Definitions

The underlying asset determines the payoff structure of a derivative contract. A number of different assets serve as underlying assets, including

• Financial assets, e.g. Shares, Government Bonds
• Commodities, e.g. Gold, Oil, Cattle
• Another derivative, e.g. Options on Futures
• An index, e.g. Options on the SP500, or the FTSE
• Interest rates, e.g. Swaps on the LIBOR rate
• Other weird stuff, e.g. Weather, Election results, etc

Since the payoffs of any derivative depend on the future values of the underlying asset, a stochastic process that captures the salient features of the underlying is paramount to price the derivative, and for risk management purposes. For this reason, a large volume of research has been dedicated in exploring different specifications and their implications for derivative pricing and hedging.

Shares and stock market indices

Options on shares and in particular indices, such as the SP500 are very popular. Index derivatives are used widely to manage the market risk, which is otherwise difficult to hedge. Loosely speaking, models for share prices and indices can be categorized in two groups

• Models with discrete state space and time
• Models with continuous state space but discrete time
• Models with continuous state space and time

In the first group, the underlying asset is restricted in taking a discrete (albeit potentially large) number of values, and the time is assumed to change in discrete steps. Examples include the binomial and trinomial lattices. Lattices are very interesting and intuitive, and have a huge paedagogical significance. In addition, other families of models use tree structures as approximations, when implemented numerically.

In the second group, the asset is allowed to take values in a continuous set, while the time is still considered discrete. The Garch models are significant examples of this approach. Having a continuous support allows one to use the substantial theory on continuous distributions to model asset returns. The discrete time steps make such models easy to calibrate using time series of the underlying asset. Unfortunately, under these models the market is always incomlete, and for this reason they have not been widely used in practice.

The last group of models has attracted most attention, since the seminal work of Black, Merton and Scholes in the early 70s. Their celebrated formula is based on the assumption of a geometric Brownian motion as the model of asset prices. A number of studies have extended their methodology to account for the stylized facts of asset returns and the implied volatility smile.

Interest rates

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