# Cap

is a fixed-income option in which the holder receives a payment when the underlying interest rate exceeds a specified level, the strike. This payment is the interest rate less the strike. These payments happen regularly, monthly, or quarterly, etc., as specified in the contract, and the underlying interest rate will usually be of the same tenor as this interval. The life of the cap will be several years. They are bought for protection against rises in interest rates. Market practice is to quote prices for caps using the Black '76 model. A contract with a single payment as above is called a caplet.

# Chooser option

is an option on an option, therefore a second-order option. The holder has the right to decide between getting a call or a put, for example, on a specified date. The expiration of these underlying options is further in the future. Other similar contracts can be readily imagined. The key to valuing such contracts is the realization that the two (or more) underlying options must first be valued, and then one values the option on the option. This means that finite-difference methods are the most natural solution method for this kind of contract. There are some closed-form formula for simple choosers when volatility is at most time dependent.

# Cliquet option

is a path-dependent contract in which amounts are locked in at intervals, usually linked to the return on some underlying. These amounts are then accumulated and paid off at expiration. There will be caps and/or floors on the locally locked-in amounts and on the global payoff. Such contracts might be referred to as locally capped, globally floored, for example. These contracts are popular with investors because they have the eternally appreciated upside participation and the downside protection, via the exposure to the returns and the locking in of returns and global floor. Because of the locking in of returns and the global cap/floor on the sum of returns, these contracts are strongly path dependent. Typically there will be four dimensions, which may in special cases be reduced to three via a similarity reduction. This puts the numerical solution on the Monte Carlo, finite difference border. Neither are ideal, but neither are really inefficient either. Because these contracts have a gamma that changes sign, the sensitivity is not easily represented by a simple vega calculation. Therefore, to be on the safe side, these contracts should be priced using a variety of volatility models so as to see the true sensitivity to the model.

# Constant Maturity Swap (CMS)

is a fixed-income swap. In the vanilla swap the floating leg is a rate with the same maturity as the period between payments. However, in the CMS the floating leg is of longer maturity. This apparently trivial difference turns the swap from a simple instrument, one that can be valued in terms of bonds without resort to any model, into a model-dependent instrument.

# Collateralized Debt Obligation (CDO)

is a pool of debt instruments securitized into one financial instrument. The pool may consist of hundreds of individual debt instruments. They are exposed to credit risk, as well as interest risk, of the underlying instruments. CDOs are issued in several tranches which divide up the pool of debt into instruments with varying degrees of exposure to credit risk. One can buy different tranches so as to gain exposure to different levels of loss. As with all correlation products they can be dangerous, so trade small.

The aggregate loss is the sum of all losses due to default. As more and more companies default so the aggregate loss will increase. The tranches are specified by levels, as percentages of notional. For example, there may be the 0-3% tranche, and the 3-7% tranche, etc. As the aggregate loss increases past each of the 3%, 7%, etc., hurdles so the owner of that tranche will begin to receive compensation, at the same rate as the losses are piling up. You will only be compensated once your attachment point has been reached, and until the detachment point. The pricing of these contracts requires a model for the relationship between the defaults in each of the underlying instruments. A common approach is to use copulas. However, because of the potentially large number of parameters needed to represent the relationship between underlyings, the correlations, it is also common to make simplifying assumptions. Such simplifications might be to assume a single common random factor representing default, and a single parameter representing all correlations.