Hull-White 2-factor Model: 3) Simulation

This post discretizes Hull-White 2-factor model and provide derivations of the simulation equations.

Earlier posts on Hull-White 2-factor model

• Hull-White 2-factor model : 1) Introduction
• Hull-White 2-factor model : 2) Zero coupon bond

We try to price an interest derivatives which have cashflows at times T₁,T₂,…,T_N. When we let f(T_j) denote a cash flow at time T_j, the price of this product is

This pricing is the risk-neutral pricing and needs cash flows and discount factors from future interest rate simulations.

Since dW₁(t) and dW₂(t) follow  independently, dx(t) and dy(t) can be discretized as follows.

Here, ϵ1 and ϵ2 are random numbers from the standard normal distribution.

Like HW 1-factor model, the same discretized time axis is used as follows 

For this time axis, stochastic process of discretized x(t) and y(t) has the following form.

Since x_t0, x_t1, x_t2, x_t3, …, and y_t0, y_t1, y_t2, y_t3, … are easily obtained from this scenario generating equation, discount factors at time T_j is

Cash flow at time T_j is

Therefore, for discount factors and cash flows from this scenario, the price of interest derivatives P₀ which has cash flows f(T₁),f(T₂),…,f(T_N) at time T₁,T₂,…,T_N respectively under the risk-neutral measure is as follows.

The present value is the average from iterating this process with a number of scenario. This is the same case with the Hull-White 1-factor model except for correlated related cross terms. Expressions for this kind of terms are presented as product (or multiplication) terms not as square terms.

This manipulation for cross terms also holds for numerical integration and calculation which are also similar to 1-factor case. So we do not repeat these similar works.

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