It is primary security, which pays off 1 unit no matter state of economy is realized at time
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.
[1] For instance, United States Treasury notes and United States Treasury bonds are often assumed to be risk-free bonds.
[2] Even though investors in United States Treasury securities do in fact face a small amount of credit risk,[3] 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.
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.
is constant (but not necessarily nonnegative) over the trading interval
The risk-free security is assumed to continuously compound in value at the rate
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
is a security paying to its holder 1 unit of cash at a predetermined date
This shows that, in the absence of arbitrage opportunities, the price of the bond satisfies
Note that for any fixed T, the bond price solves the ordinary differential equation
We consider here a risk-free bond, meaning that its issuer will not default on his obligation to pat to the bondholder the face value at maturity date.
The risk-free bond can be replicated by a portfolio of two Arrow-Debreu securities.
This portfolio exactly matches the payoff of the risk-free bond since the portfolio too pays 1 unit regardless of which state occurs.
This is because if its price were different from that of the risk-free bond, we would have an arbitrage opportunity present in the economy.
When an arbitrage opportunity is present, it means that riskless profits can be made through some trading strategy.
Let's call the price of the risk-free bond at time
refers to the fact that the bond matures at time
As mentioned before, the risk-free bond can be replicated by a portfolio of two Arrow-Debreu securities, one share of
which is a product of ratio of the intertemporal marginal rate of substitution (the ratio of marginal utilities across time, it is also referred to as the state price density and the pricing kernel) and the probability of state occurring in which the Arrow-Debreu security pays off 1 unit.
Therefore, the price of a risk-free bond is simply the expected value, taken with respect to the probability measure
Then using the previous formulas, we can calculate the bond price