Simulation of arithmetic asian option

I'm trying to implement a monte carlo simulation for asian option pricing by using a higher accuracy schemes. But i don't know exactly how to simulate (2.6), someone can help me?

• Usually the key idea to simulating arithmetic Asian options is to take the sum of lognormals with respective to time as delta time becomes small. I am guessing that is what is going here, but I am not quite sure about the definition of h. Can you further define the terms and/or let us know which text uses this approximation? Mar 17, 2018 at 1:15
• cermics.enpc.fr/~bl/PS/CERMICS-99-144b.pdf page 5. Thanks for your response. Mar 17, 2018 at 8:42

1 Answer

I am not going to answer the question, but hopefully will be of some help anyway.

I think we should clarify that this is about a way to get better MC accuracy for continuously averaged Asian options. In reality though the majority of traded options are discretely averaged (someone correct me if I'm wrong). So in the latter case, this scheme is irrelevant.

Anyway, I looked at this paper for about 15 mins (it is quite late though!) and I couldn't understand either how to simulate using that 3rd scheme (2.6). That being said, why don't you just use the previous scheme (2.5), which judging by the authors' tests is as good or better than (2.6)? (2.5) amounts to using the trapezoidal rule, so basically instead of adding $S_{t_i}$ to your running sum as you simulate a path, you add $(S_{t_i}+S_{t_{i-1}})/2$. Very simple.

Or alternatively, you can use the first scheme (2.1) twice and do a Richardson extrapolation. So first calculate the price of the Asian call with say 5 time steps and get the price $C_{coarse}$. Then price with 10 time steps to get $C_{fine}$. Then your continuously averaged Asian price is approximated by $2C_{fine}- C_{coarse}$. My brief tests show that this is slightly more accurate even than using (2.5). For this to work well, you need to use some variance reduction technique (as suggested in the paper as well), so that your simulation noise is relatively low compared to the simulation bias (the latter stemming from averaging only a few observations instead of having true continuous averaging).

As an alternative to using a control variate as in the paper, it is worth noting that Asians are hugely helped by using quasi-random numbers with the Brownian Bridge path construction technique. To see how well this works in practice you could use this tool.