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

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

Validation of derivatives pricing modelsDr Dario Cziráky

Copyright ©1999-2007Insightful 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


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


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


Alcoa Inc example


Root mean forecast error

•RMSE of the forecast+ Monte Carlo 1 = 119.78

+ Monte Carlo 2 = 156.89


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:


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


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:


CKLS model

•The model error can be defined as:

•GMM estimation can be undertaken by using the following instruments:


CKLS model

•We can now write down the non-linear error equation and the GMM moment vector:


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)


GMM functions for GB and CKLS calibration


Calibration functions


Calibration backtest


Calibration model coefficients: Backtest


Simulation functions for CKLS and GB processes


Monte Carlo simulations for year-ahead prices


Multivariate simulationz.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))








Efficient simulation: Quasi Random Numbers (QRM)


Comparing different random number generators


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 ofGovernors 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.

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

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