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Exotic options lead to partial differential equations that are not of the simple structure of the basic Black–Scholes equation (4.1). In the general case, the transformations (4.3) are no longer useful and the PDEs must be solved directly. Thereby numerical instabilities or spurious solutions may occur, which do not play any role for the methods of Chapter 4. To cope with the “new” difficulties, Chapter...
Simulation and valuation of finance instruments require numbers with specified 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,...
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 locally a finer grid such that the difference quotients provide good approximations of the differentials. On the other hand, a flat gradient may be...
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 and 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...
What do we mean by option? An option is the right (but not the obligation) tobuy 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...
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