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

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!

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

• Hello Matteo, thanks for the reply. That clarifies why $\lambda(t)$ does not depend on $T$. On the other side, do you have any idea why there is a $P(t,T)$ term in the expression for $\lambda(t)$ given in Brigo & Mercurio? Where is it coming from?
– KT8
Commented Mar 23, 2022 at 9:52
• My guess is that this is due to the "re-formulation" of the dynamics of the price when you use Ito's lemma. Following Björk, you see that you can re-define the drift and diffusion terms to get a nicer form by multiplying by the price itself $P(t,T)$. Think about the connection between a Brownian motion and a Geometric Brownian Motion. I imagine $r(t)$ remains multiplied by $P(t,T)$ because of something similar happening in the PDE. Commented Mar 23, 2022 at 11:37
• @user9875321__ I came across this post while trying to find a solution to my own problem as stated here (quant.stackexchange.com/questions/79466/…). In my problem I seem to have a market price of risk that depends on maturity time $T$ and whose limit $lim_{t \to T} \lambda(t)$ tends to some function $f(T)$ of maturity time, does this mean that it's wrong? Because this sort of implies that a bond with different maturity times will get different $\lambda$s? I would really appreciate some input, if anyone has any. Commented May 28 at 12:30
• In accordance to what above, $\lambda(t)$ does depend on $t$ because $r(t)$ and $\sigma_T(t)$ and $\alpha_T(t)$ all depend on $t$, but $\lambda(t)$ itself should remain the same for every maturity $T$. Importantly, notice that the mpr is a function of time $t$, meaning it can change over time, but does not change across maturities $T$. So my guess is you got confused between $t$ (point in time) and $T$ (maturity) Commented May 29 at 7:26