Recall a few concepts that you have seen an in earlier financial mathematics course:
(v(t)) – the current market price of a (t)-year coupon bond
(y_t) – the (t)-year spot rate of interest. This is defined through the expression
begin{eqnarray*}
v(t)(1+y_t)^t = 1 textrm{ which is the same thing as } v(t)=frac{1}{(1+y_t)^t}
end{eqnarray*}
yield curve – a plot of ({y_t}) versus (t)
term structure of interest rates describes this relationship (between ({y_t}) and (t)).
flat term structure – (v(t) =v^t = e^{-delta t}), or (y_t equiv y).
forward rates of interest – (f(t,t+k)), the (annual) interest rate contracted at time 0 earned on an investment made at time (t) that matures at time (t+k)
begin{eqnarray*}
(1+f(t,t+k))^k= frac{(1+y_{t+k})^{t+k}}{(1+y_t)^t}= frac{v(t)}{v(t+k)}
end{eqnarray*}
no-arbitrage argument – we should not be able to make money from nothing in risk free bonds by disinvesting and then reinvesting. An arbitrage opportunity exists if an investor can construct a portfolio that costs zero at inception and generates positive profits with a non-zero probability in the future, with no possibility of incurring a loss at any future time.