Exercise: Pricing a knock-out barrier option

Up-and-out call: pays $(S_T - K)^+$ if $\max_{t \le T} S_t < B$ for some barrier $B > K$ , else pays 0.

Parameters: $S_0 = 100$ , $K = 100$ , $B = 130$ , $T = 1$ , $r = 0.05$ , $\sigma = 0.2$ .

Tasks

Implement MC pricing with $M = 50$ time steps per path and $N = 10^5$ paths. Report the price and 95% CI.
Discretisation bias. Repeat with $M = 12, 50, 250$ time steps. The discrete-monitoring approximation undercounts barrier hits, so the price is biased upward (the option appears more valuable). Show this empirically.
Brownian-bridge correction. Between two grid points $S_{t_k}, S_{t_{k+1}}$ both below $B$ , the barrier may still have been crossed mid-step. The probability of hitting under a Brownian bridge is

p_{hit} = \exp\!\left(-\frac{2(B - S_{t_k})(B - S_{t_{k+1}})}{\sigma^2 S_{t_k}^2 \Delta t}\right).

Implement the Brownian-bridge correction and verify it removes most of the bias.

Closed-form Black-Scholes for the up-and-out call (Reiner-Rubinstein) gives a known answer. Compare your bridge-corrected MC against $\$ 0.95$ (approximate analytic value).