Choosing Time Steps for Local-Volatility Barrier Simulations

Summary

The document considers Monte Carlo pricing of barrier options under a Dupire local-volatility model. The questioner has calibrated smooth implied-volatility surfaces and calculates local volatility at simulation dates, but finds daily stepping slow for long maturities. The central issue is that the path grid must reflect when the barrier is observed: monthly monitoring does not require daily barrier checks, while daily monitoring may be sensitive to a coarser grid.

For monthly intervals, the response proposes replacing varying local variance over each interval with an average integrated variance, then using the corresponding constant interval volatility. For daily observations, it says retaining daily steps is the more precise approach; grouping days under a common volatility can reduce computation but sacrifices accuracy. The answer offers no benchmark, error estimate, or more advanced bridge correction, so it presents a practical approximation and trade-off rather than a validated speedup method. The appropriate grid and tolerance depend on the desired precision and monitoring schedule.

Key ideas

• Barrier monitoring frequency determines which simulation dates need explicit checks.
• For a monthly interval, integrated local variance can be averaged over the interval to define an effective volatility.
• Daily monitoring may require daily path steps when precision is important.
• Grouping days under a shared volatility can reduce computation while introducing approximation error.
• The response provides no numerical accuracy comparison or error bound.

Tags

# Monte Carlo: How to interpolate Dupire's Local Volatility


# Monte Carlo: How to interpolate Dupire's Local Volatility












I am trying to price barrier options which can have daily or monthly observations. I first calibrated by Black vols into smooth SVI vols (with linear interpolation along time in variance) to obtain arbitrage free vols.

In my initial MC implementation, I was simulating daily prices up to the option maturity by calculating the local vol using Dupire's formula on each of those dates/strikes.

But this is obviously very slow when pricing long dated options. My worry is that if I just have weekly time grids instead in my paths, I would be losing accuracy especially for the one with daily observations. And for the monthly observations, how do i speed this up since I don't really need the daily observations.

## Answer by KT8 (score 0)

https://quant.stackexchange.com/a/68806

In the case of monthly observations, I recon you can use the accumulated volatility $$\sigma^2 = \dfrac{1}{t_i - t_{i-1}} \int_{t_{i-1}}^{t_i} \sigma^2(s) \, ds$$ at every step, where you could integrate (sum) the volatilities you computed to obtain a constant volatility for each month.

Regarding options with daily observation, I think if you want to be precise there is no other way. You could of course use a constant volatility for a group of days and update it every given set of them, but you'd be losing precision. It depends on how accurate you'd like your results to be.

Maybe there are some tricks you can use, but I don't know any at this time. Hopefully another user can provide some.

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