The Black–Scholes PDE value u(t, x) equals the discounted average payoff of risk-neutral paths that start at x at time t. Drag x and t and watch the two agree.
Parameters
100
20300
0.00
00.95
100
10300
1.00
0.253
3.0%
00.1
20.0%
0.050.6
5000
20020000
7
1100
Metrics
PDE solution u(t, x)
9.4134
Monte Carlo average
9.2020 ± 0.3932
MC − PDE
-0.2113
u(t, x) against the start value x
The curve is the PDE solution at time t; the blue line averages the same draws started from each x. The vertical line marks your x.
Terminal payoff u(T, x)PDE solution u(t, x)Monte Carlo average
Discounted payoffs of the paths started at x
50.3% of the paths end in the money; the rest pay zero. The Monte Carlo average is the mean of this histogram.