Thesis Appendix B pricing errata

Exact deterministic-shift transport, forward-measure invariance, and three corrections to the 2001 thesis.

Appendix B, Section B.4 of the 2001 thesis contains three source-formula errors: the shifted caplet and floorlet omit discounting from today to the fixing date, the shifted swaplet applies one maturity's deterministic factor to both legs, and the futures recursion has a positive discount exponent. This page states the corrected formulas and the more general theorem behind them.

A deterministic-shift identity

Let $r^\phi_u=r_u+\phi(u)$, where $\phi$ is deterministic and integrable, and define

Pathwise, $D^\phi(t,u)=A(t,u)D^0(t,u)$. Under the same risk-neutral measure this gives, with no Gaussian, affine, or independence assumption,

Payoff-transport theorem

Let $u_1,\ldots,u_d\ge T$ and let $\Psi$ be any Borel payoff for which the expectations below exist. Then

Proof. Substitute $D^\phi(t,T)=A(t,T)D^0(t,T)$ and $P^\phi(T,u_j)=A(T,u_j)P^0(T,u_j)$ inside the defining expectation. Every factor $A$ is deterministic and the first can be taken outside. This also proves the path-functional version:

Thus a deterministic shift translates every rate argument and scales each bond according to its own maturity. It does not generally leave the payoff unchanged.

Sharp common-factor converse

Suppose one scalar $c$ multiplied the unshifted value of every linear portfolio of bonds maturing at $u_1,\ldots,u_d$. Applying that assertion to each single-bond portfolio forces $c=A(t,u_j)$ for every $j$. Hence a universal common factor exists if and only if all relevant maturity factors coincide.

The corrected swaplet below is the two-maturity counterexample. This transport mechanism underlies the deterministic-shift extensions of Brigo and Mercurio (2001).

Forward measures are invariant

For any bond-numeraire maturity $U$ and $s\le U$,

The last equality is the cocycle identity $A(0,s)A(s,U)=A(0,U)$. Every conditional law under a bond-forward measure is therefore unchanged by the deterministic additive shift.

If $X_T^0=P^0(T,u)/P^0(T,U)$ and $c=A(T,u)/A(T,U)$, then $X_T^\phi=cX_T^0$. At the transformed strike $K^\phi=cK^0$,

Black implied volatility is consequently invariant at matched transformed moneyness. Holding the numeric strike fixed changes moneyness and need not preserve implied volatility.

Why annuity measures generally change

For a positive bond-basket numeraire $N_s^0=\sum_jw_jP^0(s,u_j)$, $w_j>0$, the shifted density is

For universal invariance over base term-structure models, the ratio $\sum_jw_jA(s,u_j)p_j/\sum_jw_jp_j$ must be independent of every positive bond-price vector $p$. That holds exactly when all $A(s,u_j)$ are equal. Therefore

A one-bond forward measure is the automatic $d=1$ case. A swap annuity usually spans unequal payment dates, so its measure generally changes; there is no blanket shift-invariance theorem for swaption distributions or implied volatilities.

A sharp finite sensitivity bound

The qualitative failure of invariance is nevertheless uniformly controlled. Put $a_s=\min_j A(s,u_j)$, $b_s=\max_j A(s,u_j)$, and $\kappa_s=b_s/a_s$.

On $\mathcal F_s$, let $\nu=Q^{N,0}$ and $\mu=Q^{N,\phi}$. Then

Proof. The random ratio $R_s=N_s^\phi/N_s^0$ is a convex combination of the maturity factors, so $a_s\le R_s\le b_s$. Change of numeraire gives $d\mu/d\nu=R_s/m_s$, where $m_s=\mathbb E_\nu R_s$. Hence

The last expression is maximized at $m_s=\sqrt{a_sb_s}$ and gives the displayed bound. A two-point law supported at $a_s,b_s$ with mean $\sqrt{a_sb_s}$ attains equality, so no smaller bound can follow from the factor range alone. This is the same hyperbolic-tangent profile as the classical projective contraction coefficient of Birkhoff (1957).

If $L_j=\int_s^{u_j}\phi(v)\,dv$ and $\Delta_s=\max_jL_j-\min_jL_j$, then $\log\kappa_s=\Delta_s$. Therefore

For any bounded payoff $H$, this also yields $|\mathbb E_\mu H-\mathbb E_\nu H| \le\operatorname{osc}(H)\tanh(\Delta_s/4)$. Thus the annuity-measure effect is exactly zero when the factors agree and is Lipschitz in their accumulated-shift spread when they nearly agree.

Corrected contract formulas

Caplets and floorlets

Take $t\le T\le\tau<\tau+\Delta$, put $q=1+\Delta K$, and define $q^\phi=qA(T,\tau+\Delta)/A(T,\tau)$. The generalized caplet satisfies

The floorlet has the same prefactor with the positive part reversed. Printed pp. 172–173 retain $A(T,\tau)$ but omit $A(t,T)$.

Swaplets

Linearity requires the factor for each cashflow's own payment date:

Printed p. 175 instead multiplies both terms by $A(t,\tau)$.

Futures recursion

Bayes' rule for the $T_i$-bond numeraire gives

The exponent is negative. Printed p. 177 uses a positive exponent while retaining the outside bond normalization. This is the change-of-numeraire identity of Geman, El Karoui and Rochet (1995).

Exact numerical certificate

For $t=0$, $T=\tau=1$, $\Delta=1$, $K=0$, zero base rate, and $\phi=0.05$, the corrected caplet and swaplet value is $e^{-0.05}-e^{-0.10}=0.0463920065$. The printed caplet formula gives $0.0487705755$ and the printed swaplet gives zero.

The certificate checks the caplet and floorlet identities on 500 exact finite-state trees, the nonlinear three-bond transport identity, forward-law and normalized-option invariance, transformed Black implied-volatility invariance, the common-factor basket criterion, and the sharp total-variation envelope. It also supplies an annuity-measure counterexample with total-variation distance $0.00289904$ and verifies the bound on 500 random trees. Source-page evidence is tracked in Issue #69.