# 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.

• What practioners usually do in that case is turn to a "Brownian bridge" technique (quant.stackexchange.com/questions/36943/…). If you assume that the underlying dynamics is locally lognormal over each (large) time step, you can indeed calculate the conditional probability of hitting the barrier over the interval without effectively simulating the smaller time steps and correct for that. Another more subtle question would be: are you sure you want local vol to price barrier options? Nov 18, 2021 at 7:10

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.
• Hi, I can follow suggestion for a (purely) time-dependent volatility; but how would you suggest to integrate over the spot / strike domain as well, i.e. $\int\sigma(S_u,u)du$? Nov 17, 2021 at 12:02
• Naively, I'd say that you can use the value of the underlying at the beginning of every period, i.e. at $t_{i-1}$.