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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.

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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.

  1. 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.

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  • $\begingroup$ Thank you very much for your response. Very interesting. I only have a couple of questions. Could you please provide me a reference that covers all things you mentioned? For the second treatment, we have a relation between the expectation under measures P and Q. I wonder if there is a close-form formula for the covariance function (it is under both measures or only one measure) and how I can find an expression for the Radon-NIkody derivative. $\endgroup$
    – user53249
    May 10 at 6:42
  • $\begingroup$ here what I am thinking: $\endgroup$
    – user53249
    May 10 at 6:50
  • $\begingroup$ We know that for the Radon-Nikodym derivative the following relation holds: \begin{eqnarray} \frac{d\mathbb{Q}}{d\mathbb{P}}\bigg |_{\mathcal{F}_t^1} = \exp\left\{\int_{0}^{t}\gamma(s)dW(s) - \frac{1}{2}\int_{0}^{t}\gamma^2(s)ds\right \}, \end{eqnarray} where $W(t)$ stands for the standard Brownian motion, and $\gamma(t)$ for a kernel function. Suppose that the Kernel function is specified by $\gamma(t) = \frac{\lambda}{\sigma}\sqrt{r(t)}$, with $\lambda$ being the market price. $\endgroup$
    – user53249
    May 10 at 6:55
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    $\begingroup$ math.stackexchange.com/questions/3189009/… - here is how you evaluate covariance of two Ito integrals. You can express the SDF as an Ito integral by using $d(r(t)*t)=r(t)dt+tdr(t)$ $\endgroup$
    – Arshdeep
    May 10 at 8:45
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    $\begingroup$ The second component of your second integral is deterministic so can be ignored $\endgroup$
    – Arshdeep
    May 10 at 11:04

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