Bullet Bond Hull White model Yield rate Bloomberg

Bullet Bond & Hull White model

Yield rate • 從Bloomberg 得到 • Risky yield 部分只有 6 M 1 Y 3 Y 5 Y 7 Y 10 Y 20 Y 30 Y 八筆 • 線性內插成 60筆yield (0. 5 Y , 1 Y, 1. 5 Y, 2. 5 Y, ……, 29. 5 Y , 30 Y) 用void linear(double *Yield, double *Data)

Zero Rate • 用線性內插法得來的60筆yield rate bootstrap出 60筆零息利率 void bootstrap(double *Zero. Curve, double *Raw. Data)

Bootstrap

Day count (t to T) • struct date • int year; • int month; • int day; • double Cal_day(date &date 1, date &date 2) • 比方 2017/8/15 到 2045/2/3

Zero. Rate(T-t) • 抓最接近的index來內插 • 比如T-t =27. 4666666 • 找Zerorate 27. 5年與Zerorate 27年來內插

Bond price • Bullet bond price = par*(1+implied_r)^30*exp(-Zerorate*(T-t))

Call schedule 舉例

短利模型的種類 • Equilibrium model: • • 對經濟模型做出假設, 進而推出短利的隨機過程短利隨機過程決定決定利率期限結構 Rendleman & Bartter model Vasicek model Hull-White model為其推廣 • No arbitrage model: • 通常為Equilibrium model的推廣

No-arbitrage model 簡介 • No arbitrage model: • • • 包含Equilibrium model的性質市場的利率結構當作輸入的參數造出來利率結構和市場觀察到的結構相符 Ho-Lee Model Hull-White Model

No-arbitrage model •

步驟一產生的三元樹 Δt=1 year E (2, 2) B (1, 1) F (2, 1) G (2, 0) C (1, 0) D (1, -1) A (0, 0) H (2, -1) I (2, -2) 對 0 mean reversion Node R A B jmax=2 C D E F G H I 0. 000% 1. 732% 0. 000% -1. 732% 3. 464% 1. 732% 0. 000% -1. 732% -3. 464% pu pm 0. 1667 0. 1217 0. 1667 0. 2217 0. 8867 0. 1217 0. 1667 0. 2217 0. 0867 0. 6666 0. 6566 0. 0266 0. 6566 0. 6666 0. 6566 0. 0266 pd 0. 1667 0. 2217 0. 1667 0. 1217 0. 0867 0. 2217 0. 1667 0. 1217 0. 8867 Let a=0. 1 σ=0. 01

建立Hull-White tree步驟一決定分支機率: Type A node match mean and variance See the proof in next slide Ex: Node B, F j=1 Node C, G j=0 Node D, H j=-1

Match the Variance

建立Hull-White tree步驟一決定分支機率: Type B node match mean and variance Ex: Node I j=-2

建立Hull-White tree步驟一決定分支機率: Type C node match mean and variance Ex: Node E j=2

步驟二校正後的三元樹 E F B G C H A D I 平移 3. 824% Node R pu pm pd A 平移 6. 252% B C D E F G H I 3. 824% 6. 937% 5. 205% 3. 473% 9. 716% 7. 984% 6. 252% 4. 520% 2. 788% 0. 1667 0. 1217 0. 1667 0. 2217 0. 8867 0. 1217 0. 1667 0. 2217 0. 0867 0. 6666 0. 6566 0. 0266 0. 6566 0. 6666 0. 6566 0. 0266 0. 1667 0. 2217 0. 1667 0. 1217 0. 0867 0. 2217 0. 1667 0. 1217 0. 8867 平移 5. 205%

Bootstrap-1 year(程式更動部分) • void linearinterpolation. Risky(double *yield, double *data_yield)； • void bootstrap(double *Zero. Curve, double *Treasury. Yield. Raw. Data, int type, int plusday)； • void Rky. Zero. Rate. Interpolation(double* Zero_HW, double *Zero. Rate, int N, double Delta. T_first, double Delta. T)

Recovery rate定義 •

Recovery rate定義的改變 •

Reduced form model •