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What do we mean by option? An option is the right (but not the obligation) to buy or sell a risky asset at a prespecified fixed price within a specified period. An option is a financial instrument that allows —amongst other things— to make a bet on rising or falling values of an underlying asset. The underlying asset typically is a stock, or a parcel of shares of a company. Other examples of underlyings include stock indices (as the Dow Jones Industrial Average), currencies, or commodities. Since the value of an option depends on the value of the underlying asset, options and other related financial instruments are called derivatives (→ Appendix A2). An option is a contract between two parties about trading the asset at a certain future time. One party is the writer , often a bank, who fixes the terms of the option contract and sells the option. The other party ist the holder , who purchases the option, paying the market price, which is called premium . How to calculate a fair value of the premium is a central theme of this book. The holder of the option must decide what to do with the rights the option contract grants. The decision will depend on the market situation, and on the type of option. There are numerous different types of options, which are not all of interest to this book. In Chapter 1 we concentrate on standard options, also known as vanilla options . This Section 1.1 introduces important terms.

Simulation and valuation of finance instruments require numbers with speci- fied distributions. For example, in Section 1.6 we have used numbers Z drawn from a standard normal distribution, Z ~ N (0, 1). If possible the numbers should be random. But the generation of “random numbers” by digital computers, after all, is done in a deterministic and entirely predictable way. If this point is to be stressed, one uses the term pseudo-random ^1.

This chapter provides an introduction into the numerical integration of stochastic differential equations (SDEs). Again X _ t denotes a stochastic process and solution of an SDE,

We now enter the part of the book that is devoted to the numerical solution of equations of the Black-Scholes type. Here we discuss “standard” options in the sense as introduced in Section 1.1. Accordingly, let us assume the scenario characterized by the Assumptions 1.2. In case of European options the function V ( S , t ) solves the Black-Scholes equation (1.2). It is not really our aim to solve this partial differential equation because it possesses an analytic solution (#x2192; Appendix A4). Ultimately it is our intention to solve more general equations and inequalities. In particular, American options will be calculated numerically. The goal is not only to calculate single values V ( S _0, 0) —for this purpose binomial methods can be applied— but also to approximate the curve V ( S , 0), or even the surface defined by V ( S , t ) on the half strip S > 0, 0 ≤ t ≤ T .

The finite-difference approach with equidistant grids is easy to understand and straightforward to implement. The resulting uniform rectangular grids are comfortable, but in many applications not flexible enough. Steep gradients of the solution require a finer grid such that the difference quotients provide good approximations of the differentials. On the other hand, a flat gradient may be well modeled on a coarse grid. Such a flexibility of the grid is hard to obtain with finite-difference methods.

In Chapter 4 we discussed the pricing of vanilla options (standard options) by means of finite differences. The methods were based on the simple partial differential equation (4.2),