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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.

share|improve this question
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? – nsw 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

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