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How to Implement Market Models Using VBA by François Goossens

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CHAPTER 4

Numerical Solutions

This chapter handles common numerical methods used in finance to tackle path-dependent contingent claims and multi-asset instruments. Most of them consist in transferring a continuous-time valuation issue into a discrete time one, which is more tractable. Here is a list of problems solved by discretization of time: American-style options, early termination issues, exotic contingent claims, basket options…

We will review successively:

  1. finite differences
  2. binomial and trinomial trees
  3. Monte-Carlo scenarios
  4. simulation and regression.

At the end of this chapter, we will develop a recipe to price double-barrier options as a series of single barriers: the benefit of this method is that it allows stochastic volatility models to be taken into account when pricing a double-barrier.

4.1 FINITE DIFFERENCES

4.1.1 Generic equation

The rationale is to find close solutions to differential equations, using finite differences to approximate derivatives. This method applies to contingent claims satisfying the following generic equation, in a complete market (in which every contingent claim can be perfectly hedged with liquid assets):

image

To justify this equation, let's consider the matter from the market-maker's point of view. In theory, this portfolio must be self-financing. Concretely, it means that the changes in the values of the contingent claim + the hedge must be ...

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