## Simplified Bonds Price and Greeks

We present and derive analytical formulae for the price and greeks of a simplified bond.

### Price

The price of a bond with maturity T and face value F, paying a coupon of c with a frequency f per year, can be written as:

$$Price = \sum_{i=1}^{n=fT}{\frac{c}{{\left( 1+r/f \right)}^i}}+\frac{F}{{\left( 1+r/f \right)}^n}$$ $$\quad \quad = c \sum_{i=1}^{n}{z^i}+F z^n$$ $$\quad \quad= c \frac{z-z^{n+1}}{1-z}+F z^n$$

Where r represent the flat spot rate, and we let $$z= \frac{1}{1+r/f}$$ to simplify the formula.

We also used the sum of series formula (see wikipedia for its derivation):

$$\sum_{i=1}^{n}{z^i}=\frac{z-z^{n+1}}{1-z}$$