Risk-free bond

A risk-free bond is a theoretical bond that repays interest and principal with absolute certainty. The rate of return would be the risk-free interest rate. It is primary security, which pays off 1 unit no matter state of economy is realized at time t + 1 {displaystyle t+1} . So its payoff is the same regardless of what state occurs. Thus, an investor experiences no risk by investing in such an asset. A risk-free bond is a theoretical bond that repays interest and principal with absolute certainty. The rate of return would be the risk-free interest rate. It is primary security, which pays off 1 unit no matter state of economy is realized at time t + 1 {displaystyle t+1} . So its payoff is the same regardless of what state occurs. Thus, an investor experiences no risk by investing in such an asset. In practice, government bonds of financially stable countries are treated as risk-free bonds, as governments can raise taxes or indeed print money to repay their domestic currency debt. For instance, United States Treasury notes and United States Treasury bonds are often assumed to be risk-free bonds. Even though investors in United States Treasury securities do in fact face a small amount of credit risk, this risk is often considered to be negligible. An example of this credit risk was shown by Russia, which defaulted on its domestic debt during the 1998 Russian financial crisis. In financial literature, it is not uncommon to derive the Black-Scholes formula by introducing a continuously rebalanced risk-free portfolio containing an option and underlying stocks. In the absence of arbitrage, the return from such a portfolio needs to match returns on risk-free bonds. This property leads to the Black-Scholes partial differential equation satisfied by the arbitrage price of an option. It appears, however, that the risk-free portfolio does not satisfy the formal definition of a self-financing strategy, and thus this way of deriving the Black-Sholes formula is flawed. We assume throughout that trading takes place continuously in time, and unrestricted borrowing and lending of funds is possible at the same constant interest rate. Furthermore, the market is frictionless, meaning that there are no transaction costs or taxes, and no discrimination against the short sales. In other words, we shall deal with the case of a perfect market. Let's assume that the short-term interest rate r {displaystyle r} is constant (but not necessarily nonnegative) over the trading interval [ 0 , T ∗ ] {displaystyle } . The risk-free security is assumed to continuously compound in value at the rate r {displaystyle r} ; that is, d B t = r B t   d t {displaystyle dB_{t}=rB_{t}~dt} . We adopt the usual convention that B 0 = 1 {displaystyle B_{0}=1} , so that its price equals B t = e r t {displaystyle B_{t}=e^{rt}} for every t ∈ [ 0 , T ∗ ] {displaystyle tin } . When dealing with the Black-Scholes model, we may equally well replace the savings account by the risk-free bond. A unit zero-coupon bond maturing at time T {displaystyle T} is a security paying to its holder 1 unit of cash at a predetermined date T {displaystyle T} in the future, known as the bond's maturity date. Let B ( t , T ) {displaystyle B(t,T)} stand for the price at time t ∈ [ 0 , T ] {displaystyle tin } of a bond maturing at time T {displaystyle T} . It is easily seen that to replicate the payoff 1 at time T {displaystyle T} it suffices to invest B t / B T {displaystyle B_{t}/B_{T}} units of cash at time t {displaystyle t} in the savings account B {displaystyle B} . This shows that, in the absence of arbitrage opportunities, the price of the bond satisfies B ( t , T ) = e − r ( T − t )       ,       ∀ t ∈ [ 0 , T ]   . {displaystyle B(t,T)=e^{-r(T-t)}~~~,~~~forall tin ~.} Note that for any fixed T, the bond price solves the ordinary differential equation d B ( t , T ) = r B ( t , T ) d t       ,       B ( 0 , T ) = e − r T   . {displaystyle dB(t,T)=rB(t,T)dt~~~,~~~B(0,T)=e^{-rT}~.}