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Estimating Conditional Expectations with Monte Carlo and a Fixed Path History

Article Quant Q&A · Author: Tipeg
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Summary

The document asks how to estimate a risk-neutral conditional expectation of a discounted payoff when two state variables follow stochastic differential equations driven by the same Wiener process. The requested calculation uses Euler discretization and considers a range of strikes and a specified maturity. The questioner proposes fixing a simulated path up to the conditioning time, then repeatedly simulating the remaining increments and averaging the resulting payoffs.

This setup points to the central Monte Carlo idea: conditioning on the filtration means treating the information available up to that time as fixed, while averaging over possible future paths. The document asks whether the estimate depends on the sampled history and how one should interpret the conditional value across histories. It provides no numerical experiment or resolution, and leaves implementation details such as the number of paths and time steps unspecified.

Key ideas

  • The state processes share a Wiener driver, creating dependence between their simulated paths.
  • The target is a risk-neutral conditional expectation of a discounted terminal payoff.
  • A proposed simulation fixes the Brownian history up to the conditioning time and resamples future increments.
  • The document asks how estimates vary across histories but gives no numerical results or answer.

Tags

StatisticsDerivatives pricingBacktesting
Full text
# How to simulate a conditional expectation given a filtration


# How to simulate a conditional expectation given a filtration












I had a question regarding how to simulate a certain conditional expectation. I am given two processes $X_1(t), X_2(t)$ which both follow their own SDE, but both are of the form \begin{equation*} dX_i(t) = \mu_i X_i (t) dt + \sigma_i X_i(t) dW^{\mathbb{P}}(t), \qquad X_i(t_0)=X_{i, 0} \end{equation*} with known constants and the same Wiener process $W^{\mathbb{P}}(t)$. Then I am also given a known risk-free interest rate $r$ and a known range of strike price $K=[0, ..., 10]$ and maturity time $T$.

Given that $M(T)=e^{r T}$, I am asked to use Euler discretisation to find \begin{equation*} V(t) = \mathbb{E}^{\mathbb{Q}}\left[\frac{f(X(T), Y(T), K)}{M(T)}|\mathcal{F}(t)\right] \end{equation*} for every value of $K$ in the range. Here $f$ is just some continuous function, which is also known but not really relevant for my question.

I think I know how to start: I first just replace $\mu_i$ by $r$ since we are dealing with $\mathbb{Q}$. I also already implemented something to calculate $X(t), Y(t)$ for any $t$ given a Wiener process $W(t)$ up to time $T$.

My question is about how to interpretation behind implementing the expected value conditioned on $\mathcal{F}(t)$. Would this mean I just randomly generate a Wiener process $W(t)$ up to a time $t$, then fix this Wiener process, and then simulate the latter half up to time $T$ differently many times and calculate the average of $\frac{f(X(T), Y(T), K)}{M(T)}$? If so, wouldn't this depend a lot on the choice of the Wiener process up to time $t$? So, how does one calculate an expected value (like above), where $\mathcal{F}(t)$ is given?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.