On Stiffness in Affine Asset Pricing Models

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On Stiffness in Affine Asset Pricing Models

• By Shirley J. Huang and Jun Yu
• University of Auckland
• Singapore Management University

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Outline of Talk

• Motivation and literature
• Stiffness in asset pricing
• Simulation results
• Conclusions

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Motivation and Literature

• Preamble
• around 1960, things became completely
  different and everyone became aware that world
  was full of stiff problems Dahlquist (1985)

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Motivation and Literature

• When valuing financial assets, one often needs to
  find the numerical solution to a partial
  differential equation (PDE) see Duffie (2001).
• In many practically relevant cases, for example,
  when the number of states is modestly large,
  solving the PDE is computationally demanding and
  even becomes impractical.

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Motivation and Literature

• Computational burden is heavier for econometric
  analysis of continuous-time asset pricing models
• Reasons
• Transition density are solutions to PDEs which
  have to be solved numerically at every data point
  and at each iteration of the numerical
  optimizations when maximizing likelihood
  (Lo1988).
• Asset prices themselves are numerical solutions
  to PDEs.

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Motivation and Literature

• The computational burden in asset pricing and
  financial econometrics has prompted financial
  economists econometricians to look at the class
  of affine asset pricing models where the
  risk-neutral drift and volatility functions of
  the process for the state variable(s) are all
  affine (i.e. linear).

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Motivation and Literature

• Examples
• Closed form expression for asset prices or
  transition densities
• Black and Scholes (1973) for pricing equity
  options
• Vasicek (1977) for pricing bonds and bond options
• Cox, Ingersoll, and Ross (CIR) (1985) for pricing
  bonds and bond options
• Heston (1993) for pricing equity and currency
  options

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Motivation and Literature

• Nearly closed-form expression for asset prices
  in the sense that the PDE is decomposed into a
  system of ordinary differential equations (ODEs).
  Such a decomposition greatly facilitates
  numerical implementation of pricing (Piazzesi,
  2005).
• Duffie and Kan (1996) for pricing bonds
• Chacko and Das (2002) for pricing interest
  derivatives
• Bakshi and Madan (2000) for pricing equity
  options
• Bates (1996) for pricing currency options
• Duffie, Pan and Singleton (2000) for a general
  treatment

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Motivation and Literature

• If the transition density (TD) has a closed form
  expression, maximum likelihood (ML) is ready to
  used.
• For most affine models, TD has to be obtained via
  PDEs.
• Duffie, Pan and Singleton (2000) showed that the
  conditional characteristic function (CF) have
  nearly closed-form expressions for affine models
  in the sense that only a system of ODEs has to be
  solved
• Singleton (2001) proposed CF-based estimation
  methods.
• Knight and Yu (2002) derived asymptotic
  properties for the estimators. Yu (2004) linked
  the CF methods to GMM.

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Motivation and Literature

• AD Asset Price, TD Transition density, CF
  Charateristic function

Affine Asset Pricing Models
AP is Obtained via ODE TD is Obtained via PDE CF
is Obtained via ODE
Closed Form AP Closed Form TD
Closed Form AP TD is Obtained via PDE CF is
Obtained via ODE
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Motivation and Literature

• The ODEs found in the literature are always the
  Ricatti equations. It is generally believed by
  many researchers that these ODEs can be solved
  fast and numerically efficiently using
  traditional numerical solvers for initial
  problems, such as explicit Runge-Kutta methods.
  Specifically, Piazzesi (2005) recommended the
  MATLAB command ode45.

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Motivation and Literature

• Ode45 has high order of accuracy
• It has a finite region of absolute stability
  (Huang (2005) and Butcher (2003)).
• The stability properties of numerical methods are
  important for getting a good approximation to the
  true solution.
• At each mesh point there are differences between
  the exact solution and the numerical solution
  known as error.
• Sometimes the accumulation of the error will
  cause instability and the numerical solution will
  no longer follow the path of the true solution.
• Therefore, a method must satisfy the stability
  condition so that the numerical solution will
  converge to the exact solution.

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Motivation and Literature

• Under many situations that are empirically
  relevant in finance the ODEs involve stiffness, a
  phenomenon which leads to certain practical
  difficulties for numerical methods with a finite
  region of absolute stability.
• If an explicit method is used to solve a stiff
  problem, a small stepsize has to be chosen to
  ensure stability and hence the algorithm becomes
  numerically inefficient.

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Motivation and Literature

• To illustrate stiff problems, consider
• with initial conditions

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Motivation and Literature

• This linear system has the following exact
  solution
• The second term decays very fast while the first
  term decays very slowly.

