Chapter 8.P2, Problem 1P

Show that Euler’s method applied to the differential equation

$\frac{dS}{dt}=\mu S$ (i)

yields Eq. (1) in the absence of random disturbances, that is, when $\sigma =0$.

$S_{n+1}=S_n+\mu S_n\Delta t+\sigma S_n{\epsilon }_{n+1}\sqrt{\Delta t},S_0=s$, (1)

To determine

To prove: The Euler’s method applied to the differential equation $\frac{dS}{dt}=\mu S$ yield equation $S_{n+1}=S_n+\mu S_n\Delta t$ if the random disturbance is absent in the equation $S_{n+1}=S_n+\mu S_n\Delta t+\sigma S_n{\epsilon }_{n+1}\sqrt{\Delta t},S_0=s$, where $S_n=S\left(t_n\right)$ is the stock price at time $t_n=n\Delta t$, $n=0,\mathrm{....},N-1$, $\Delta t=T/N$, $\mu$ is the annual growth rate of the stock and $\sigma$ is a measure of the stock’s annual price volatility or tendency to fluctuate.

Explanation of Solution

Given information:

The differential equation $\frac{dS}{dt}=\mu S$ and the equation $S_{n+1}=S_n+\mu S_n\Delta t+\sigma S_n{\epsilon }_{n+1}\sqrt{\Delta t},S_0=s$, where $S_n=S\left(t_n\right)$ is the stock price at time $t_n=n\Delta t$, $n=0,\mathrm{....},N-1$, $\Delta t=T/N$, $\mu$ is the annual growth rate of the stock and $\sigma$ is a measure of the stock’s annual price volatility or tendency to fluctuate.

Formula used:

Euler’s method: Suppose the solution of the initial value problem $\{\begin{array}{c}\frac{dy}{dt}=f\left(t,y\right)\\ y\left(t_0\right)=y_0\end{array}$ is denoted $y=\varphi \left(t\right)$ and have a sequence of points $t_0<t_1<t_2<\cdot \cdot \cdot <t_n<\cdot \cdot \cdot$ for $n=0,1,2,\mathrm{...},$ then the Approximation of $y=\varphi \left(t\right)$ at $t=t_{n+1}$ is $y_{n+1}=y_n+f\left(t_n,y_n\right)\left(t_{n+1}-t_n\right)$, the linear approximation of $\varphi \left(t\right)$ on the interval $\left[t_n,t_{n+1}\right]$ is $y\left(t\right)=y_n+f\left(t_n,y_n\right)\left(t-t_n\right)$.

So, the approximation for $t$ is $y_{n+1}=y_n+f\left(t_n,y_n\right)h$.

Proof:

Let the differential equation is $\frac{dS}{dt}=\mu S$.

By using Euler’s method, $y_{n+1}=y_n+f\left(t_n,y_n\right)h$.

Here $h=\Delta t$.

The solution of the differential equation $\frac{dS}{dt}=\mu S$ is $S_{n+1}=S_n+\mu S\Delta t$.

A discrete model for change in the price of a stock over a time interval $\left[0,T\right]$ is $S_{n+1}=S_n+\mu S_n\Delta t+\sigma S_n{\epsilon }_{n+1}\sqrt{\Delta t},S_0=s$.

If the value of the random disturbance is absent in the discrete model that is the value of $\sigma =0$.

Thus the discrete model becomes,

$S_{n+1}=S_n+\mu S_n\Delta t+\left(0\right)S_n{\epsilon }_{n+1}\sqrt{\Delta t}$

$=S_n+\mu S\Delta t+0$

$\Rightarrow S_{n+1}=S_n+\mu S\Delta t$

Therefore, the differential equation $\frac{dS}{dt}=\mu S$ yields the equation $S_{n+1}=S_n+\mu S_n\Delta t$ if the random disturbance is absent in the discrete model.