We denote discount factor $D(t)$, and zero coupon bond $B(t,T)$ as: $$B(t,T) =\dfrac{1}{D(t)} E_t[D(T)]$$ here $E_t[X] = E[X|\mathcal{F}(t)].$

And we define

Zero curve $Z(t,T)$ $$B(t, T)\cdot e^{Z(t,T)(T-t)} = 1$$

Libor $L(t,T)$ $$B(t, T)\cdot (1 + (T-t) L(t, T)) = 1.$$ Denote $E^{T}[\ ]$ the $T$-forward measure i.e use $B(t,T)$ as numeraire.

I remember that Libor and zero curve should be martingale under the $T$-forward measure. But we see the representations, equivalently $\dfrac{1}{B(t,T)}$ should be martingale under the $T$-forward. This is the $T$-forward price of $1,$ but discounted value of $1$ is not martingale under original measure i.e $$E_t[D(T)\cdot 1] \neq D(t)\cdot1.$$ We can see that forward rate $$B(t,T-\delta) = (1 + (T-t)F(t,T-\delta,T))B(t,T)$$ is really $T$-forward martingale, since $\frac{B(t,T-\delta)}{B(t,T)}$ is $T$-forward martingale, equivalently $D(t)B(t,T-\delta)$ is martingale under the original measure. So, I really confuse here. Can anyone tell where is the mistake?


1 Answer 1


Your definition of Libor is invalid as you make it cover the period $t, T$.

A Libor with tenor $\delta$ that fixes on $T$ (or to be accurate usually 2 days before $T$) covers the period $T, T+\delta$. Thus the forward Libor rate $L(t, T, T+\delta)$ is computed as $$ B(t, T+\delta) (1 + \delta L(t, T, T+\delta)) = B(t, T) $$ hence $$ L(t, T, T+\delta) = \frac{1}{\delta}\left(\frac{B(t, T) }{B(t, T+\delta)} -1 \right) $$ is a martingale under the $T+\delta$ forward measure.

  • $\begingroup$ yeah, this is the definition in shreve's book and I have mentioned it on the bottom, here I call it forward Libor which is truly martingale. $\endgroup$
    – A.Oreo
    May 31, 2017 at 6:59
  • 1
    $\begingroup$ Not sure what you want to get to but the rate which you wrote $L(t, T)$ and that covers the running period $t, T$ cannot be a martingale under a risk neutral measure for any numeraire. Also $1/B(t, T)$ cannot be a martingale under the $T$ forward measure because $1$ is the not the price of a self financed asset. $\endgroup$ May 31, 2017 at 7:10
  • $\begingroup$ Actually, we question comes from the cap option, $(L(T,T+\delta) - K)^+$, here if we choose the forward Libor(your Libor), then we can transform into the Black-Scholes call, but if we use the my Libor, then it becomes a put on bond value at $T$ maturity at $T+\delta,$ i,e $(K^*-B(T,T+\delta))^+.$ Then I don't know how to solve it, since we don't know the dynamic of $B(t,T+\delta)$ under the $T+\delta$-forward measure. $\endgroup$
    – A.Oreo
    May 31, 2017 at 7:25
  • $\begingroup$ pls see me new quesiton quant.stackexchange.com/questions/34474/cap-option-on-libor $\endgroup$
    – A.Oreo
    May 31, 2017 at 7:36
  • $\begingroup$ As noted the forward Libor is a martingale under the caplet payment date $T+\delta$ forward measure, which is why Black & Scholes is applicable, using the forward libor volatility. If you want to model directly the zero coupon bond $B(t,T+\delta)$ then you have to use an HJM approach for specifying the entire yield curve dynamics. $\endgroup$ May 31, 2017 at 7:38

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