Problem14.7. Create a VBA function CIR_Caplet_MC that values a caplet in the CIR model using Monte Carlo, without calibrating the model to the current yield curve. Simulate the CIR process as described in Sect. 4.5 for the Heston model. The inputs should be R ̄, r(0), κ, θ, σ, T1, T2, N and M, where T1 is the reset date for the caplet, T2 is the payment date for the caplet, N is the number of periods between 0 and T1, and M is the number of simulations. The payoff of the caplet is as in Prob. 14.3, where P(T1,T2) is the function of r(T1) and T2 − T1 given in (14.18) with φ = 0.
Problem14.8. Modify the function in the preceding exercise to create a function CIR_Calibrated_Caplet_MC that values a caplet in the CIR model using Monte Carlo, with the model calibrated to the market. Look up market dis- count bond prices from a function such as DiscountBondPrice in Prob. 12.1. To compute φ(ti) for i = 1,…,N from market bond prices, use (14.19) for dates ti and ti+1 as in Sect. 13.8.
3) CIR model and HJM
(a) Derive the dynamics of the forward rates of the CIR model using the discount bond price given by equation (14.18) in the textbook.
(b) Show that the dynamics of the forward rates you derived above satisfy the HJM equation.
4) Create a VBA function to compute the credit spreads of the Merton model. Assume that the face value of the debt is $100, 5% risk-free rate, and 20% volatilty of the asset, plot the credit spreads from 0 to 10 years for the current asset value of 90, 100, 120, and 150, respectively.
Create a VBA function to compute the credit spreads of the Merton model with Vasicek interet rate. Assume that the face value of the debt is $100, risk-free rate is 5%, and volatilty of the asset is 20%, asset value is $120, κ = 0.4, θ = 5%, σr = 0.015. Plot the credit spreads from 0 to 10 years for the correlation of 0.5, 0, and -0.5, respectively. Discuss the results.