Market price of risk ($\lambda$) - Brigo and Mercurio

Question

In page 52 of Interest Rate Models by Brigo and Mercurio the following is stated:

Precisely, let us assume that the instantaneous spot rate evolves under the real-world measure $Q_0$ according to $dr(t) = \mu (t, r(t))dt + \sigma(t, r(t))dW_0(t)$, where $\mu$ and $\sigma$ are well-behaved functions and $W_0$ is a $Q_0$-Brownian motion. It is possible to show (they cite Björk 1997 here) the existence of a stochastic process $\lambda$ such that if $$dP(t, T) = \mu^T (t, r(t))dt + \sigma^T (t, r(t))dW_0(t),$$ then $$λ(t) = \dfrac{\mu^T (t, r(t)) − r(t)P(t, T)}{\sigma^T (t, r(t))}.$$ for each maturity T, with $\lambda$ that may depend on $r$ but not on $T$.

They then move to the risk-free measure $Q$ via definition of a Radon-Nikodym derivative, and they claim that the process $r$ evolves under $Q$ according to $$dr(t)=\Big[\mu(t, r(t)) − \lambda(t) \sigma (t, r(t)) \Big]dt + σ(t, r(t))dW(t).$$

I think I know how the Girsanov change of measure works, or at least I am able to follow it when it concerns stock underlyings. However, I am unable to see where the $P(t,T)$ in the definition is coming from, and why they claim $\lambda(t)$ depends only on $t$ and not on both $t$ and $T$. Moreover, in page 80 of Björk (Proposition 3.1) the following appears, while they discuss the locally risk-free portfolios:

$$\dfrac{\alpha_T(t) - r(t)}{\sigma_T(t)} = \lambda(t),$$ holds for all $t$ and for every choice of maturity time $T.$

So, apparently, the definition of $\lambda(t)$ does depends on the maturity.

I would appreciate if someone could help me clarify this a bit. Thanks!

Answer 1

From Bjork, 2004, Arbitrage Theory in Continuous Time, Ch. 21 Short Rate models: The price of a bond with maturity $T$ can be seen as a function of the short rate $r_t$: $p(t,T)=F(t,r(t); T)$. Apply Ito's lemma on this. You get a bunch of terms that can be then collected into a drift term (multiplying $dt$) and a diffusion. By renaming them $\mu^T$ and $\sigma^T$ you get the bond price dynamics. Now imagine to do this for 2 bonds with different maturities, i.e. $T$ and $S$ and that you want to build a risk-free portfolio $V$ combining these two. Such portfolio will be defined by $w_T\sigma_T+w_S\sigma_S=1$ where $w_T+w_S=1$ are portfolio weight. The first condition is required to have a risk-free portfolio (essentially: zero diffusion). Now, solve the system and plug the solution back into the dynamics of $V$. By setting its drift to zero (no-arbitage), you then get that

$$\frac{\alpha_S\sigma_T-\alpha_T\sigma_S}{\sigma_T-\sigma_S}=r(t), \; \forall t$$ or, alternatively, that:

$$\frac{\alpha_S-r(t)}{\sigma_S}=\frac{\alpha_T-r(t)}{\sigma_T}$$

The last line coincide with your $\lambda(t)$. This says that the market price of risk must coincide across bonds with different maturities and thus, ultimately, does not depend on the maturity. In other words: if you compute the market price of risk using data for a bond with maturity $T$, then you have it also for a bond with maturity $S$. Hope this helps