em_bs

README

How It Works

In this example, we solve the following SDE with the following approximation method.

SDE: Black-Scholes

X(t,x) = x + \int_0^t\mu X(s,x)\,ds+\int_0^t\sigma X(s,x)\,dB(s),

where $x, \mu, \sigma\in\mathbb{R}$.

Method: Euler-Maruyama

Letting $\Delta t>0$ and $n\in\mathbb{Z}_{>0}$ be specified, define $\widehat{X}(0,x):=x$ and

\widehat{X}(t_{k}, x) := \widehat{X}(t_{k-1}, x)
+ \mu \widehat{X}(t_{k-1}, x)\,\Delta t + \sigma \widehat{X}(t_{k-1}, x)\, \sqrt{\Delta t}\,Z_{k},

for $k=1,2,\cdots,n$. Here, $t_{k}=\frac{k}{n}t$ and $Z_{k}$, $k=1,\cdots,n$ are i.i.d. standard normal random variables.