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Motivation and Literature

• This feature can be captured by the Jacobian
  matrix
• It has two very distinct eigenvalues, -1 and
  -1000. The ratio of them is called the stiffness
  ratio, often used to measure the degree of
  stiffness.

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Motivation and Literature

• The system can be rotated into a system of two
  independent differential equations
• If we use the explicit Euler method to solve the
  ODE, we have

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Motivation and Literature

• This requires 0lthlt0.002 for a real h (step size)
  to fulfill the stability requirement. That is,
  the explicit Euler method has a finite region of
  absolute stability (the stability region is given
  by 1zlt1). For this reason, the explicit Euler
  method is not A-Stable.

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Motivation and Literature

• For the general system of ODE
• Let be the Jacobian matrix. Suppose
  eigenvalues of J are
• If
  we say the ODE is stiff. R is the stiffness
  ratio.

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Motivation and Literature

• The explicit Euler method is of order 1. Higher
  order explicit methods, such as explicit
  Runge-Kutta methods, will not be helpful for
  stiff problems. The stability regions for
  explicit Runge-Kutta methods are as follows

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Motivation and Literature
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Motivation and Literature

• To solve the stiff problem, we have to use a
  method which is A-Stable, that is, the stability
  region is the whole of the left half-plane.
• Dalhquist (1963) shows that explicit Runge-Kutta
  methods cannot be A-stable.
• Implicit methods can be A-stable and hence should
  be used for stiff problems.

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Motivation and Literature

• To see why implicit methods are A-stable,
  consider the implicit Euler method for the
  following problem
• The implicit Euler method implies that

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Motivation and Literature

• So the stability region is

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Motivation and Literature

• Higher order implicit methods include implicit
  Runge-Kutta methods, linear multi-step methods,
  and general linear methods. See Huang (2005).

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Stiffness in Asset Pricing

• The multi-factor affine term structure model
  adopts the following specifications
• Under risk-neutrality, the state variables
  follows
• The short rate is affine function of Y(t)
• The market price of risk with factor j is

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Stiffness in Asset Pricing

• Hence the physical measure is also affine
• Duffie and Kan (1996) derived the expression for
  the yield-to-maturity at time t of a zero-coupon
  bond that matures at in the Ricatti form,
• with initial conditions A(0)0, B(0)0.

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Stiffness in Asset Pricing

• Dai and Singleton (2001) empirically estimated
  the 3-factor model in various forms using US
  data.
• Using one set of their estimates, we obtain
• Using another set of their estimates, we obtain

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Stiffness in Asset Pricing

• The stiffness ratios are 9355.6 and 52.76
  respectively. Hence the stiff is severe and
  moderate.
• However, in the literature, people always use the
  explicit Runge-Kutta method to solve the Ricatti
  equation.

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Stiffness in Parameter Estimation

• Based on the assumption that the state variable
  Y(t) follow the following affine diffusion under
  the physical measure
• Duffie, Pan and Singleton (2000) derived the
  conditional CF of Y(t1) on Y(t)
• where

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Stiffness in Parameter Estimation

• Stiffness ratios implied by the existing studies
• Geyer and Pichler (1999) 2847.2.
• Chen and Scott (1991, 1992) 351.9.
• Dai and Singleton (2001) ranging from 28.9 to
  78.9.

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Comparison of Nonstiff and Stiff Solvers

• Compare two explicit Runge-Kutta methods (ode45,
  ode23), an implicit Runge-Kutta method (ode23s),
  and an implicit linear multistep method (ode15s).
• Two experiments
• Pricing bonds under the two-factor square root
  model
• Estimating parameters in the two-factor square
  root model using CF

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Comparison of Nonstiff and Stiff Solvers

• The true model
• The parameters for market prices of risk are
• Hence
• The stiffness ratios are 3333.3 and 1200
  respectively.

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Simulation Results

• Bond prices with 5, 10, 20, 40-year maturity

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Simulation Results
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Simulation Results

• Parameter estimation 100 bivariate samples, each
  with 300 observations on 6-month zero coupon bond
  and 300 observations on 10-year zero coupon bond,
  are simulated and fitted using the CF method.

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Simulation Results
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Simulation Results
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Conclusions

• Stiffness in ODEs widely exists in affine asset
  pricing models.
• Stiffness in ODEs also exists in non-affine asset
  pricing models. Examples include the quadratic
  asset pricing model (Ahn et al 2002).
• Stiff problems are more efficient solved with
  implicit methods.
• The computational gain is particularly
  substantial for econometric analysis.