I recently came across a question whether a Monte Carlo simulation should represent a forward curve at each tenor. I encountered an approach at a bank which I would consider as somehow strange.
Approach 1: Let’s take GBPEUR as an underlying. To keep things as simple as possible I would choose a Geometric Brownian Motion with constant drift and diffusion calibrated to forward volatilities on GBPEUR. In the next step I would choose an Euler Scheme, simulate 100000 paths in order to solve the stochastic differential equation numerically. By taking the mean over the paths at each tenor I get a point which is close to the forward curve at this tenor. But in my opinion it’s nearly impossible to calibrate the Geometric Brownian Motion in a way that it exactly represents the forward GBPEUR curve on each tenor. Nevertheless that’s the way a simulation is done!
So what I now saw implemented in a financial institution is the following.
Approach 2: The guy took each tenor of the forward curve and multiplied it with a normally distributed random number. I would say this should be represented like this:
F_t=F_t N(0,σ) F_t GBPEUR forward curve at tenor t N(0,σ) a normally distributed random number with zero mean and σ the volatility
Then he took 100000 draws and said that’s my Monte Carlo simulation and of course trivially his simulation represented in the mean the whole forward curve.
So here are my questions:
- Do you agree that the approach 1 is the right one? Of course there are technically more advanced ones, but this is the way it is done.
- Do you agree that it is nearly impossible to represent the whole forward curve with a Monte-Carlo simulation regardless of how complex the underlying sdes are? Otherwise we would have a perfectly calibrated model.
- Approach 2 is awfully wrong and has nothing to do with a Monte Carlo Simulation.
Thanks in advance