Solution of Stochastic Differential Equations in Finance

Milstein Method

w₀ = X₀
w_i+1 = w_i + a(w_i, t_i)∆t_i + b(w_i, t_i)∆W_i
+ 1 ^∂b 2
₂ b(w_i, t_i) _∂x (w_i, t_i)(∆W_i − ∆t_i) (14)
The Milstein Method has order one. Note that the Milstein Method is identical to the Euler-Maruyama Method if there is no X term in the diffusion part b(X, t) of the equation. In case there is, Milstein will in general converge to the correct stochastic solution process more quickly than Euler-Maruyama as the step size ∆t_i goes to zero.
For comparison of the Euler-Maruyama and Milstein methods, we apply
them to the Black Scholes stochastic differential equation


dX = µX dt + σX dW_t. (15)
We discussed the Euler-Maruyama approximation above. The Milstein Method becomes


w₀ = X₀ (16)
w_i+1 = w_i + µw_i∆t_i + σw_i∆W_i + ¹ σ(∆W ² − ∆t_i)
2 ⁱ
Applying the Euler-Maruyama Method and the Milstein Method with de- creasing stepsizes ∆t results in successively improved approximations, as Table 1 shows:
The two columns represent the average, over 100 realizations, of the error
|w(T )−X(T )| at T = 8. The orders 1/2 for Euler-Maruyama and 1 for Milstein are clearly visible in the table. Cutting the stepsize by a factor of 4 is required to reduce the error by a factor of 2 with the Euler-Maruyama method. For the Milstein method, cutting the stepsize by a factor of 2 achieves the same result. The data in the table is plotted on a log-log scale in Fig. 3.
The Milstein method is a Taylor method, meaning that it is derived from a truncation of the stochastic Taylor expansion of the solution. This is in many cases a disadvantage, since the partial derivative appears in the approximation method, and must be provided explicitly by the user. This is analogous to Tay- lor methods for solving ordinary differential equations, which are seldom used in practice for that reason. To counter this problem, Runge-Kutta methods were developed for ODEs, which trade these extra partial derivatives in the Taylor expansion for extra function evaluations from the underlying equation.