For interest rate models, in which the evolution of the short rate is given by a stochastic differential equation, i.e.
the valuation process can be easily performed using the Monte Carlo technique.
The k-th sample path of the interest rate process can be simulated using an Euler discretization:
where z is a standard normal random variable.
The valuation of interest rate derivatives using Monte Carlo (MC) Simulation can be performed using the following steps:
- Generate a time discretization from 0 to the maturity T of the financial derivative, which includes all relevant cash flow dates.
- Generate MxN standard normal random numbers (M=number of paths, N = number of time steps per path)
- Starting from r(0), simulate the paths according to the formula above for k=1..M.
- Calculate the cash flows CF of the Instrument at the corresponding cash flow dates
- Using the generated paths, calculate the discount factors DF to the cash flow dates and discount the cash flows to time t0
- Calculate the fair value of the interest rate derivative in each path by summing the discounted cashflows from step 5.:
- Calculate the fair value of the interest rate derivative as the arithmetic mean of the simulated fair values of each path, i.e.
The only difference of QMC simulation is to use deterministic low discrepancy points, instead of the random points used in step 2 of the Monte Carlo algorithm. These points are chosen to be better euidistributed in a given domain by avoiding large gaps between the points. The advantage of QMC Simulation is, that it can result in better accuracy and faster convergence compared to the Monte Carlo Simulation technique.
The following picture shows the dependence of the MC/QMC valuation result on the number of chosen pahts for a vanilla floater, which matures in 30 years, pays annually the Euribor 12M reference rate. The time steps in the simulation method is chosen to be 1 day. One can see that using QMC, a much lower number of paths is needed to achieve an accurate price.
