1	"Bates Model" Bates Model: A Model Analysis
2	"Bates Model"
3	"Prospect"
4	Jump process Jumps Homogeneous Poisson process Distributions for Jump Size Log Normal Double exponential
5	Poisson process
6	Poisson process as martingale
7
8	Jump Size distributions
9	Toy implementation Toy implementation Bates Model (Log-Jump)
10	Bates model: Analytical exp. Analytical expressions A. Sepp (2004) developed closed forms to evaluate vanilla options (characteristic functions) www.javaquant.net/papers/seppthesis.pdf Closed forms + optimization algorithm = calibration machinery. QL: implemented and tested this features
11	Googling QL NX and QL have similar design Smart pointers Object Handlers Lazy objects, observers, observables QuantLib-addin, QuantLibXL Webpage: http://quantlib.org
12	Googling QL: test examples Vanilla priced with Bates Engine and BS Rho = 0, Lambda = 0, Nu = 0 Vanilla Bates Engine and Merton76 Xi = 0, Rho = 0 -> BS + Merton76 Calibration of Bates Model to DAX options www.javaquant.net/papers/stochjumpvols.pdf A little less conversation, and a little bit more of action…..(some code)
13	Testing QL Bates and Pricing Cliquets Options Vanilla Cliquet Market Data Simple Realistic Methods Analytical Monte Carlo NX Models BS, Dupire, SABR and Heston Calibration algorithm: GA (Heston) QL Models Heston, Bates Calibration algorithm: L-M (Heston, Bates)
14	Vanillas: Test1 Compare Analytical and MC Bates Engines to known results (Nx, BBG) FX PUT option DomYC = 5.215%, ForYC = 4.44%, Vol = 8.844%, Spot = 1.4436, Strike= 1.4675, Maturity = 20days, Notional = EUR1,000,000, Model = BS 1) Analytical BS Engines: Nx = USD27,002.53 BBG = USD27,002.53 QL = USD27,064.22 2) QL Bates Analytical Engine, Parameters->Black Scholes QL = USD27,064.22 3) Nx Heston Pricer, Parameters->Black Scholes Nx ForwardMC = USD27,001.71. Nx BackwardPDE = USD27,001.59 4) QL Bates European MC Engine, Parameters->Black Scholes. QL = USD27,366.36 (tolerance 5%) Conclusion: Bates analytical and MC engine reproduces well known vanilla results
15	Cliquet: Test2 Compare Nx and QL Prices, flat YC-Div-Vol Test 2 Model: Heston, constant parameters Yield Curve: Flat Dividends: Flat Volatility: Flat Methods: Nx and QL Monte Carlo. NumeriX Forward MC Heston result = 67.967 bps BS result = 68.248 bps QuantLib Forward MC Bates = 68.130 bps Conclusion: More than flat term needed to reproduce market quote
16	Cliquet: Test3 and Test3b Compare Heston Calibration parameters (Nx and QL) and Price results Test 3 and Test3b Model: Heston, constant parameters Yield Curve: Flat Dividends: Flat Volatility: Realistic Calibration Methods: Genetic Algorithm (Nx) Levenberg-Marquardt (QL). 1) Nx Heston, calibrated parameters will be used by QL. Nx Heston = 124.1592 bps Kappa = 3.992337 Vol^2 = 0.03894531 Sigma = 0.78870795 Rho =-0.63044743 Theta = 0.05841538 2) QL Bates (own calibration) QL Bates = 97.331 bps Kappa = 3.76631 Vol^2 = 0.0346356 Sigma = 0.790490 Rho = -0.9999999 Theta = 0.0639668 Conclusion: Different calibration algorithms (NX-GA,QL-LM) may give different parameters
17	Cliquet: Test3a To obtain Price using Bates MC engine (QL) using market vol, flat YC&Div Test 3a Model: Bates, constant parameters Yield Curve: Flat Dividends: Flat Volatility: Realistic Methods: QL Monte Carlo. 1) QL Bates. This time keeping own parameters. QL Bates = 164.31 bps nu = -0.08147 delta = 0.13959 lambda = 0.820949 theta = 0.058415 kappa = 1.112211 xi = 0.51393561 rho = -0.99999999 v0^2 = 0.0233669 Conclusion: Vol Skew seems to improve results
18	Cliquet: Test4 Get Price using Bates MC Engine for full market data Test 4 Model: Bates, constant parameters Yield Curve: Realistic Dividends: Continuous rate Volatility: Realistic Methods: QL Monte Carlo. 1) QL Bates. Keeping own calibration parameters. QL Bates = 187.80 bps nu = -0.0786981 delta = 0.138449 lambda = 0.816217 theta = 0.055030 kappa = 1.0326335 xi = 0.513779 rho = -0.99999999 v0^2 = 0.0235143 Conclusion: Realistic Market Data improve results, still quite off Market Quote.
19	Cliquet: Test5 Get Price using Bates MC Engine and Heston Nx MC, No Dividends Test 5 Model: Bates, constant parameters Yield Curve: Realistic Dividends: Zero Volatility: Realistic Methods: QL Monte Carlo. 1) Nx Heston. QL Bates = 127.22 bps 1) QL Bates. Keeping own calibration parameters. QL Bates = 244.74 bps Conclusion: Getting the Forward crucial when pricing Cliquets
20	Conclusions Bates Model might improve Cliquet results, though other factor might be critical (forward curve). Bates Model is superior to BS, Dupire, SABR and Heston for short term options (skew). Jumps are a requirement when pricing certain skew sensitive options.