Simplified Bonds Price and Greeks

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

Theta

To compute theta, the first derivative of price with respect to time, we will need to make an assumption about how the bond price is calculated between coupons. We assume that the price after a passage of time t is calculated using compounding method:

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

Where \(ft\) can be interpreted as the fraction of coupon period that has elapsed. For example, if a bond pays coupon every 6 month (f=2), then the first coupon as at time t=0 will be due in 0.5 (=6/12) years. After 3 months, t=0.25(=3/12), and half of the first coupon period would have elapsed (0.25/0.5=1/2), which is precisely ft=2*0.25=0.5.

Differentiating both sides with respect to t, we get:

$$ \frac{\partial Price(t)}{\partial t} = \frac{\partial}{\partial t}\left[ \sum_{i=1}^{n}{\frac{c}{{\left( 1+r/f \right)}^{i-ft}}}+\frac{F}{{\left( 1+r/f \right)}^{n-ft}} \right] $$ $$ \frac{\partial Price(t)}{\partial t} = \sum_{i=1}^{n}{ \frac{\partial}{\partial t} \frac{c}{{\left( 1+r/f \right)}^{i-ft}}}+ F \frac{\partial}{\partial t} \frac{1}{{\left( 1+r/f \right)}^{n-ft}} $$

Note that t is in the numerator of each expression to be differentiated, so we need to rearrange it before we apply the differentiation rules. Let

$$ y=\frac{1}{{\left( 1+r/f \right)}^{i-ft}} $$ $$ ln(y)= -(i-ft) ln {\left( 1+r/f \right)}$$ $$ \frac{\partial ln(y)}{\partial t}=f \; ln {\left( 1+r/f \right)}$$ $$ \frac{1}{y} \frac{\partial y}{\partial t}=f \; ln {\left( 1+r/f \right)}$$ $$ \frac{\partial y}{\partial t}= f \; \frac{ln {\left( 1+r/f \right)}}{{\left( 1+r/f \right)}^{i-ft}}$$

Thus:

$$ \frac{\partial Price(t)}{\partial t} = \sum_{i=1}^{n}{ \frac{\partial}{\partial t} \frac{c}{{\left( 1+r/f \right)}^{i-ft}}}+ F \frac{\partial}{\partial t} \frac{1}{{\left( 1+r/f \right)}^{n-ft}} $$ $$ = \sum_{i=1}^{n}{ \frac{f \; ln {\left( 1+r/f \right)}}{{\left( 1+r/f \right)}^{i-ft}}}+ F \frac{f \; ln {\left( 1+r/f \right)}}{{\left( 1+r/f \right)}^{n-ft}} $$ $$ = f \; ln{\left( 1+r/f \right)} \left[ \sum_{i=1}^{n}{\frac{c}{{\left( 1+r/f \right)}^{i-ft}}}+\frac{F}{{\left( 1+r/f \right)}^{n-ft}} \right] $$ $$ = f \; ln{\left( 1+r/f \right)} \quad Price $$

Note the above formula holds between coupon dates. On each coupon date, the price of the bond will reduce by the amount of coupon that the holder will receive in cash.