Solution of Stochastic Differential Equations in Finance

dX = a(t, X) dt + b(t, X) dW_t (1)
Notice that the SDE (1) is given in differential form, unlike the derivative form of an ODE. That is because many interesting stochastic processes, like Brow- nian motion, are continuous but not differentiable. Therefore the meaning of the SDE (1) is, by definition, the integral equation

X(t) = X(0) +
_t
a(s, y) ds +
0
_t
b(s, y) dW_s,
0

where the meaning of the last integral, called an Ito integral, will be defined next.
Let c = t₀ < t₁ < . . . < t_n−1 < t_n = d be a grid of points on the interval [c, d]. The Riemann integral is defined as a limit

_d
f (x) dx = lim


n

i
^J f (t^l )∆t_i,

_c ∆t→0 _i=1

i
where ∆t_i = t_i − t_i−1 and t_i−1 ≤ t^l ≤ t_i. Similarly, the Ito integral is the limit

_d
f (t) dW_t = lim


n

−
^J f (t_i
1)∆W_i

_c ∆t→0 _i=1

i
where ∆W_i = W_ti − W_ti−1 , a step of Brownian motion across the interval. Note a major difference: while the t^l in the Riemann integral may be chosen at any point in the interval (t_i−1, t_i), the corresponding point for the Ito integral is required to be the left endpoint of that interval.

_d
Because f and W_t are random variables, so is the Ito integral I =
_c f (t) dW_t. The differential dI is a notational convenience; thus
_d

I =
is expressed in differential form as
f dW_t
c

dI = f dW_t.
The differential dW_t of Brownian motion W_t is called white noise. A typical solution is a combination of drift and the diffusion of Brownian motion.
To solve SDEs analytically, we need to introduce the chain rule for stochas- tic differentials, called the Ito formula:
If Y = f (t, X), then

2
dY = ^∂f (t, X) dt + ^∂f (t, X) dx + ^{1 ∂
f} (t, X) dx dx (2)

∂t ∂x
2 ∂x²

where the dx dx term is interpreted by using the identities


dt dt = 0
dt dW_t = dW_t dt = 0
dW_t dW_t = dt (3)

The Ito formula is the stochastic analogue to the chain rule of conventional calculus. Although it is expressed in differential form for ease of understanding, its meaning is precisely the equality of the Ito integral of both sides of the equation. It is proved under rather weak hypotheses by referring the equation back to the definition of Ito integral (Oksendal, 1998).

Some of the important features of typical stochastic differential equations can be illustrated using the following historically-pivotal example from fi- nance, often called the Black-Scholes diffusion equation:

⁽ dX = µX dt + σX dW_t X(0) = X₀
(4)

with constants µ and σ. Although the equation is comparatively simple, the fact that it can be exactly solved led to its central importance, by making a closed-form formula available for the pricing of simple options (Black and Scholes, 1973).

The solution of the Black-Scholes stochastic differential equation is geo- metric Brownian motion
¹ 2
X(t) = X₀e^{(µ− 2 σ )t+σWt} . (5)

2
To check this, write X = f (t, Y ) = X₀e^Y , where Y = (µ − ¹ σ²)t + σW_t. By

2
the Ito formula,
dX = X₀e^Y dY + ¹ e^Y dY dY

2
where dY = (µ − ¹ σ²) dt + σ dW_t. Using the differential identities from the

Ito formula,
and therefore
dY dY = σ² dt,

dX = X₀e^Y (r − ¹ σ²) dt + X₀e^Y σ dW_t + ¹ σ²e^Y dt
2 2
= X₀e^Y µ dt + X₀e^Y σ dW_t
= µX dt + σX dW_t

as claimed.

Fig. 1 shows a realization of geometric Brownian motion with constant drift coefficient µ and diffusion coefficient σ. Similar to the case of ordinary differen- tial equations, relatively few stochastic differential equations have closed-form solutions. It is often necessary to use numerical approximation techniques.

2. Numerical methods for SDEs.

The simplest effective computational method for the approximation of or- dinary differential equations is Euler’s method (Sauer, 2006). The Euler- Maruyama method (Maruyama, 1955) is the analogue of the Euler method for ordinary differential equations. To develop an approximate solution on the interval [c, d], assign a grid of points

c = t₀ < t₁ < t₂ < . . . < t_n = d.
Approximate x values


w₀ < w₁ < w₂ < . . . < w_n

will be determined at the respective t points. Given the SDE initial value problem

⁽ dX(t) = a(t, X)dt + b(t, X)dW_t X(c) = X_c
we compute the approximate solution as follows:

Euler-Maruyama Method

w₀ = X₀

(6)

where
w_i+1 = w_i + a(t_i, w_i)∆t_i+1 + b(t_i, w_i)∆W_i+1 (7)


^∆ti+1 ^{= t}i+1 ^{− t}i
∆W_i+1 = W (t_i+1) − W (t_i). (8)

The crucial question is how to model the Brownian motion ∆W_i. Define N (0, 1) to be the standard random variable that is normally distributed with mean 0 and standard deviation 1. Each random number ∆W_i is computed as


∆W_i = z_i ∆t_i (9)
where z_i is chosen from N (0, 1). Note the departure from the deterministic ordinary differential equation case. Each set of {w₀, . . . , w_n} produced by the Euler-Maruyama method is an approximate realization of the solution stochastic process X(t) which depends on the random numbers z_i that were chosen. Since W_t is a stochastic process, each realization will be different and so will our approximations.
As a first example, we show how to apply the Euler-Maruyama method to the Black Scholes SDE (4). The Euler-Maruyama equations (7) have form
w₀ = X₀ (10)
w_i+1 = w_i + µw_i∆t_i + σw_i∆W_i.
We will use the drift coefficient µ = 0.75 and diffusion coefficient σ = 0.30, which are values inferred from the series of market close share prices of Google, Inc. (NYSE ticker symbol GOOG) during the 250 trading days in 2009. To calculate the values µ and σ², the mean and variance, respectively, of the daily stock price returns were converted to an annual basis, assuming independence of the daily returns.
An exact realization, generated from the solution (5), along with the cor- responding Euler-Maruyama approximation, are shown in Fig. 1. By corre- sponding, we mean that the approximation used the same Brownian motion realization as the true solution. Note the close agreement between the solution and the approximating points, plotted as small circles every 0.2 time units. In addition, the original time series of Google share prices is shown for com- parison. Both the original time series (grey curve) and the simulation from
(5) (black curve) should be considered as realizations from the same diffusion process, with identical µ, σ and initial price X₀ = 307.65.

price


600
300

0 time (years) 1