Keystone Research

August 17th, 2010

Keystone Research

¡ INTRODUCTION

¡ DESCRIPTION OF CALIBRATION

¡ ALGORITHMIC SETUP

¡ NUMERICAL RESULTS

¡ CONCLUSIONS

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Model calibration is a key step before using any financial market model for pricing and hedging purposes

Introduction

n If the model does not allow an analytic calibration, usually a least squares fit iscomputed with suitable optimization methods

nLarge derivative houses frequently need to calibrate hundreds of underlyings tomarket data

nAlgorithm speed and robustness are necessary to obtain accurate and stable PnL and Greeks in a front office environment

à Modern algorithms are needed for the solution of calibration problems. This is especially true in less efficient markets in Asia, e.g. Japan, Korea

Alos/Ewald (2005), Andersen/Andreasen (2000), Egger/Engl (2005), Hamida/Cont (2005), Mikhailov/Noegel (2003)

Selected approaches in the literature:

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Goal: Choose the model parameters x such that the model matches the market pricesCj

obs of n given standard calls with strikes Kj and maturities Tj

s.t.

Description of Calibration

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It is possible to derive a semi-closed form solution for the price of a standard calloption C by solving the partial differential equation

with suitable final and boundary conditions.

Description of the Calibration Problem

Hence the calibration problem can be rephrased as a (deterministic) nonlinearleast squares problem

with residual function

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An analytic computation of the derivatives of yields

where JR(x) denotes the Jacobian of R(x).

Algorithmic Setup

Since the residuals Rj(x) in the optimal point are usually quite small, we can approximate

the Hessian of f by

à We get a very good approximation of the second derivative by solely making use offirst order information (Gauss Newton approximation)

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Idea: Combine Gauss-Newton approximation of the Hessian with a feasible point trustregion SQP algorithm developed by Wright and Tenny

To preserve feasibility of the iterates we project the solution of (QP) onto the feasibleset after each iteration

Algorithmic Setup

where H is a Gauss-Newton-approximation of the Hessian of the Lagrangian

To compute a stationary point the SQP algorithm successively solves

x

Feasible set

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Projection Theorem:The set of Heston constraints is equivalent to

with an explicit solution of the associated projection problem.

Furthermore, if additional lower and upper boundsare imposed on the parameters, we can solve theprojection problem via the semidefinite program

Algorithmic Setup

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While do

Properties / Advantages:§ Algorithm makes full use of the structure of the problem (Gauss Newton

approximation of the Hessian / projections via SDPs)§ Closed form is only evaluated for parameters inside the feasible set§ Convergence to a Karush-Kuhn-Tucker point is guaranteed (if TOL=0)§ Trust region strategy implicitely regularizes the ill-posed inverse problem

1. Solve (QP) to obtain search direction dk

2. Project xk+dk onto the feasible set by solving the SDP3. Pursue a trust region step size strategy to achieve convergence4. Update the Lagrange multipliers and the Hessian (via Gauss-Newton)

Algorithmic Setup

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Numerical Results

Example: (taken from Andersen, Brotherton-Ratcliffe, 1998)§ Risk-free interest rate r = 6%§ Dividend yield = 2.62%§ Implied volatilities for a set of 100 European call options on the S&P 500 index

Goals:§ Analyze algorithm performance for Heston‘s model with const./TD parameters§ Compare robustness/performance of algorithm to other methods§ Derive ex-ante parameters of Asian markets, e.g. KOSPI, TOPIX, via cross

correlations and local volatilities

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Algorithm output for the calibration of the constant parameter Heston model to the dataset taken from Andersen and Brotherton (1997).

Calibration error at optimal solution

à The calibration usually takes less than one second on a desktop PCà A combination of nonlinear and semidefinite programming leads to a robust and

rapidly converging algorithm

Numerical Results

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Many practitioners apply derivative-free global optimization algorithms, because

àThe calibration results of the FPSQP code are much more stable, although the DSSA algorithm took 40 times longer (Ø 3100 calls of f) to solve the problem

Numerical Results

à But Gauss-Newton methods may actually be more robust in finance applications:

Benchmark: Direct search simul. annealing algorithm of Hedar and Fukushima

Statistics of optimal solutions for 100 randomly chosen start points of the algorithms

§ They are easy to apply§ Ill-conditioning and/or local minima can lead to instable parameters

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Numerical Results

Calibration results for an increasing number m of model parameters:

Results for the regularized calibration with an increasing number m of parameters:

à The time-dependent calibration problem is much more ill-conditioned and requiresadditional regularization, e.g. with

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In addition to improving the condition of the optimization problem the regularization termalso leads to smoother optimal solutions:

Solution of unregularized problem Solution of regularized problem

à The time-dependent parameters reflect the curvature of the volatility surface

Numerical Results

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The feasible point FPSQP algorithm can also be applied to§ Calibration of other models (local volatility, jump diffusions etc.)§ Minimization of other residuals like differences of implied volatilities

Numerical Results

Example: Calibration of the Bates model

where § (Nt)t is a Poisson process with jump intensity § Yi are iid, denoting the relative jump size§ : jump mean, : jump vol§ : Drift adjustment for jump part

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The additionally introduced jump parameters are constrained by lower/upper boundsà The projection is a simple extension of the Heston projectionà Convergence results etc. are also applicable in the Bates case

Implied vol fit of Heston model withtime-dependent parameters

Implied vol fit of time-dependentBates model

Sample results for the dataset of Andersen and Brotherton (1997):

Numerical Results

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Conclusions

nThe proposed feasible point trust region Gauss Newton SQP algorithm combines speed and robustness with an implicit regularization of the calibration problem

nFor most models of practical interest the projection onto the feasible set can either be derived analytically or computed numerically

n If model parameters are time-dependent, additional regularization is needed

n Further regularization is needed for Asian markets via cross correlations and local volatilities

n In the Heston/Bates models, the projection can be computed by solving a semidefinite programming problem