Choosing Error Metrics for Monte Carlo Interest Rate Pricing

Summary

The document asks how to benchmark a Monte Carlo interest rate pricing model and choose time-step and path counts that meet an acceptable error target with minimal computation. It defines a converged model value by comparing prices across increasingly refined settings, then measures error as the difference between a candidate price and that converged value. It considers scaling this error by notional, price, vega, or delta.

The examples explain why no single normalization is obviously suitable: notional scaling ignores the influence of the forward rate and volatility, while value scaling can become unstable for low-value or exotic contracts. Sensitivity-based measures also present difficulties when the relevant Greek is small. The document offers no benchmark data, recommended tolerance, or final metric, so it frames an open practitioner question rather than establishing a pricing standard. The discussion is specific to convergence assessment for Monte Carlo pricing, illustrated with a European swaption; acceptable error will need a defined context and practical convention.

Key ideas

• The document defines pricing error as the absolute difference between a candidate Monte Carlo value and a value treated as converged.
• Normalizing error by notional does not account for how market levels and volatility affect price magnitude.
• Normalizing by model value can be problematic for low-value or exotic contracts.
• Delta or vega scaling may be unsuitable when the relevant sensitivity is small.
• The document asks for practitioner tolerances but does not provide an answer or empirical benchmark.

Tags

# Practitioner's criterion for MC pricing convergence


# Practitioner's criterion for MC pricing convergence












Let's say I have some Interest Rates (IR) pricing model which relies on Monte Carlo pricing and I'd like to benchmark its quality and find out optimal settings (time steps & iterations) per asset class, which yield minimum computational effort yet providing acceptable error. Which metrics should/may be used for these purposes and which error is tolerated by practitioners?

To give an example, let's say we are pricing an European swaption: $$NPV(ts,p) = \text{Net present Value for}~ts~\text{time steps and}~p~\text{paths.}$$ $$NPV_{ref} = \text{Reference NPV given by a Vanilla model.}$$ $$NPV_{conv} = NPV(ts^*,p^*):\forall ts > ts^* \& p > p^* ~|NPV(ts^*,p^*) - NPV(ts,p)| < \epsilon$$ $$Err(ts,p) = |NPV(ts,p) - NPV_{conv}|$$

An obvious metric is Error relative to nominal: $Err_{Nom}(ts,p)=\frac{Err(ts,p)}{Nominal}$. But the problem is that it's not adjusted to current spot rate and volatility. Obviously, an ATM swaption with forward 3% costs much more than ATM swaption with 30bps fwd ceteris paribus, which changes magnitude of errors. Same with volatility.

Another metric one might think of could be Error to value: $Err_{val}(ts,p) = \frac{Err(ts,p)}{NPV_{conv}}$. But this metric fails for more exotic payoffs and contracts with low NPV.

I also thought about some fwd or vol weighted estimators. For example, $Err_{Vega}(ts,p)=\frac{Err}(ts,p){\nu}$ or $Err_{Delta}(ts,p)=\frac{Err(ts,p)}{\Delta}$. But for some payoffs, delta or vega could be small; moreover, I have no idea what is tolerable error in this case.

Would be grateful for any ideas and recommendations.

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.