1. Intro to FIQT - Eurodollar Futures
Convexity Adjustment
(A time scope graph is needed here...)
There exists a convexity adjustment between LIBOR forward contracts and Eurodollar futures. Hos is this derived is shown below
Assume \(\delta(t, T)\) is the forward price at time \(t\) which expires at time \(T\), and \(L(T)\) is the LIBOR rate at time \(T\), \(r(t)\) is the risk-free rate process. We can easily know the rational price of zero coupon bond at time \(t\) which expires at time \(T\), or \(T\)-forward numeraire is \[ p(t, T) = E_t^Q[e^{-\int_{t}^{T}r(u)du}] \] For a LIBOR forward contract that was entered at time \(t\), its contact value is 0, which is \[ \begin{aligned} 0 &= E_t^Q \left[ e^{-\int_{t}^{T}r(u)du} [\delta(t, T) - L(T)] \right] \\ &= E_t^{Q}\left[ e^{-\int_{t}^{T}r(u)du} \delta(t, T) - e^{-\int_{t}^{T}r(u)du} L(T)\right]\\ &= E_t^{Q}\left[ e^{-\int_{t}^{T}r(u)du} \delta(t, T) \right] - E_t^{Q}\left[ e^{-\int_{t}^{T}r(u)du} \right] E_t^Q [L(T)] - cov \left( e^{-\int_{t}^{T}r(u)du}, L(T) \right) \\ &= \delta(t, T) \cdot p(t, T) - p(t, T) \cdot E_t^Q [L(T)] - cov \left( e^{-\int_{t}^{T}r(u)du}, L(T) \right) \\ \end{aligned} \] where we apply \(cov(X, Y) = E[X Y] - E[X] E[Y]\), so we have \[ \delta(t, T) = E_t^Q [L(T)] + \frac{1}{p(t,T)} cov \left( e^{-\int_{t}^{T}r(u)du}, L(T) \right) \]
NPV Effect
Consider the value of a forward contract at \(t' > t\) under CSA, a contract that was entered at time \(t\), so the difference in contract values on \(t'\) and \(t\) that exchanges hands at \(t'\) is equal to \[ V(t') - V(t) = E_{t'} \left( e^{-\int_{t'}^{T}r_c(u)du} \right) (F_{CSA}(t', T) - F_{CSA}(t, T)) \] while the difference of futures contract will not be discounted, which is \[ F(t') - F(t) = F_{CSA}(t', T) - F_{CSA}(t, T) \]
Let's take a look at another example in Piterbarg (2010).