Errata for the 2001 thesis
Corrections to Chapter 3 and Appendix B of the thesis listed on the papers page.
Page numbers are the printed ones; the page of the PDF is ten higher. The site's own formulas do not use the expressions corrected here.
Chapter 3: the switching rate
The generator printed on p. 85 has off-diagonal rate $\lambda/2$, so the regime contrast decays like $e^{-\lambda s}$ and a regime lasts $2/\lambda$ on average. On p. 125, $\lambda = 5$ is described as five switches a year, a rate $\lambda$ out of each state and a contrast decaying like $e^{-2\lambda s}$. The simulations on pp. 126 and 128 follow the second convention, as does this site.
Chapter 3: the dynamic mean Vasicek formula
Eq. (3.7) on p. 106 replaces the level and the variance by their means given the starting regime. A coefficient at calendar time $s$ is weighted by the remaining maturity, so with $b(u) = (1 - e^{-\kappa u})/\kappa$ and rate $\lambda$ out of each state the exponent is
$$a_{\mathrm{dmv}}(T) = \kappa\int_0^T\big(\bar\theta \pm \tilde\theta\,e^{-2\lambda s}\big)\,b(T - s)\,ds - \frac12\int_0^T\big(\bar s \pm \tilde s\,e^{-2\lambda s}\big)\,b(T - s)^2\,ds ,$$with $\tilde\theta$ and $\tilde s$ the half-differences of the level and the variance, and the upper sign for a start in the first regime. As printed, Eq. (3.7) pairs $e^{-\lambda s}$ with $b(s)$. The remaining maturity is needed under either convention for $\lambda$.
The two examples have levels 0.35 and 0.15 and a negligible volatility. The size of the antisymmetric correction, $|P_1 - P_2|$ over twice the mean Vasicek price, is:
| pages | $\lambda$, $\kappa$, $T$ | Eq. (3.7) as printed | corrected | two-state system | thesis simulation |
|---|---|---|---|---|---|
| 125–126 | 5, 2, 15 | 0.0057143 | 0.0100002 | 0.0101419 | 0.010020 ± 0.0003 |
| 127–128 | 1, 25, 5 | 0.0956252 | 0.0500184 | 0.0509952 | 0.0516 ± 0.0004 |
The printed column reproduces the dynamic-mean entries of the thesis tables, 0.0057143 and 0.0956. With the p. 85 generator the two-state system would give 0.0206 and 0.1023, far from the simulations.
Two smaller points in the second example. The table of time scales at the foot of p. 127 is carried over from the first example and should read $1/\kappa = 0.04$, $1/\lambda = 1$, $T = 5$. The antisymmetric entries on p. 128 are magnitudes: the first regime has the higher level and the cheaper bond, as the negative entries on p. 126 show.
Appendix B.4: the deterministic shift
The short rate is $\mathit{fudge}(u) + \beta\cdot X(u)$, and $\Lambda(t, u) = \exp\big(-\int_t^u \mathit{fudge}\big)$ scales the bond from $t$ to $u$.
- Caplet and floorlet, pp. 172–173. The shifted formulas carry $\Lambda(T, \tau)$ and need $\Lambda(t, T)\,\Lambda(T, \tau) = \Lambda(t, \tau)$: the discount from today to the quote day is missing.
- Swaplet, p. 175. The second bond matures at $\tau + \Delta\tau$ and takes $\Lambda(t, \tau + \Delta\tau)$, so $$\mathit{swaplet}^\Lambda = \Lambda(t, \tau)\,B(t, \tau) - (1 + \Delta\tau K)\,\Lambda(t, \tau + \Delta\tau)\,B(t, \tau + \Delta\tau) .$$
- Futures, p. 177. Under the risk-neutral expectation the recursion is $f_{i-1} = B(T_{i-1}, T_i)^{-1}\,\mathbb E\big[e^{-\int r}f_i\big]$, with a negative exponent. If the expectation is already under the forward measure, neither the discount nor $B^{-1}$ appears.
A check with $X = 0$ and a shift of 5%: $t = 0$, $T = \tau = 1$, $\Delta\tau = 1$, $K = 0$. The caplet and the swaplet are both worth $e^{-0.05} - e^{-0.10} = 0.046392$. The printed caplet formula gives 0.048771, too high by the factor $e^{0.05}$, and the printed swaplet formula gives zero.
Certificate: verify_thesis_errata.py.