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

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

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.

• 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. May 10 at 6:42
• here what I am thinking: May 10 at 6:50
• 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. May 10 at 6:55
• 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 $d(r(t)*t)=r(t)dt+tdr(t)$ May 10 at 8:45
• The second component of your second integral is deterministic so can be ignored May 10 at 11:04