Notation
Bonds and Forward Rates
Zero Conpon Bond and its Forward Price
The most basic instrument in the interest rate modelling is zero coupon bond $P(t, T)$ which deliver $1 for certain at time $T$. $P(t, T)$ represents the price at time t.
Also, $P(t, T, T+\tau)$ (A.K.A. Forward price for the zero conpon bond) represents the current value that we purchase a zero coupon bond at time $T$ which deliver $1 at time $T+\tau$. This can be replicated by just using zero coupon bonds.
| Cash Flow | $t$ | $T$ | $T+\tau$ |
|---|---|---|---|
| Long 1 unit of $P(t, T+\tau)$ | $-P(t, T+\tau)$ | 0 | 1 |
| Short $P(t, T+\tau) / P(t, T)$ unit of $P(t, T)$ | $P(t, T+\tau) / P(t, T) * P(t, T)$ | $-P(t, T+\tau) / P(t, T) * 1$ | 0 |
| Total Cash Flow | 0 | $-P(t, T+\tau) / P(t, T)$ | 1 |
Expressed this in mathmatical words:
\[P(t, T, T+\tau) = P(t, T+\tau) / P(t, T)\]Forward Rate
Based on the forward price of zero conpon bond, we further define the forward rate:
- Continuous compounded forward yield $y(t, T, T+\tau)$
- Simple(discrete) forward rate $L(t, T, T+\tau)$
- Instantaneous forward rate $f(t, T)$
- Spot rate / Short rate
Relationship between Forward Bond Price and Forward Rate
The relationship between instantaneous forward rate and forward bond price is given by:
\[P(t, T, T+\tau) = \exp({-\int_T^{T+\tau}f(t,u)du})\]Such that,
\[f(t,T) = -\frac{\partial \ln P(t,T)}{\partial T} = -\frac{1}{P(t,T)}\frac{P(t,T+\tau)-P(t,T)}{\tau}\]Rearrange and apply Taylor Expansion we got,
\[P(t,T, T+\tau) = 1-\tau f(t,T) = \frac{1}{1+\tau f(t,T)}\]We also notice the relationship between continuous compounded forwrad yield is as below:
\[P(t, T, T+\tau) = \exp({-y(t, T, T+\tau) * \tau}) = \exp({-\int_T^{T+\tau}f(t,u)du}) \\ y(t, T, T+\tau) * \tau = \int_T^{T+\tau}f(t,u)du\]Furture Rate
Entering a Eurodollar futures, investor will pay $1-L(T,T,T+\tau)$ at time $T$ per unit of notional.
At time 0, to long a Eurodollar future doesn’t trigger upfront cost, but with an implicit obligation to pay $1-F(0,T,T+\tau)$ at time $T$ per unit of notional.
We define $F(t, T, T+\tau)$ is the simple future rate for the period $[T, T+\tau]$ observed at time $t$. Note that, future rate is marked to market (MTM) each day. The day’s change in the future rate will immediately reflect on the holder’s account.
Assume the holder take the future position for $\Delta$ days from time $0$, the holder will experience a cashflow of:
\[(1-F(\Delta,T,T+\tau)) - (1-F(0,T,T+\tau)) = F(0,T,T+\tau) - F(\Delta,T,T+\tau)\]If the future is hold through time 0 to $T$, the total amount of cash flow would be:
\[F(0,T,T+\tau) - F(T,T,T+\tau) = F(0,T,T+\tau) - L(T,T,T+\tau)\]Also, instantaneous future rates is defined as below:
\[\lim_{\tau\to 0}F(t, T, T+\tau) = q(t, T)\]FX Forwards
Two economics are considered (i.e. domestic and foreign). Let $P_d(t, T)$ and $P_f(t, T)$ denotes the domestic and foreign zero coupon bond respectively. Use $X(t)$ to denotes the exchange rates at time t. Thus, the value to domestic investor for one unit of foreign bond can be expressed as：
\[\widetilde{P_d(t, T)} = X(t)P_f(t,T)\]The forward FX rate at time $T$ observed at time t is expressed as $X_T(t)$ and has the following equation for non-aribitrage:
\[X_T(t) = \frac{\widetilde{P_d(t, T)}}{P_d(t, T)}=X(t)\frac{P_f(t,T)}{P_d(t, T)}\]