# 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.

• When you ask about time steps and iterations, it implies you care about computational effort. Otherwise, you would just use as many iterations and time steps as possible to get a better estimate... But, the metrics you list don't seem to rely on computational effort as far as I can tell. So, maybe I've misunderstood your question? Could you perhaps clarify a little? – ch-pub Aug 28 '14 at 23:27
• @nsw I wasn't clear about what are optimal parameters: I want to find minimal $\{ts,p\}$ (minimize computational effort, as you said) for each derivative type such that MC yields "good" price approximation relative to $NPV_{ref}$, if available, otherwise $NPV_{conv}$. My question is which metric should I use to say that approximation is "good" enough for practical purposes. One thing I also thought about, is to use half of quoted volatility spread multiplied by vega as a proxy for acceptable error. Hope I was clear with this. – Bruno Aug 29 '14 at 8:26