立于利率水平。SuperCurve®运用偏移参数对收益率变量进行二次调整。
The volatility of interest rates is more or less proportional to the level of the rates when the 
inflation and the rates are high, but independent of their level when the inflation is low and rates are low as well, 
if not negative, as we often see today. SuperCurve® uses shift parameter for second change of variable in the rate level. 

 

脊最小二乘准则和振荡控制

Ridge Least Square Criterion and Oscillations Control

 

SuperCurve®使用最小化最小二乘标准,目标函数包含两项,
第一项用于确保每只债券的理论价格可以与其真实价格相吻合,第二项是脊正则化,目标是确保收益率曲线的规律性。
SuperCurve® minimizes the Least Square Criterion, 
the first term of the objective function aims at fitting the theoretical price of each bond to its real price. 
The second term is ridge regularization to ensuring the regularity of the yield curve.

 

有效统计和误差消除

Robust Statistics and Error Elimination

 

中国固定收益市场的价格和收益率数据往往远离实际市场情况,要么是因为报价长期没有更新,要么是因为这些
是虚假的或预先安排的交易,它们的价格不符合当前的市场情况,这会使得一些价格无法由参数法或者半参数法拟合。
SuperCurve®用M估计优化最小二乘目标函数,以减少估值偏差的影响;系统将过滤掉价格不匹配的债券,并且用其他债券重新拟合曲线。
Price and yield data in the Chinese fixed-income market are often away from the actual market situation, either because quotes have not 
been updated for a long time or because these are fake or pre-arranged trades at prices that are not in line with the current market situation. 
To address this issue, robust techniques are used. First, replacing the simple square in the objective function of Least Squire Criterion by an 
M-estimator reduces the influence of poorly fitted prices. Then by filtering out bonds whose price cannot be 
correctly matched and re-running the fitting procedure without these bonds.

 

实时拟合

Real-Time Fitting

 

不同剩余期限的信息输入是无序的,因而需要使用特定技术将其进行组合,以使新的价格及收益率与之前的信息保持一致。
基于贝叶斯统计,每次在收到一个或多个新报价时, 我们引入卡尔曼滤波这一自适应的方法能够高效地更新整条零息曲线,
并实时调整系数以匹配新的报价。同时,这也可以有效地解决了数据缺失问题。
Real-time curve fitting is a more significant challenge. Information on various parts of the maturity range arrives in a 
disorganized manner. It needs specific techniques to be combined to merge new prices and yields coherently 
with past information. Based on Bayesian statistics, SuperCurve® adapts Kalman filtering method to optimally update the whole 
zero-coupon curve each time one or several new quotes are received. This can efficiently solve the problem of missing data. 

 

拟合IRS曲线

Fitting the IRS Curve

 

鉴于利率互换可视作一只具有相同期限的虚拟债券,我们可以按照债券曲线的拟合方式来拟合IRS零息曲线。
因为SuperCurve®只拟合为数不多的期限对应的利率互换收益率,
我们可以调整收益率曲线系数,以确保这些参考互换收益率与通过定价模型计算的理论值完全匹配。
We use this equivalence to calibrate the IRS zero-coupon curves in the same way we calibrate the bond curves. 
Thanks to the fact that we must only fit a small number of swap rates at fixed maturities, we can adjust the yield curve coefficients and 
ensure that these reference swap rates match their theoretical values from the model.

 

非国债曲线的拟合

Fitting Curves Other Than T-Bonds

 

国债收益率的任何波动都会即时影响到其他类型债券或利率互换的收益率水平,为确保不同收益率曲线的拟合保持一致性,
SuperCurve®运用了不同品种曲线系数,例如(αcdb, βcdb)和(αTB, βTB)之间的偏移系数,
实现拟合国开债、农发和口行债,以及利率互换(基准利率为SHIBOR3M和FR007)。
Any fluctuation on the T-bond yield would tend to be immediately reflected on the other bond or IRS. SuperCurve® use the shift in coefficients between (αcDb,βcDb) and (αTb,βTb) . So it can be applied to CDB, EIBC & ADBC and IRS (both SHIBOR3M and FR007).

 

税收调整

Tax Adjustment

 

与其他债券不同,国债息票免税。存在税前和税后收益差。SuperCurve®在给国债定价时,考虑了免税因素对理论收益率的影响。
Contrary to other bonds, the coupons paid by Treasury Bonds (TB) are not taxable. This induces a price difference between these bonds and other taxable bonds, the higher the coupon, the larger the difference. SuperCurve® incorporate the tax-exempt feature into the TB fitting curve. 

 

 

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