Controladores

In order to solve for the transition equation, we exploit the operator approximation to transform the Kolmogorov forward equation

∂t[p] = (A +J)[p] (A.6)

into the ODE homogeneous system

d

dtp(t) = (A+J)·p(t)⇒p(t) = exp [(A+J)t]·p(0) (A.7)

In the case of the likelihood estimation exercise, the PIDE approximation is formally solved via the exponentiation of the system matrix, as stated in Eq. (A.7). The vectorp(t)contains the stack of the

grid-points attand where the initial conditionp(0)is a representation of the delta-functional. Care has

been taken in the stabilisation of the approximation of the jump-diffusion operator.

First, the one step ahead likelihood has been centred and scaled, such that the grid upon which the solu- tion is constructed stretches around the initial condition and is extended or contracted proportionally to the conditional variance of the state vector, while the number of the grid points is kept constant: centring grants stability in those regions of the domain that would otherwise be extreme, whereas scaling, under model-tailored solutions, grants improved sensitivity to model parameters and further better behaviour around the tails. We exploit, where possible, closed form solutions for the calculation of the conditional expected variances. The exception is represented by the CEV specifications, for which we adopt again the approximation in Eq. (A.20).

Second, the initial condition which is the delta function is modelled as a simple pulse, as the pretty coarse grid we are using cannot justify more articulated proxies. This choice is a consequence of the fact that we do not embrace the strategy of fixing the initial condition covariance matrix at the terminal time values, cfr. for instancePoulsen (1999), as we have found that the shape of the distribution tends to be overblown, when compared to the simulated distribution. Furthermore, at the border of the grid set which defines the solution, we impose that the function is equal to zero. Although the solution usually integrates almost to one, we adjust this feature to make it exact. We use two dimensional trapezoidal rule for numerical integration for the model of interest.

A.2 The PIDE solution

In recent years, the finite different method (FDM) has received renewed attention in continuous-time financial econometrics, since the seminal papers ofLo(1988),Pedersen(1995b) andPoulsen(1999). Ex- amples are Jensen and Poulsen (2002), Lindström(2007), Hurn et al. (2010), Lux (2012). The model problem we tackle is represented by the forward equation (A.6), which is employed in the likelihood esti- mation of Chapter3. Nonetheless, in terms of abstract operators, the option pricing problem of Chapter4 involving the backward equation, can also be regarded under the same formulation. The techniques de- scribed in this section are adapted and applied to both those exercises.

The construction of the PIDE solution ofp(x, v, t)involves the approximation of the integral-differential

operator with the purpose of obtaining a linear system. The solution function is accomplished on a grid of points that are stacked into the time dependent vector

f(t) =vec         p(x1, v1, t) . . . p(xn, v1, t) p(x1, v2, t) . . . p(xn, v2, t) ... . . . ... p(x1, vm, t) . . . p(xn, vm, t)         .

The approach we adopt is to apply the FDM to the differential operator, whereas the numerical integration corresponds to a linear operator that transformf into the sought primitive function. The references for

the finite difference method are represented, for instance, byTavella and Randall(2000),Duffy (2006). In particular, the former reference contains a detailed explanation on how to construct a finite difference operator with a given order of precision, to estimate the partial derivative off, locally. The strategy

consists of taking Taylor expansions of the function in a neighbourhood of each point of the defined grid and then estimating the partial derivative as a linear combination of enough function values to obtain the following

∂•fk= X

i

αifi+R

where R is the residual with the target order of precision. In practice, we can obtain a linear system,

whereby the weights αi are such that annihilate the unnecessary Taylor terms and sum up to one. It

is implicit that the higher the dimension of the system, the more complex the expression to obtain a finite difference operator would result. Whereby the partial differential operators in one dimension are relatively easy to obtain, the mixed derivatives involve instead many alternative formulations. The dif- ferential operators approximations define a banded matrix that approximates the action of the PIDE operatorA in the space dimension. The main problem in constructing the approximation matrix A is

obtaining a stable matrix; that is, a matrix that does not explode under exponentiation. This problem is complicated by the presence of the integral operator, which breaks the banded structure of the system matrixA+J. The general rule we follow is exploiting the Gershgorin circle theorem, see Duffy(2006),

sampling from an allowed range of model parameters. The approximation of the various configurations of the integral operatorJ with the linear discrete operator J is constructed as a matrix of weights that

are obtained either applying the trapezoidal rule, as in the Chapter3 exercise, or analytically integrat- ing1 _{a linear or exponential interpolation within the context of the experiments of Chapter4. The only}

jump size convolution that exploit the linear interpolation is the case of the Lomax jump, whereas all other cases require the exponential interpolation approach. Some assumptions are taken to enclose the integration within the border of the grid over which the numerical solution is constructed. In the case of the likelihood, we assume that over an incremental contour the solution decreases linearly to zero. In the case of the option pricing function, we let the function be constant outside the grid and simply integrate the convolution from the function boundary to the jump domain closure. The latter approach amounts to adding a weight to the corresponding matrix slot, according to the analytic solution of the tail integral. 1_{In the case of the joint jump in return and in volatility, the convolution operator is defined on a two-dimensional plane.} The grid points define a network whereby the integration of the interpolating function is performed over the various types of rectangles that patch the integrating function domain.

A.3. THE NONLINEAR FILTER 158