核心算法 / Core algorithm
零息收益率曲線的半參數表
Semi-Parametric Representation of the Zero-Coupon Yield Curve
為了適應中國固定收益市場的復雜情況，SuperCurve®選擇使用半參數法來擬合零息收益率曲線。我們審慎選擇樣條曲線
表示方法、樣條數量和擬合算法，使得系統能夠提供符合一致性的穩定定價，并持續反映當前市場狀況。
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.
剩余期限和利率的對數標度
Logarithmic Scale for the Maturity and the Rates
當通貨膨脹和利率都很高時，利率的波動或多或少都會與利率水平成正比；但在通脹率和利率都很低而又不至于為負數時,
利率波動又會獨立于利率水平。SuperCurve®運用偏移參數對收益率變量進行二次調整。
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.
脊最小二乘準則和振蕩控制
Ridge Least Square Criterion and Oscillations Control
SuperCurve®使用最小化最小二乘標準,目標函數包含兩項，第一項用于確保每只債券的理論價格可以與其真實價格相吻合，
第二項是脊正則化，目標是確保收益率曲線的規律性。
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.
有效統計和誤差消除
Robust Statistics and Error Elimination
中國固定收益市場的價格和收益率數據往往遠離實際市場情況，要么是因為報價長期沒有更新，要么是因為這些是虛假的
或預先安排的交易，它們的價格不符合當前的市場情況，這會使得一些價格無法由參數法或者半參數法擬合。SuperCurve®
用M估計優化最小二乘目標函數，以減少估值偏差的影響；系統將過濾掉價格不匹配的債券，并且用其他債券重新擬合曲線。
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 Square
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.
實時擬合
Real-Time Fitting
不同剩余期限的信息輸入是無序的，因而需要使用特定技術將其進行組合，以使新的價格及收益率與之前的信息保持一致。
基于貝葉斯統計，每次在收到一個或多個新報價時，我們引入卡爾曼濾波這一自適應的方法能夠高效地更新整條零息曲線，
并實時調整系數以匹配新的報價。同時，這也可以有效地解決了數據缺失問題。
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.
擬合IRS曲線
Fitting the IRS Curve
鑒于利率互換可視作一只具有相同期限的虛擬債券，我們可以按照債券曲線的擬合方式來擬合IRS零息曲線。因為
SuperCurve®只擬合為數不多的期限對應的利率互換收益率，我們可以調整收益率曲線系數，以確保這些參考互換收益率
與通過定價模型計算的理論值完全匹配。
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.
非國債曲線的擬合
Fitting Curves Other Than Treasury Bonds
國債收益率的任何波動都會即時影響到其他類型債券或利率互換的收益率水平，為確保不同收益率曲線的擬合保持一致性，
SuperCurve®運用了不同品種曲線系數，例如(αcdb，βcdb)和(αTB，βTB)之間的偏移系數，實現擬合國開債、農發和口行債，
以及利率互換（基準利率為SHIBOR3M和FR007）。
Any fluctuation on the Treasury Bonds 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).
稅收調整
Tax Adjustment
與其他債券不同，國債息票免稅。存在稅前和稅后收益差。SuperCurve®在給國債定價時，考慮了免稅因素對理論收益率
的影響。
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.
