Interest Rate Models and Term Structure
This domain tests spot, forward and yield curves, short-rate models and no-arbitrage term-structure modelling.
Term-structure basics
- The discount factor P(t,T) is the value at t of 1 payable at T. The continuously compounded spot rate R(t,T) satisfies P(t,T) = exp(-R(t,T)(T - t)).
- The instantaneous forward rate f(t,T) satisfies P(t,T) = exp(-integral_t^T f(t,s) ds). Forward rates can be extracted from discount factors.
- Short-rate models describe dr_t. Vasicek: dr = a(b - r)dt + sigma dW. CIR: dr = a(b - r)dt + sigma sqrt(r) dW. Both are affine models, giving bond prices of the form P(t,T) = A(t,T) exp(-B(t,T)r_t).
- Hull-White adds a time-dependent drift to Vasicek so the model fits the initial yield curve. HJM models the whole forward curve and imposes a no-arbitrage drift condition.
Worked example
Suppose P(0,1) = 0.95 and P(0,2) = 0.90. The one-year forward rate from time 1 to time 2 satisfies P(0,2) = P(0,1) x (1 + f)^{-1}, so f = P(0,1)/P(0,2) - 1 = 0.95/0.90 - 1 = 0.05556, or 5.56% per year. With continuous compounding, f = ln(0.95/0.90) = 0.05407, or 5.41%. The compounding convention changes the numerical answer.
Traps that cost marks
Do not confuse spot rates with forward rates. Check whether rates are continuously compounded or annually compounded. Vasicek permits negative rates; CIR keeps rates non-negative only if its parameters satisfy the required condition. Calibration is to market prices, not to real-world expectations. In HJM, the drift is not free: no-arbitrage restricts it.