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* Subordinated bonds are those that have a lower priority than other bonds of the issuer in case of liquidation. In case of bankruptcy, there is a hierarchy of creditors. First the liquidator is paid, then government taxes, etc. The first bond holders in line to be paid are those holding what is called senior bonds. After they have been paid, the subordinated bond holders are paid. As a result, the risk is higher. Therefore, subordinated bonds usually have a lower credit rating than senior bonds. The main examples of subordinated bonds can be found in bonds issued by banks, and asset-backed securities. The latter are often issued in tranches. The senior tranches get paid back first, the subordinated tranches later.

* Perpetual bonds are also often called perpetuities. They have no maturity date. The most famous of these are the UK Consols, which are also known as Treasury Annuities or Undated Treasuries. Some of these were issued back in 1888 and still trade today, although the amounts are now insignificant. Some ultra-long-term bonds (sometimes a bond can last centuries: West Shore Railroad issued a bond which matures in 2361 (i.e. 24th century)) are virtually perpetuities from a financial point of view, with the current value of principal near zero.

* Bearer bond is an official certificate issued without a named holder. In other words, the person who has the paper certificate can claim the value of the bond. Often they are registered by a number to prevent counterfeiting, but may be traded like cash. Bearer bonds are very risky because they can be lost or stolen. Especially after federal income tax began in the United States, bearer bonds were seen as an opportunity to conceal income or assets. U.S. corporations stopped issuing bearer bonds in the 1960s, the U.S. Treasury stopped in 1982, and state and local tax-exempt bearer bonds were prohibited in 1983.

* Registered bond is a bond whose ownership (and any subsequent purchaser) is recorded by the issuer, or by a transfer agent. It is the alternative to a Bearer bond. Interest payments, and the principal upon maturity, are sent to the registered owner.

* Municipal bond is a bond issued by a state, U.S. Territory, city, local government, or their agencies. Interest income received by holders of municipal bonds is often exempt from the federal income tax and from the income tax of the state in which they are issued, although municipal bonds issued for certain purposes may not be tax exempt.

* Book-entry bond is a bond that does not have a paper certificate. As physically processing paper bonds and interest coupons became more expensive, issuers (and banks that used to collect coupon interest for depositors) have tried to discourage their use. Some book-entry bond issues do not offer the option of a paper certificate, even to investors who prefer them.

* Lottery bond is a bond issued by a state, usually a European state. Interest is paid like a traditional fixed rate bond, but the issuer will redeem randomly selected individual bonds within the issue according to a schedule. Some of these redemptions will be for a higher value than the face value of the bond.

* War bond is a bond issued by a country to fund a war.

* Serial bond is a bond that matures in installments over a period of time. In effect, a $100,000, 5-year serial bond would mature in a $20,000 annuity over a 5-year interval.

* Revenue bond is a special type of municipal bond distinguished by its guarantee of repayment solely from revenues generated by a specified revenue-generating entity associated with the purpose of the bonds. Revenue bonds are typically “non-recourse,” meaning that in the event of default, the bond holder has no recourse to other governmental assets or revenues

The Black–Scholes model of the market for a particular equity makes the following explicit assumptions:

* It is possible to borrow and lend cash at a known constant risk-free interest rate.
* The price follows a Geometric Brownian motion with constant drift and volatility.
* There are no transaction costs.
* The stock does not pay a dividend (see below for extensions to handle dividend payments).
* All securities are perfectly divisible (i.e. it is possible to buy any fraction of a share).
* There are no restrictions on short selling.
* There is no arbitrage opportunity

From these ideal conditions in the market for an equity (and for an option on the equity), the authors show that “it is possible to create a hedged position, consisting of a long position in the stock and a short position in calls on the same stock, whose value will not depend on the price of the stock.”

The Black–Scholes model disagrees with reality in a number of ways, some significant. It is widely employed as a useful approximation, but proper application requires understanding its limitations – blindly following the model exposes the user to unexpected risk.

Among the most significant limitations are:

* the underestimation of extreme moves, yielding tail risk, which can be hedged with out-of-the-money options;
* the assumption of instant, cost-less trading, yielding liquidity risk, which is difficult to hedge;
* the assumption of a stationary process, yielding volatility risk, which can be hedged with volatility hedging;
* the assumption of continuous time and continuous trading, yielding gap risk, which can be hedged with Gamma hedging.

In short, while in the Black–Scholes model one can perfectly hedge options by simply Delta hedging, in practice there are many other sources of risk.

Results using the Black–Scholes model differ from real world prices due to simplifying assumptions of the model. One significant limitation is that in reality security prices do not follow a strict stationary log-normal process, nor is the risk-free interest actually known (and is not constant over time). The variance has been observed to be non-constant leading to models such as GARCH to model volatility changes. Pricing discrepancies between empirical and the Black–Scholes model have long been observed in options that are far out-of-the-money, corresponding to extreme price changes; such events would be very rare if returns were lognormally distributed, but are observed much more often in practice.

Nevertheless, Black–Scholes pricing is widely used in practice, for it is easy to calculate and explicitly models the relationship of all the variables. It is a useful approximation, particularly when analyzing the directionality that prices move when crossing critical points. It is used both as a quoting convention and a basis for more refined models. Although volatility is not constant, results from the model are often useful in practice and helpful in setting up hedges in the correct proportions to minimize risk. Even when the results are not completely accurate, they serve as a first approximation to which adjustments can be made.

One reason for the popularity of the Black–Scholes model is that it is robust in that it can be adjusted to deal with some of its failures. Rather than considering some parameters (such as volatility or interest rates) as constant, one considers them as variables, and thus added sources of risk. This is reflected in the Greeks (the change in option value for a change in these parameters, or equivalently the partial derivatives with respect to these variables), and hedging these Greeks mitigates the risk caused by the non-constant nature of these parameters. Other defects cannot be mitigated by modifying the model, however, notably tail risk and liquidity risk, and these are instead managed outside the model, chiefly by minimizing these risks and by stress testing.

Additionally, rather than assuming a volatility a priori and computing prices from it, one can use the model to solve for volatility, which gives the implied volatility of an option at given prices, durations and exercise prices. Solving for volatility over a given set of durations and strike prices one can construct an implied volatility surface. In this application of the Black–Scholes model, a coordinate transformation from the price domain to the volatility domain is obtained. Rather than quoting option prices in terms of dollars per unit (which are hard to compare across strikes and tenors), option prices can thus be quoted in terms of implied volatility, which leads to trading of volatility in option markets.