Improving Monte Carlo Estimates for Continuously Averaged Asian Options
Summary
The reply discusses ways to reduce discretization error when Monte Carlo simulation prices a continuously averaged Asian option. One suggestion is to replace a simple running sum of simulated asset prices with a trapezoidal approximation, averaging adjacent observations. Another is Richardson extrapolation: price with a coarse time grid and a finer grid, then combine those estimates to reduce bias. The response reports that brief tests found this combination slightly more accurate than the trapezoidal method, while stressing that the evidence is limited.
Variance reduction is important because simulation noise should be small relative to the bias from approximating continuous averaging with a finite number of observations. The reply mentions control variates and quasi-random numbers with Brownian Bridge path construction as possible aids. It also cautions that these continuous-averaging schemes do not directly apply to discretely averaged options, which it says are common in practice.
Key ideas
- A trapezoidal rule can approximate continuous averaging by using the mean of neighboring simulated prices.
- Richardson extrapolation combines estimates from coarse and fine time grids to reduce discretization bias.
- Variance reduction helps expose the bias caused by approximating continuous averaging with finite time steps.
- Quasi-random numbers with Brownian Bridge path construction are suggested as a variance-reduction approach.
- The discussed schemes target continuously averaged options and may not suit discretely averaged contracts.
Tags
Full text
# Simulation of arithmetic asian option
# 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?
## Answer by Yian Pap (score 1)
https://quant.stackexchange.com/a/38841
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.