# Hull-White Monte Carlo simulation - mean reversion function

Quite new to implementing Hull white model in Monte Carlo simulation, hope to get help for 1. how to get the function $$\theta$$ in the following formula (the function used to match initial term structure)? 2.In pricing swaptions, floating is libor fwd and discounting is OIS, does this mean two curves need to be simulated jointly? 3. is it ok to get the $$\theta, \alpha$$ and $$\sigma$$ from HWTree and use in MC simulation? thanks

Given a initial discount bond $$P^M(0, T)$$ curve, the expression for $$\theta(t)$$ in the Hull White Short Rate model is a know result given by:

$$\theta(t) = \frac{1}{\kappa} \cdot f'(0, t) + f(0, t) + \frac{1}{2} \cdot \left( \frac{\sigma}{\kappa} \right)^2 \cdot \left( 1 - e^{-2 \kappa t} \right).$$

I have used a notation where the spot rate dynamics is given by:

$$dr(t) = \kappa \cdot (\theta(t) - r(t)) \cdot dt + \sigma \cdot dW(t).$$

Note that $$f(t)$$ is the instantaneous forward rate, given by:

$$f(t, T) = - \frac{\partial}{\partial T} \ln \left( P(t, T) \right).$$

• so theta function comes directly from the initial term structure, and pretty has nothing to do with the simulated path, right? In practice, are those sigma and mean reversion calibrated to swaptions directly through the MC simulation or use lattice approach? I assume the lattice approach is more efficient to get those parameters than simulation, right? thanks! Oct 15 '20 at 3:41
• $\sigma$ is obtained by means of swaption pricing. However, I am not sure how is $\kappa$ obtained (I suppose that also by means of swaption pricing). On the other hand, MC simulations + pricing can be fast, it depends on many things: which MC scheme are you using (a simple Euler scheme or a high order adaptive time step scheme), parallelization, etc. You can also use a lattice approach, of course. You can even use analytical expressions to get some useful seeds for the other two approaches. Hope it helps! Thank you! Oct 15 '20 at 4:00
• I just wanted to clarify that $\theta(t)$ has an analytical solution for this particular model, but it could be obtained by means of an optimization. In order to do that, you would have to achieve pricing using either a MC method or a lattice, and match the current zero coupon bond prices. However, for this particular model, you can avoid that step. Finally, please notice that since $\theta(t)$ depends on $f(t)$ and its derivative, it could be really ugly if $f(t)$ is not correctly bootstrapped. Oct 15 '20 at 4:05
• I can first setup the model in Excel/VBA using lattice approach, calibrate using proper swaption, and then bring the parameters (sigma and alpha) and the thetas to MC simulation. is this a valid approach (even seems quite redundant)? BTW, if solely using MC simulation is below formula correct for theta? Oct 15 '20 at 14:52
• @ rvignolo; thank you for your help, btw, it's for internal review concept soundness of a third party HW1F for CVA, so transparency is quite important. Has the mainstream moved away from HW1F to G2++ or more complicated models? thanks Oct 15 '20 at 15:15