Why Monte Carlo Payoff Variance Depends on the Payoff Structure
Summary
The document describes a question about pricing instruments by simulating share prices with geometric Brownian motion, calculating a payoff on each path, and averaging many simulated payoffs. The author reports that estimates can have very different dispersion even with a similar simulation horizon and volatility. One example is an option with accelerated exercise after the share price exceeds a threshold for consecutive days, making the payoff depend on the path as well as the final price.
The central lesson is that Monte Carlo precision depends on the distribution of the payoff being averaged, not just on the simulated asset process. Rare paths with very large payoffs can create high payoff variance and noisy estimates; path-dependent triggers can also change how often such outcomes occur. The document asks for an explanation but does not provide one or establish the cause of the reported cases. It also gives a simulation expression whose time scaling and drift convention merit checking before drawing conclusions. No payoff distribution, model calibration, or convergence analysis is supplied.
Key ideas
- Monte Carlo pricing averages payoffs computed from simulated asset paths.
- The variance of the price estimate depends on the payoff distribution, including its tail behavior.
- A trigger requiring consecutive threshold crossings makes the example payoff path-dependent.
- The document reports widely varying dispersion but does not resolve its cause or provide convergence evidence.
- The stated simulation formula should be checked for consistent time and drift scaling.
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Full text
# Monte Carlo Simulation of GBM Process has a Very High Variance - Explanation Needed as to why?
# Monte Carlo Simulation of GBM Process has a Very High Variance - Explanation Needed as to why?
I use Geometric Brownian Motion (GMB) to simulate a share price from March 24, 2020 to March 24 as follow:
\begin{equation} S_t=S_{t-1}exp((rf-0.6\sigma^2)*(2)+\sigma*sqrt(2)*\mathcal{N}(0,1)) \end{equation}
Then I use the simulated share price to calculate the payoff of the financial instrument that I am valuing (how I use the share price to compute can be different but it mostly checking against some fixed number i.e. I do not use the share price for any complicated/random calculations). Then I re run the simulation 100,000 times to get the average value of payoff. In essence, doing a Monte Carlo Simulation. In some cases the standard deviation of my 100,000 simulation average is very large (mean \$5 and stdev=100,000) in other cases it is more acceptable (mean=$2 and stdev=0.5). I know the standard deviation increases with time step and volatility, however for a particular fixed time step (10 years and similar volatility) I see both high and low standard deviations.
My question is can someone explain why this is the case. How does the standard deviation of Monte Carlo Simulations is affected when using GBM simulated share price to calculate payoff?
Concrete Example: I use Monte Carlo Simulation to value options that have an accelerated maturity clause (share price greater the \$5 for 10 consecutive days then immediate exercise otherwise normal option). In order to value this I use GBM to simulate share price and check if the accelerated maturity condition is met. The standard deviation of the payoff in some simulations is very low relative to the mean (mean \$2 with stdev 0.5) in some cases the it is extremely high (\$50 mean with 10,000 st dev).
Can someone please explain why the standard deviation is high in some cases and not in others when I use the same GBM to simulate share price and use that to compute some sort of payoff.
I am grateful for any help. Thank you.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.