A risk-free bond is a theoretical
bond
Bond or bonds may refer to:
Common meanings
* Bond (finance), a type of debt security
* Bail bond, a commercial third-party guarantor of surety bonds in the United States
* Chemical bond, the attraction of atoms, ions or molecules to form chemical ...
that repays
interest
In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is distin ...
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
. 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 bond
A government bond or sovereign bond is a form of bond issued by a government to support public spending. It generally includes a commitment to pay periodic interest, called coupon payments'','' and to repay the face value on the maturity dat ...
s 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
A credit risk is risk of default on a debt that may arise from a borrower failing to make required payments. In the first resort, the risk is that of the lender and includes lost principal and interest, disruption to cash flows, and increased ...
,
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
The Russian financial crisis (also called the ruble crisis or the Russian flu) began in Russia on 17 August 1998. It resulted in the Russian government and the Russian Central Bank devaluing the ruble and defaulting on its debt. The crisis had ...
.
Modelling the price by Black-Scholes model
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
In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between t ...
, 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
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, t ...
''
is constant (but not necessarily nonnegative) over the trading interval