Brownian-Bridge Simulation for Floating Lookback Puts

Summary

The document addresses a mismatch between an analytic price and a Monte Carlo estimate for a floating lookback put under geometric Brownian motion. The question gives a volatility, maturity, interest rate, time step, and spot, and reports that the simulation produces a lower estimate than the analytic formula. The response points to Brownian-bridge simulation as a way to estimate the path maximum more accurately.

A floating lookback payoff depends on an extreme value along the asset path. If a simulation records prices only at discrete time steps, it can miss maxima that occur between observations, biasing the estimated payoff. A Brownian bridge uses the conditional distribution of the path between simulated endpoints to account for possible in-between extrema. The answer is brief and supplies no derivation, implementation details, or numerical reconciliation, so it identifies a likely source of discretization error rather than confirming the reported prices.

Key ideas

• A floating lookback put’s payoff depends on the maximum reached along the underlying path.
• Discrete-time path simulation can miss maxima between observation dates.
• Brownian-bridge methods can account for extrema between simulated endpoints.
• The response suggests a technique but does not derive it or verify the reported price difference.

Tags

# Floating lookback put, MC vs analytic


# Floating lookback put, MC vs analytic












I am attempting to price a floating lookback put using the analytic formula. (eg. can be found in Shreve's vol II stochastic calculus section 7.4 or on Wikipedia) and wish to confirm the result by using an MC estimator with geometric Brownian motion paths. Unfortunately, I obtain different results (analytic : 0.1429 vs ~ 0.13 using MC simulation) which I don't expect.

My parameters are the following vol: 0.2, T (expiry): 1, r: 0.05, dt: 0.01, t: 0, spot: 1.

Please find below my code:

Thank you very much for any help.

## Answer by AkhiCTropChaud (score 0, accepted)

https://quant.stackexchange.com/a/64239

You can simulate the maximum using the brownian bridge as explained in the chapter 8 of this course https://www.lpsm.paris/documents/71/probnum_gilp_pf17_wCJtiAO.pdf

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