Monte Carlo and PDE Methods for Options Probabilities
Summary
The document compares approaches for estimating the probability that an option finishes in the money or reaches a strike. It explains that under Black–Scholes, the in-the-money event can be treated as a discontinuous indicator payoff, distinct from the usual call payoff. Analytical formulas or PDE methods may work in simpler settings, while changing the model or claim can make closed-form solutions unavailable and require numerical methods.
Monte Carlo estimates probabilities by simulating possible outcomes and counting how often the event occurs. The answer highlights its flexibility across different claims and its usefulness in high-dimensional models, where it can outperform PDE solvers. Its estimate is subject to sampling error, so precision depends on the number and quality of simulations. It is also poorly suited to optimization problems such as pricing American-style claims directly; the answer says these generally require a dynamic programming approach with simulation within the iterations. The discussion is an overview, not an implementation recipe, and it supplies no specific model calibration or accuracy comparison.
Key ideas
- An in-the-money probability can be expressed as the expected value of an indicator payoff.
- Black–Scholes and PDE methods may become difficult when the model or contingent claim changes.
- Monte Carlo estimates event probabilities by simulating outcomes and counting how often the event occurs.
- Simulation is flexible and can be advantageous in high-dimensional problems, but its estimates carry sampling uncertainty.
- American-style claims require a dynamic programming treatment rather than direct application of ordinary Monte Carlo.
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# Monte Carlo Options Probability Calculation
# Monte Carlo Options Probability Calculation
I have a fairly simple problem for an application I am writing currently. How do you calculate the options probability of being in the money or touching a certain strike price. I know there are at least two ways of doing it. One would be to derive it from the options pricing and the Black Scholes formula, but the other which interests me probably more would be to run a monte carlo simulation given the target strike, current strike, IV and days to expiration. What do you think are the advantages/disadvantages of each method, which one is easier to implement methematically/programatically and what is the math in layman terms of the easier to implement method. Any help will be appreciated!
I just want to point out that i come from a developer background and not math/stats one so if you can please provide any answers in very plain terms that would be great. Thanks!
## Answer by SBF (score 1)
https://quant.stackexchange.com/a/9026
As Matt has mentioned, although BS allows for the explicit formula of the price in case of European call and put, things get much tougher in case you change the contingent claim - not to say the model itself. For example, the probability of a European call option being in the money can be regarded as a contingent claim given by $f(S_t):=1\{S_t\geq K\}$ where $1\{A\}$ here stands for the Indicator function of a set. In contrast to the normal payoff $g(S_t) = (S_t - K)^+$, $f(S_t)$ is a discontinuous function of the stock price which already gives you a new level of complexity. It shall not be a big problem for a parabolic PDE such as the one in BS case - and even an analytical solution may be known. However, if your model is different from BS - there are rarely analytical solutions to corresponding PDEs, so in the best case one hope to find an appropriate numerical solver.
Monte Carlo can actually outperform PDE numerical solvers when it comes to large-dimensional models. In addition, you can use the very same samples to price different stuff - that is if you are given a time horizon of a problem, you can run couple of millions of simulations and price call, put, lookback, barrier etc. You shall bear in mind that Monte Carlo gives you a result only up to some level of confidence - that is you may be extremely unlucky, and your outcomes may appear to be completely wrong, but often one tries pushing the confidence to be around $1-10^{-6}$ or even closer to $1$ - this will require running quite some samples, of course, but besides that Monte Carlo is a very flexible method. I would say a more fundamental drawback of it is that it does not fit well optimization problems - as an example, you can't use Monte Carlo directly to price American-style contingent claims unless you express the problem as a dynamic programming and run 1-step Monte Carlo simulations for each iteration.
This is by no means a complete answer, and I agree with Matt that a book would be more comprehensive, but I hope it gives you at least a brief overview.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.