Instantaneous Forward LIBOR rate formula under the real-world measure: A fundamental question

Question

We know how the formula of an instantaneous forward LIBOR rate looks like:

\begin{eqnarray} L(t, t, T) = \frac{1}{\Delta}\left(\frac{1}{P(t, T)} -1\right) \end{eqnarray} where $P(t, T)$ stands for the zero-coupon bond price at time $t$, with $T$ being the maturity time (the time at which our contract is terminated). Mathematically, the corresponding relation is given by:

\begin{eqnarray} P(t, T) = \mathbb{E}^{\mathbb{Q}}[D(t, T) | \mathcal{F}_t] \end{eqnarray} where the expectation is taken with respect to a risk-netral measure equivalent to the real-world measure $\mathbb{P}$, and $D(t, T)$ is the discount factor between $t$ and $T$ (Let's say it is characterized by a CIR model).

My question here is: what if we want to write a formulation for the instantaneous forward LIBOR rate under the real-world measure $\mathbb{P}$. More precisely, suppose that we specify by $P^{A}(t, T)=\mathbb{E}^{\mathbb{P}}[D(t, T)| \mathcal{F}_t]$ the actuarial value of a zero-coupon bond at time t with maturity time T, and $\Delta = T-t$. Then, is it still possible to write down

\begin{eqnarray} L(t, t, T) = \frac{1}{\Delta}\left(\frac{1}{P^{A}(t, T)} -1\right) \end{eqnarray}

Please let me know what you think. Thank you in advance.

Answer 1

I will show you two different treatments, the first from the classic utility theory and the other from financial economics.

1. Consider a risk averse market operating on a concave, monotonic and increasing utility function. Under some regularity (Von-Neumann) conditions, this is without loss of generality. Such a utility function, is unique upto a linear transformation. Then,

$E_Q(X)=P(t,T)$ and

$U(P(t,T))=E_P(U(X))$ imply that

$U(E_Q(X))=E_P(U(X))$

Let $Y=U(X)$ so that $E_P(Y)=U(E_Q(U^{-1}(Y))$

Also $P_A(t,T)=E_P(Y)$

so that we need to relate $U(E_Q(U^{-1}(Y))$ and $E_Q(Y)$. Observe on account of U being concave,

$U(E_Q(U^{-1}(Y))>E_Q(U(U^{-1}(Y))=E_Q(Y)$

and thus the only relationship we can be sure of is:

$E_P(Y)>E_Q(Y)$ where the difference is the well known 'Jensen gap' which depends heavily on the market utility function. You can now plug $Y$ as the stochastic discount factor.

2. The expectations under equivalent measures $Q$ and $P$ are related by:

$E_Q(X)=E_P(X)+cov(X,dQ/dP)$

where $dQ/dP$ is the Radon Nikodym derivative. You can plug $X$ as the stochastic discount factor and thus be able to express the LIBOR forward in terms of the $P$ measure expectation.