Monte Carlo Modeling of Forward Curves Under Risk-Neutral Dynamics

# Monte Carlo Simulation and forward curves


# Monte Carlo Simulation and forward curves












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

## Answer by AFK (score 0, accepted)

https://quant.stackexchange.com/a/30636

You are confusing Monte Carlo which is a numerical method and the modelling that comes before that.

The 2 approaches use Monte Carlo simulation but it is the modelling that is different.

In approach 2, the forwards for different expiries are modelled as completely independent. The corresponding process is a white noise. This is not realistic at all but this is not an issue in itself if what you are interested in does not depend on the covariance between rates. For example, you could get the right value for a portfolio of vanilla options at the expiries you simulate. Obviously you would't need a MC simulation for that and a much more serious issue is that your FX forward can actually be negative but I hope you get my point.

## Answer by Quantuple (score 2)

https://quant.stackexchange.com/a/30633

Assume deterministic interest rates to keep notations uncluttered.

In the absence of arbitrage opportunity, under the risk-neutral measure $\Bbb{Q}$ associated to a risk-free money market account denominated in EUR, you suggest to model the GBP/EUR spot exchange rate dynamics using the following SDE: $$ \frac{dX_t}{X_t} = (r^{EUR}_t - r^{GBP}_t)dt + \sigma dW_t^{\Bbb{Q}},\quad X(0)=X_0$$ Solving the equation yields the unique solution: \begin{align} X_T &= \underbrace{X_0 \exp\left( \int_0^T (r^{EUR}_t - r^{GBP}_t) dt \right)}_{=X(0,T)} \underbrace{\exp(- \frac{1}{2}\sigma^2 T) \exp(\sigma W_T^\Bbb{Q})}_{= \mathcal{E}[W_t]} \tag{0} \end{align} with $X(0,T)$ denoting the forward exchange rate and $\mathcal{E}[X_t] = \exp(X_t - \frac{1}{2}\langle X \rangle_t)$ the stochastic exponential of a continuous semi martingale $X_t$. From the properties of the latter exponential, it is straightforward to see that: $$ X(0,T) = \Bbb{E}^\Bbb{Q} \left[ X_T \vert \mathcal{F}_0 \right] \tag{1} $$

When you are evaluating a forward price using Monte Carlo simulations you are numerically evaluating $(1)$ as: $$\tilde{X}(0,T) = \frac{1}{M} \sum_{m=1}^M X_T^{(m)} \tag{2} $$ where $\{ X_T^{(m)} \}_{m=1}^M$ are iid samples of the random variable $X_T$ simulated under the risk-neutral measure $\Bbb{Q}$, conditionally on the knowledge of $X_0$.

Assuming you manage to sample from the true distribution of $X_T$ (which is possible under our working modelling assumptions), then central limit theorem (CLT) tells you that your Monte Carlo estimation of the forward exchange rate, although unbiased, will have a variance $\propto M^{-1/2}$. This is probably what you mean by "you can never exactly represent the forward price". But I don't understand the approach 2 you mention: do you mean a plus sign instead of a multiplication?

That being said, you do not need Monte Carlo to compute forward prices: in general, those are computable in closed form as I just showed. Also, yes, you can perfectly calibrate a GBM to match a given forward cruve as long as your model can accomodate a term structure for the risk-neutral drift. It is basically equivalent to matching the first moment of the conditional distributions.