Why the price of a zero-coupon bond maturing at time T is e^{-RT}?

[Kafuka]

The following content is from OPTIONS, FUTURES,AND OTHER DERIVATIVES 9th by John C. Hull P87

where R is the zero rate for a maturity of T. The value of [math]R_F[/math] obtained in this way is known as the instantaneous forward rate for a maturity of T. This is the forward rate that is applicable to a very short future time period that begins at time T. Define P(0,T) as the price of a zero-coupon bond maturing at time T. Because [math]P(0,T)=e^{-RT}[/math],the equation for the instantaneous forward rate can also be written as[math]R_{F}=-\frac{\partial}{\partial T}\ln P(0,T)[/math]

My question is :Why the price of a zero-coupon bond maturing at time T is [math]e^{-RT}[/math].

It looks more like the spot discount rate in period T, rather than the price. Even if we were to make a connection to the price, it would look more like the present price rather than the price at time T

For example:[math]p_0 = 100*e^{-RT}[/math]

 

[_quanty_]

[math]e^{-rT}[/math] is the present value of a ZCB with face $1 which matures at time T. It is not the price at time T.

The price at time T of a ZCB with face $1 maturing at time T is $1. How much would you pay to receive $1 today? You'd pay $1.

The price today (time 0) is e^(-rT) by a no-arbitrage argument. Suppose the spot rate is r. Then if I deposit $1 in a bank account at time 0 then I will have e^{rT} dollars in that bank account at time T, assuming continuous compounding. Maybe an easier way to see that is to deposit e^{-rT} dollars in the bank account today. Then I'll have e^{-rT}*e^{rT} = $1 in the bank account at time T. So a ZCB paying $1 at time T must have current price e^{-rT} or else I could arbitrage between the ZCB and the bank account since they have the same payoff.

In general, the price at time t of a ZCB maturing at time T with face F is [math]F*e^{-r(T-t)}[/math].

 

[achirikhin]

The following content is from OPTIONS, FUTURES,AND OTHER DERIVATIVES 9th by John C. Hull P87

where R is the zero rate for a maturity of T. The value of [math]R_F[/math] obtained in this way is known as the instantaneous forward rate for a maturity of T. This is the forward rate that is applicable to a very short future time period that begins at time T. Define P(0,T) as the price of a zero-coupon bond maturing at time T. Because [math]P(0,T)=e^{-RT}[/math],the equation for the instantaneous forward rate can also be written as[math]R_{F}=-\frac{\partial}{\partial T}\ln P(0,T)[/math]

My question is :Why the price of a zero-coupon bond maturing at time T is [math]e^{-RT}[/math].

It looks more like the spot discount rate in period T, rather than the price. Even if we were to make a connection to the price, it would look more like the present price rather than the price at time T

For example:[math]p_0 = 100*e^{-RT}[/math]

When the curve is flat, forward equals instantaneous short.

Economically, zerobonds trade term yield or instantaneous forward.

 

[Kafuka]

[math]e^{-rT}[/math] is the present value of a ZCB with face $1 which matures at time T. It is not the price at time T.

The price at time T of a ZCB with face $1 maturing at time T is $1. How much would you pay to receive $1 today? You'd pay $1.

The price today (time 0) is e^(-rT) by a no-arbitrage argument. Suppose the spot rate is r. Then if I deposit $1 in a bank account at time 0 then I will have e^{rT} dollars in that bank account at time T, assuming continuous compounding. Maybe an easier way to see that is to deposit e^{-rT} dollars in the bank account today. Then I'll have e^{-rT}*e^{rT} = $1 in the bank account at time T. So a ZCB paying $1 at time T must have current price e^{-rT} or else I could arbitrage between the ZCB and the bank account since they have the same payoff.

In general, the price at time t of a ZCB maturing at time T with face F is [math]F*e^{-r(T-t)}[/math].

Thank you for your answer. I'm going to think carefully about your answer.

 

[Kafuka]

When the curve is flat, forward equals instantaneous short.

Economically, zerobonds trade term yield or instantaneous forward.

Thank you for your answer. I'm going to think carefully about your answer.