Solution of Stochastic Differential Equations in Finance

Weak Order 2 Taylor Method

w₀ = X₀
w_i+1 = w_i + a∆t_i + b∆W_i + ¹ bb_x(∆W ² − ∆t_i)
2 ⁱ
+ a_xb∆Z_i + ¹ (aa_x + ¹ a_xxb²)∆t²
2 2

2
+ (ab_x + ¹ b_xxb²)(∆W_i∆t_i − ∆Z_i) (18) where ∆W_i is chosen from ^√∆t_iN (0, 1) and ∆Z_i is distributed as in the above Strong Order 1.5 Method.
A second approach is to mimic the idea of Runge-Kutta solvers for ordinary differential equations. These solvers replace the explicit higher derivatives in the Ito-Taylor solvers with extra function evaluations at interior points of the current solution interval. Platen (1987) proposed the following weak order 2 solver of Runge-Kutta type:

Weak Order 2 Runge-Kutta Method

w₀ = X₀

2
w_i+1 = w_i + ¹ [a(u) + a(w_i)]∆t_i

where
1

1
+ ₄ [b(u₊) + b(u₋) + 2b(w_i)]∆W_i

^— i

4
+ [b(u₊) − b(u )](∆W ² − ∆t)/ ∆t_i (19)
u = w_i + a∆t_i + b∆W_i

u₊ = w_i + a∆t_i + b ∆t_i u₋ = w_i + a∆t_i − b ∆t_i.
0
10
−1
10

error
−2
10
−3
10
−4
¹⁰ −2 −1 0
10 10 10

time step t

Fig. 4. The mean error of the estimation of E(X(T )) for SDE (15). The plot compares the Euler-Maruyama method (circles) which has weak order 1, and the weak order 2 Runge-Kutta type method (squares) given in (19). Parameter used were X(0) = 10, T = 1, µ = −3, σ = 0.2.
Fig. 4 compares the Euler-Maruyama method, which converges with order 1 in the weak sense, to the Weak Order 2 Runge-Kutta-Type Method. Note the difference between strong and weak convergence. In the previous Fig. 3, which considers strong convergence, the mean error of the estimate of a point X(T ) on the solution curve was plotted. In Fig. 4, on the other hand, the mean error of the estimate of the expected value E[X(T )] is plotted, since we are comparing weak convergence of the methods. The weak orders are clearly revealed by the log-log plot.

Several other higher-order weak solvers can be found in Kloeden and Platen (1992). Weak Taylor methods of any order can be constructed, as well as Runge-Kutta analogues that reduce or eliminate the derivative cal- culations. In addition, standard Richardson extrapolation techniques (Sauer, 2006) can be used to bootstrap weak method approximations of a given order to the next order. See Kloeden and Platen (1992) for full details.
Weak solvers are often an appropriate choice for financial models, when the goal is to investigate the probability distribution of an asset price or interest rate, or when Monte-Carlo sampling is used to price a complicated derivative. In such cases it is typical to be primarily interested in one of the statistical moments of a stochastically-defined quantity, and weak methods may be simpler and still sufficient for the sampling purpose. In the next section we explore some of the most common ways SDE solvers are used to carry out Monte-Carlo simulations for derivative pricing.

5.