Transcript Calibration Illustration.pptx

Regime Calibration
Calibration Steps 1
• Define, vector 𝑉1, indicates when the regime switch take place. Define, vector 𝑉2, indicates what the new regime target is when the previous
regime is replaced.
• Find V1
• Define a vector 𝑃 𝑖 by
𝑃 𝑖 = 1,
𝑃 𝑖 = 0.5,
𝑃 𝑖 = −0.5,
𝑃 𝑖 = −1,
𝑖𝑓 W < ∆𝑙𝑛 𝑅𝑖
𝑖𝑓 0 ≤ ∆𝑙𝑛 𝑅𝑖 ≤ 𝑊
𝑖𝑓 − 𝑊 ≤ ∆𝑙𝑛 𝑅𝑖 ≤ 0
𝑖𝑓 ∆𝑙𝑛 𝑅𝑖 < −𝑊
• Define another vector 𝐶𝑉 𝑗 by
𝑖+12
𝐶𝑉 𝑖 = 1,
𝑖𝑓 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑉𝑎𝑙𝑢𝑒 [
𝑃 𝑗 ] ≥ 𝐶𝑟𝑖𝑡𝑒𝑟𝑖𝑎
𝑗=𝑖
𝑖+12
𝐶𝑉 𝑖 = 0,
𝑖𝑓 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑉𝑎𝑙𝑢𝑒 [
𝑃 𝑗 ] < 𝐶𝑟𝑖𝑡𝑒𝑟𝑖𝑎
𝑗=𝑖
• If 𝐶𝑉 𝑖 = 1, then we say that in ith month, there could be a regime switching, in another word, a regime switching candidates. If 𝐶𝑉 𝑖 =
0, then we say there is no regime switching in ith month. The Criteria is chosen to make the regime switching candidates just more than
40% of all rates.
Regime Calibration
Calibration Steps 2
• Find V2
• Starting from first rates, for each of next 40 regime-switching candidates:
• Assume this is the regime-switching point, trying different target of interest rate (discretely) until we find one that yields minimum
SSE by formula:
𝑙𝑛 𝑟𝑡 = 1 − 1 − 𝐹
𝑆𝑆𝐸 =
𝑑𝑡
𝑙𝑛 𝑇𝑡 + 1 − 𝐹
𝑑𝑡
2
𝑙𝑛 𝑟𝑡 − 𝑙𝑛 𝑟𝑡
𝑙𝑛 𝑟𝑡−𝑑𝑡
• Choose the global minimum SSE generated from 40 regime-switching candidates, and make the first regime switching point.
• Use the new switching point as the starting point, loop above.
• For Reversion F
• We use the F from moment calibration work, which is 0.386806339
(1)
(1)
• Ideally F should be involved into calibration in the way: 𝐹 (1) → 𝑉1 , 𝑉2
we have better pre-determined F, it’s not necessary to so.
(2)
(2)
→ 𝐹 (2) → 𝑉1 , 𝑉2
• Eventually using the determined V1 to calibrate parameters in Gamma(a, b) by MLE method.
(3)
(3)
→ 𝐹 (3) → 𝑉1 , 𝑉2
→ ⋯ , However, since
Moment Calibration
Calibration Step 1 -Volatility Parameters
• Match moments of history delta ln rate by assuming the model follows a pure Di-GenGamma distribution
• Algorithm
• Conduct three equations by matching three moments, dlnM 𝑗
𝑑𝑙𝑛𝑀𝑗 (𝛼, 𝛽, 𝜏) = 𝑑𝑙𝑛𝑀𝑗 , 𝑗 = 1,2,3
• In order to solve the equation, take the LS-sum
𝑗=1
𝑑𝑙𝑛𝑀𝑗 (𝛼, 𝛽, 𝜏) = −𝑑𝑙𝑛𝑀𝑗
𝑆𝐷𝑗
2
• Use Excel-Solver to find values of 𝛼, 𝛽, 𝜏 and implement into our model. Manually adjust parameters through test and trials
Moment Calibration
Calibration Step 2 - Drift Parameters
• Algorithm
• For 𝑖𝑡ℎ drift parameter δi , test 100 equally spaced values on interval [li, ui] with 1000 scenarios
• For each moment of history rate Mj, fit a linear regression with respect to δi :
𝑀𝑖𝑗 = 𝑎𝑖𝑗 δ𝑖 + 𝑏𝑖𝑗 𝑤𝑖𝑡ℎ 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝛽𝑖𝑗
• Conduct a LS-Sum
𝑗=1
𝑀𝑖𝑗 − 𝑀𝑗
𝑆𝐷𝑗
2
∗ 𝐼(𝛽𝑖𝑗 ) ,
𝑤ℎ𝑒𝑟𝑒 𝑆𝐷𝑗 𝑖𝑠 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑖 𝑡ℎ 𝑚𝑜𝑚𝑒𝑛𝑡,
0, 𝑖𝑓𝛽𝑖𝑗 < 0.5
& 𝐼 𝛽𝑖𝑗 =
1, 𝑖𝑓𝛽𝑖𝑗 > 0.5
•
• Use Excel-Solver to find the that δi minimize the sum, and move on to next (𝑖 + 1)𝑡ℎ parameter
Loop above until converge
Distance Calibration
Calibration Step
• Assume the dlnRate of our model follows a shape adjusted Di-GenGamma distribution -- Di-Gengamma(a,b,t)
• Set the mode of Di-Gengamma(a,b,t) to be within 0.004
• Loop
• Match PDF(History) vs. PDF(Di-Gengamma)
• Minimizing SSE by using Excel-Solver on parameters Di-Gengamma(a,b,t)
• Using the calibrated Di-Gengamma(a,b,t) to run model, and get PDF(Model)
• Match PDF(Model) vs. PDF(Di-Gengamma)
• Introduce shape adjustment parameters a*(index+b), we will only change the index (dlnr value/PDF bin) so that the parameters
a,b,t for shape adjusted Di-GenGamma distribution will be the same with our model
• Minimizing SSE by using Excel-Solver on shape adjustment parameters a,b
• Using the shape adjusted index (dlnr value or pdf bin) to loop above until converge
• The convergence means both a,b and a,b,t converges
• The convergence is fast