1. Copyright © 1999-2007 Insightful Corporation. All Rights Reserved. Validation of derivatives pricing models Dr Dario Cziráky

2. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 2 Pricing models: Validation or re-pricing calculation? Validation of pricing models in practice usually implies an independent process of re-calculation, sometimes re-simulation of pricing models used by the front office Alternative calibration and simulation methods are not normally considered Historical performance backtesting is rarely used

3. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 3 Assumptions about the underlying process Whether we are pricing simple options on equities, interests rates, or exotic products, the source of uncertainly rests in the underlying equity or interest rate process Therefore, modelling the underlying process correctly is essential, and vice versa, getting the process wrong will be the main source of pricing errors

4. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 4 Analytical vs. Monte Carlo pricing Many pricing formulas and analytical results are valid only under certain restrictive assumptions about the underlying processes Monte Carlo simulations can be applied to any process Therefore, Monte Carlo simulations are ideal tool for pricing models validation

5. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 5 Alcoa Inc example

6. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 6 Root mean forecast error RMSE of the forecast + Monte Carlo 1 = 119.78 + Monte Carlo 2 = 156.89

7. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 7 What caused the difference? Monte Carlo 1 modelled the equity process by geometric Brownian motion: Monte Carlo 2 modelled the equity process by generalised CIR process:

8. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 8 Validation backtesting Estimate alternative pricing models for a given window, e.g. 250 trading days Roll the estimation by one day across the available historical sample Obtain a matrix with estimated coefficients for every day across the backtest sample Run Monte Carlo pricing from each day and estimate the 250 days a head price Compare forecasting performance of alternative pricing models Compare convergence rates for different Monte Carlo random number generators

9. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 9 CKLS model The CKLS model (Chan, Karolyi, Longstaff and Sanders, 1992) is a generalisation of Vasicek and CIR models and is given by the continuous-time interest rate diffusion Euler discretisation implies the following moment conditions for the CKLS model:

10. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 10 CKLS model The model error can be defined as: GMM estimation can be undertaken by using the following instruments:

11. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 11 CKLS model We can now write down the non-linear error equation and the GMM moment vector:

12. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 12 Simulation of general diffusion processes ou.names = c("kappa", "theta", "sigma") ou.eu.aux1 <- euler.pcode.aux(ndt=25,t.per.sim=1/52, X0 = 0.1, z = z, drift.expr = expression(kappa*(theta - X)), diffuse.expr = expression(sigma),rho.names = ou.names)

13. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 13 GMM functions for GB and CKLS calibration

14. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 14 Calibration functions

15. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 15 Calibration backtest

16. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 16 Calibration model coefficients: Backtest

17. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 17 Simulation functions for CKLS and GB processes

18. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 18 Monte Carlo simulations for year-ahead prices

19. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 19 Multivariate simulation z.mat <- rmvnorm(25*(500 + 1000), mean=colMeans(IRR.ts), cov=var(IRR.ts)) kappa =.4; theta =.08; sigma =.1 sim.cir1 <- CIR.gensim(rho = c(kappa, theta, sigma), n.sim = 1000, n.burn = 500, aux = CIR.aux(X0 = theta, ndt = 25, z = z.mat[,1],t.per.sim = 1/12)) sim.cir2 <- CIR.gensim(rho = c(kappa, theta, sigma), n.sim = 1000, n.burn = 500, aux = CIR.aux(X0 = theta, ndt = 25, z = z.mat[,2],t.per.sim = 1/12)) sim.cir3 <- CIR.gensim(rho = c(kappa, theta, sigma), n.sim = 1000, n.burn = 500, aux = CIR.aux(X0 = theta, ndt = 25, z = z.mat[,3],t.per.sim = 1/12)) sim.cir4 <- CIR.gensim(rho = c(kappa, theta, sigma), n.sim = 1000, n.burn = 500, aux = CIR.aux(X0 = theta, ndt = 25, z = z.mat[,4],t.per.sim = 1/12))

20. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 20 Efficient simulation: Quasi Random Numbers (QRM)

21. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 21 Comparing different random number generators

22. Copyright ©1999-2007 Insightful Corporation. All Rights Reserved. 22 References Bluhm, C., Overbeck, L., and Wagner, C. (2003), An Introduction to Credit Risk Modeling. Chapman & Hall. Chan, K.C., G.A. Karolyi, F.A. Longstaff, and A.B. Sanders (1992). An Empirical Comparison of Alternative Models of the Term Structure of Interest Rates. Journal of Finance, 47, 1209-1227. Diebold, F.X. and Li, C. (2003), Forecasting the Term Structure of Government Bond Yields. NBER Working Paper, No. 10048. El Karoui, N., Frachot, A. and Geman, H. (1998), On the Behavior of Long Zero Coupon Rates in a No Arbitrage Framework. Review of Derivatives Research. 1, 351 – 369. Fisher, M., Nychka, D., and Zervos, D. (1995), Fitting the Term Structure of Interest Rates with Smoothing Splines. Finance and Economics Discussion Series, 1995-1. Board of Governors of the Federal Reserve System. London, J. (2005), Modeling Derivatives in C++. Hoboken: John Wiley. Nelson, C.R. and Siegel, A.F. (1987), Parsimonious Modeling of Yield Curves, Journal of Business, 60(4), 473 – 489. Scherer, B. and Martin, R.D. (2005), Introduction to Modern Portfolio Optimization With NUOPT and S-Plus. New York: Springer. Svensson, L.E.O. (1994), Estimating and Interpreting Forward Interest Rates: Sweden 1992 – 1994. NBER Working Paper No. 4871. Zivot, E. and Wang, J. (2006), Modeling Financial Time Series with S-Plus. New York: Springer.

23. Copyright © 1999-2007 Insightful Corporation. All Rights Reserved. Questions