Interest Rates
0.1 [[yield-curve]]
Assume that a zero-coupon bond that pays 1 at time \(T\) has price \(p(t,T)\) contracted at time \(t\). Naturally, for all \(t\), \(p(t,t)=1\) and \(p(t,T)\) is well-defined for all \(t\in \mathbb{R}\). Assume that \(p(t,T)\) is also well-defined for all \(T\in \mathbb{R}\) and is differentiable w.r.t \(T\).
If the accumulation of interest can be described by a constant continuous-time interest rate on \([t,T]\) denoted \(y(t,T)\), then \[ p(t,T)e^{y(t,T)(T-t)}=1, \] solves \[ y(t,T)=- \frac{1}{T-t}\log p(t,T), \] called the yield curve viewed as a function in \(T\) after fixing \(t\).
0.2 [[libor-rates]]
Consider time \(t\) with two maturity dates \(S<T\). We consider the forward interest rate of \([S,T]\) contracted at time \(t\). A riskless portfolio can be constructed with no initial endowment that at time \(t\) one can short 1 unit of bond that matures at \(S\). This brings immediately cashflow \(p(t,S)\) and a debt returning 1 at time \(S\).
However, one can long \(\frac{p(t,S)}{p(t,T)}\) unit of bond that matures at \(T\) using the \(p(t,S)\) immediate cashflow, this brings a cashflow of \(\frac{p(t,S)}{p(t,T)}\) at time \(T\). In all, this portfolio has no immediate cashflow, but we have to pay 1 at time \(S\) and receive \(\frac{p(t,S)}{p(t,T)}\) at time \(T\). If there exists a constant simple interest rate on \([S,T]\) denoted \(L(t;S,T)\), then \[ 1\cdot[1+L(t;S,T)(T-S)]= \frac{p(t,S)}{p(t,T)}, \] solves \[ L(t;S,T)= \frac{1}{T-S}\left( \frac{p(t,S)}{p(t,T)}-1 \right), \] called the simple forward rate on \([S,T]\) contracted at \(t\), also called the LIBOR Forward Rate. If one uses the continuous-time compounded interest rate mode instead and denote the rate as \(R(t;S,T)\) then \[ 1\cdot e^{R(t;S,T)(T-S)}= \frac{p(t,S)}{p(t,T)} \] solves \[ R(t;S,T)= \frac{1}{T-S}\log\left( \frac{p(t,S)}{p(t,T)}\right). \] The spot rates on \([S,T]\) is just the forward rate contracted at time \(S\) i.e. by setting \(t=S\), we get the spot rates \[ \begin{align} L(S;S,T) & = \frac{1}{T-S}\left( \frac{1}{p(S,T)} -1\right) \\ R(S;S,T) & = \frac{1}{T-S}\log\left( \frac{1}{P(S,T)} \right) \end{align} \] where \(L(S;S,T)\) is called the LIBOR spot rate.
0.3 Instantaneous Forward Rate
The instantaneous forward rate \(f(t,T)\) describes the forward rate in \([T,T+d]\) contracted at \(t\) when \(dt\) is small enough. All instantaneous rates are built upon the continuously compounded interest model. From the definition we hav that \[ f(t,t)=\lim_{ dt \to 0 }R(t,T, T+dt)=\lim_{ dt \to 0 }\frac{1}{dt} \log\left( \frac{p(t,T)}{p(t,T+dt)} \right)=- \frac{{\partial \log p(t,T)}}{\partial T} . \] ### Instantaneous Short Rate The instantaneous short rate \(r(t)\) describes the spot rate in \([t,t+dt]\) contracted at \(t\) when \(dt\) is small enough. Clearly plugging \(T=t\) into the instantaneous forward rate \(f(t,T)\) gives \[ r(t)=f(t,t). \] We can interpret this as the spot interest rate in a future infinitesimal time interval, so it is exactly the risk-free interest rate \(r\) we refer to in the context of options pricing.
When one opens a savings account, denote the amount of money in the account at time \(t\) as \(S_{t}^0\), then it is clear that \[ dS_{t}^0=r_{t}S_{t}^0dt, \] where \(r_{t}=r(t)\) is the short rate. As a result, \(S_{t}^0\) can be represented in terms of the short rate as \[ S_{t}^0=\exp\left( \int _{0}^tr_{s} \, ds \right), \] which tells us how to discount using the short rate.
0.4 LIBR FORWARD RATE
Consider time \(t\) with two maturity dates \(S<T\). We consider the forward interest rate of \([S,T]\) contracted at time \(t\). A riskless portfolio can be constructed with no initial endowment that at time \(t\) one can short 1 unit of bond that matures at \(S\). This brings immediately cashflow \(p(t,S)\) and a debt returning 1 at time \(S\).
However, one can long \(\frac{p(t,S)}{p(t,T)}\) unit of bond that matures at \(T\) using the \(p(t,S)\) immediate cashflow, this brings a cashflow of \(\frac{p(t,S)}{p(t,T)}\) at time \(T\). In all, this portfolio has no immediate cashflow, but we have to pay 1 at time \(S\) and receive \(\frac{p(t,S)}{p(t,T)}\) at time \(T\). If there exists a constant simple interest rate on \([S,T]\) denoted \(L(t;S,T)\), then \[ 1\cdot[1+L(t;S,T)(T-S)]= \frac{p(t,S)}{p(t,T)}, \] solves \[ L(t;S,T)= \frac{1}{T-S}\left( \frac{p(t,S)}{p(t,T)}-1 \right), \] called the simple forward rate on \([S,T]\) contracted at \(t\), also called the LIBOR Forward Rate.
0.5 LIBOR SPOT RATE
If one uses the continuous-time compounded interest rate mode instead and denote the rate as \(R(t;S,T)\) then \[ 1\cdot e^{R(t;S,T)(T-S)}= \frac{p(t,S)}{p(t,T)} \] solves \[ R(t;S,T)= \frac{1}{T-S}\log\left( \frac{p(t,S)}{p(t,T)}\right). \] The spot rates on \([S,T]\) is just the forward rate contracted at time \(S\) i.e. by setting \(t=S\), we get the spot rates \[ \begin{align} L(S;S,T) & = \frac{1}{T-S}\left( \frac{1}{p(S,T)} -1\right) \\ R(S;S,T) & = \frac{1}{T-S}\log\left( \frac{1}{P(S,T)} \right) \end{align} \] where \(L(S;S,T)\) is called the LIBOR spot rate.
1 Yield Curve
Assume that a zero-coupon bond that pays 1 at time \(T\) has price \(p(t,T)\) contracted at time \(t\). Naturally, for all \(t\), \(p(t,t)=1\) and \(p(t,T)\) is well-defined for all \(t\in \mathbb{R}\). Assume that \(p(t,T)\) is also well-defined for all \(T\in \mathbb{R}\) and is differentiable w.r.t \(T\).
If the accumulation of interest can be described by a constant continuous-time interest rate on \([t,T]\) denoted \(y(t,T)\), then \[ p(t,T)e^{y(t,T)(T-t)}=1, \] solves \[ y(t,T)=- \frac{1}{T-t}\log p(t,T), \] called the yield curve viewed as a function in \(T\) after fixing \(t\).