Monte Carlo Methods for Down-and-Out Call Options
Summary
The document concerns estimating the value and Greeks of a European down-and-out call under geometric Brownian motion using Monte Carlo simulation. The response does not derive the requested estimators in detail, but gives several practical modeling and simulation pointers. In log-price coordinates, Euler stepping is exact for geometric Brownian motion, so that representation avoids discretization error in the underlying diffusion.
The answer warns that approximating continuous barrier monitoring with discrete time steps converges slowly and may require many steps. It suggests sampling the barrier hitting time rather than relying on naive step-by-step monitoring, and points to analytic formulas and Monte Carlo references for further development. A second response mentions both crude and sequential Monte Carlo estimators from a cited paper, recommending the crude estimator as a likely starting point. These are guidance points rather than a complete recipe: the document supplies no formulas for the price or Greeks, and estimator performance depends on monitoring assumptions and implementation.
Key ideas
- The target is a European call whose payoff is canceled if the underlying breaches a lower barrier.
- For geometric Brownian motion, Euler discretization in log coordinates is exact.
- Discrete monitoring approximations to continuous barrier monitoring can converge slowly.
- Sampling the barrier hitting time may be preferable to checking only at time steps.
- Crude and sequential Monte Carlo estimators are mentioned, but the document does not derive them.
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Full text
# European call down and out option (geometric Brownian motion, Monte Carlo, Euler) # European call down and out option (geometric Brownian motion, Monte Carlo, Euler) I need to estimate the expected value and the Greeks of an European call down and out option, assuming geometrical Brownian motion of the asset, with Monte Carlo simulation employing Euler discretization scheme. Given are the strike K,S0, Barier(B), volatility (v), risk free rate(r) and time to end (T). I can program the code, but I cant find the mathematics behind the calculations. Can someone show them to me? Thank you in advance. ## Answer by Mark Joshi (score 1) https://quant.stackexchange.com/a/19031 first, there is a formula for the continuously monitored case. second, if you use log coordinates the Euler discretization is exact so this should be done. third, the convergence for discretely monitored to continuously is actually very slow so you will need a lot of steps. fourth, it's actually better to draw the hitting time to the barrier rather than stepping naively. fifth, as has already been noted this is a big question so hard to answer well. sixth, see my paper http://ssrn.com/abstract=1441142 for further discussion, also Glasserman's book is good. ## Answer by torbonde (score 0) https://quant.stackexchange.com/a/19476 I know this is almost a month old, but... Unless it is a homework assignment, you could have a look at this paper by Del Moral and Shevchenko, which gives a different estimator than the one you're probably using. It gives both a crude Monte Carlo-estimator and a sequential Monte Carlo-estimator. You probably just want the crude Monte Carlo one. Eq. (27) follows from Eq. (8) and standard MC arguments, as can be found in Glasserman's book. Eq. (8) follows from the tower property, as far as I recall.
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