Define strong and weak convergence order for a discretisation of an SDE, say which governs the bias of a Monte Carlo price of a European call, and explain why a barrier option obeys neither directly.

Define strong and weak convergence order for a discretisation of an SDE, say which governs the bias of a Monte Carlo price of a European call, and explain why a barrier option obeys neither directly.

Approach: Write both definitions with the error measured in the right place, then note that a barrier payoff depends on the path between grid points rather than on the marginal at the grid points.

Strong convergence of order gamma means E|X_T - X_T^h| ≤ C h^gamma, weak convergence of order beta means |E[f(X_T)] - E[f(X_T^h)]| ≤ C h^beta for smooth f, and the European call price is governed by the weak order, giving an O(h) discretisation bias for Euler. Neither order applies to a barrier option directly because both are statements about the terminal value, while the barrier payoff depends on the continuously monitored extremum of the path. A discretely simulated path can pass through the barrier between grid points and be recorded as surviving, so the survival probability is overstated and the effective order of the bias falls to h^{1/2}. The standard repair is a Brownian bridge correction, which computes the conditional probability of a crossing on each step given the two endpoints, exp(-2(b - x_n)(b - x_{n+1})/(sigma^2 h)) for a constant barrier b, and restores order h.

Follow-up: What is the one step Brownian bridge crossing probability, and how does it change for a time dependent barrier?

Key concepts: strong convergence, weak convergence, discretisation bias, Brownian bridge.