On Stiffness in Affine Asset Pricing Models By Shirley J. Huang and Jun Yu University of Auckland & Singapore Management University

Outline of Talk Motivation and literature Stiffness in asset pricing Simulation results Conclusions

Motivation and Literature Preamble: “…around 1960, things became completely different and everyone became aware that world was full of stiff problems…” Dahlquist (1985)

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.

Motivation and Literature Computational burden is heavier for econometric analysis of continuous-time asset pricing models Reasons: 1. 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 (Lo;1988). 2. Asset prices themselves are numerical solutions to PDEs.

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).

Motivation and Literature Examples: 1. 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

Motivation and Literature 2. “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

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.

Motivation and Literature AD: Asset Price, TD: Transition density, CF: Charateristic function Closed Form AP Closed Form TD Closed Form AP TD is Obtained via PDE CF is Obtained via ODE Affine Asset Pricing Models AP is Obtained via ODE TD is Obtained via PDE CF is Obtained via ODE

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.

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.

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.

Motivation and Literature To illustrate stiff problems, consider with initial conditions

Motivation and Literature This linear system has the following exact solution: The second term decays very fast while the first term decays very slowly.

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

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

Motivation and Literature This requires 0<h<0.002 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 |1+z|<1). For this reason, the explicit Euler method is not A-Stable.

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.

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

21 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.

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

Motivation and Literature So the stability region is

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

Stiffness in Asset Pricing The multi-factor affine term structure model adopts the following specifications: 1. Under risk-neutrality, the state variables follows 2. The short rate is affine function of Y(t) 3. The market price of risk with factor j is

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.

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

Stiffness in Asset Pricing The stiffness ratios are and 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.

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(t+1) on Y(t) where

Stiffness in Parameter Estimation Stiffness ratios implied by the existing studies: Geyer and Pichler (1999): Chen and Scott (1991, 1992): Dai and Singleton (2001): ranging from 28.9 to 78.9.

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: 1. Pricing bonds under the two-factor square root model 2. Estimating parameters in the two-factor square root model using CF

Comparison of Nonstiff and Stiff Solvers The true model The parameters for market prices of risk are Hence The stiffness ratios are and 1200 respectively.

Simulation Results Bond prices with 5, 10, 20, 40-year maturity 5-year bond10-year bond cpu (s) Yield (%) Step size cpu (s) Yield (%) Step size Exact ode ode ode23s ode15s

Simulation Results 20-year bond40-year bond cpu (s) Yield (%) Step size cpu (s) Yield (%) Step size Exact ode ode ode23s ode15s

36 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.

Simulation Results k1mu1sigma1lamda1Itercpu(s) true ode ode ode23s ode15s

Simulation Results k2mu2sigma2lamda2Itercpu(s) true ode ode ode23s ode15s

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.