Exercise: Discrete vs continuous monitoring

The Broadie-Glasserman-Kou correction for daily-monitored barrier options says: replace the contract barrier

B

with the effective continuous barrier

B_{\text{eff}} = B \cdot e^{\beta \sigma \sqrt{\Delta t}}, \quad \beta \approx 0.5826,

(for an up-and-out, where you shift the barrier up to compensate; for a down-and-out, shift down:

B_{\text{eff}} = B \cdot e^{-\beta\sigma\sqrt{\Delta t}}

Tasks

Theoretical motivation. Why is $B_{\text{eff}} > B$ for an up-and-out? Hint: think about which paths are "missed" by daily monitoring vs continuous.
Numerical comparison. For a down-and-out call with $S_0 = 100, K = 100, B = 80, T = 1, r = 0.05, \sigma = 0.2$ :
- Compute the continuous-barrier price (Reiner-Rubinstein).
- Compute the discrete-barrier price by Monte Carlo with daily monitoring ( $M = 252$ ).
- Apply the BGK correction: compute Reiner-Rubinstein with $B_{\text{eff}} = 80 \cdot e^{-0.5826 \cdot 0.2/\sqrt{252}}$ .
Verify the corrected analytic price matches the discrete MC within MC noise.
Why daily matters more than weekly. The shift $\sigma\sqrt{\Delta t}$ is larger for monthly monitoring than daily. Compute the BGK shift for weekly ( $\Delta t = 7/365$ ) and monthly ( $\Delta t = 30/365$ ). Tabulate the price differences.