Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit


STA 2503 Project–3 : Stochastic Interest Rates and Swaptions


        In this project you will investigate several aspects of stochastic interest rates. Let denote the short rate of interest and suppose that r satisfies the SDE

where are independent risk-neutral Brownian motions, and denotes the long-run interest rate which itself is stochastic. This is a type of two-factor interest rate model.

        For base parameters use the following:

1. This is an affine model. Let the T-maturity bond price process be denoted  As the model is affine, there exist deterministic functions and such that

Determine the functions A, B and C by either solving for the distribution of or using the PDE approach and obtain the term structure of interest rates with the base parameters.


2. Use an Euler-scheme to generate simulations of risk-neutral interest rate paths, and use the paths to obtain Monte Carlo estimate of bond yields (with confidence bands) to compare with analytical formula you derived in Q1 using the base parameters.


3. Investigate what the various parameters in the model do to the term structure using the analytical formula.


4. Using a bond of maturity as a numeraire asset, it is possible to determine an analytic expression for the price of a bond option paying Derive this formula, and use Monte Carlo simulations under both the risk-neutral and forward-neutral measures to check your results for a collection of strikes α K, for α = 0.95, 0.96, . . . , 1.05, = 3, and = 5.


5. Suppose we have an IRS with tenure structure τ = {3, 3.25, . . . , 6} where 3 is the first reset date (no payment) and the first payment is at 3.25 and every 0.25 after that. Using simulations, determine the Black implied volatility of a swaption with strike equal to today’s swap-rate. The black implied volatility is the volatility in an LSM model that makes the LSM price equal the price you obtained. How does the Black implied volatility change as a function of strike?