Interest Rate Models, MFD Flashcards
Describe payoffs for Forward Rate Agreements
Cash flow in arrears: N(F_t - L_t_i)delta
If cash flow right at the time of realization, it is discounted by the LIBOR rate
Define an interest rate future (forward)
A loan rate using the LIBOR rate at a future date
State disadvantages of the risk-neutral measure under a stochastic interest rate setting
Unbiased estimator of the forward rate: F_t1 != E^Q[L_2]
Spot rates cannot be factored out of pricing expectations
Pricing formula for FRA is non linear under Q
State the definition of the forward measure
Pi^ij = (1/B^s_t1)(phi^ij)
Define numeraire
A unit of commerce in which prices are measured
State the equation for B(t, T) under the Classical Approach
B(t, T) = E^Q[exp(-int_t^T r_s ds)]
State the equation for B(t, T) under the HJM approach
B(t, T) = exp(-int_t^T F(t, s) ds)
State the disadvantages of using a geometric SDE to model the spot rate
If mu not equal 0, the model will diverge as t approaches infinity
A constant volatility model may be oversimplified
Parameter inaction is difficult with 2 parameters for n+1 equations
Wiener process may not be appropriate if there are jumps
State three advantages of using a mean reverting SDE over a geometric SDE
More free parameters (k+1)
Under the right conditions, the model will not explode
Prevents negative interest rates under small enough time steps
State advantages of the HJM approach compares to the Classical Approach
No need to model the expected rate of change of the spot rate
Impose multivariate Markov assumption, which is supported by empirical data
More general model
State disadvantages of the HJM Approach compared to the Classical Approach
Less practical
May explode in finite time
Not as well understood as classical spot rate modeling
Compare the B-S PDE with the Bond Pricing PDE
B-S
- assumes constant risk free rate
- randomness driven by the stock price
- don’t need to explicitly model the drift of the stock
- risk free portfolio created with lending and borrowing
- option price only depends on relevant volatilities
Bond
- does not assume constant risk free rate
- randomness driven by the spot rate process
- need to explicitly model the drift of the spot rate process
- risk free portfolio created with weights in bonds
- calibration and estimation more challenging with the drift term and market price of risk
- simplifying assumption that all bonds are driven by the same Wiener process
Define the generator of the Ito diffusion
The expected instantaneous growth rate
The expected rate of change of f in the limit
