TABLE OF CONTENTS The key concept in the theory of pricing derivative claims ( i.e. contingent claims) is that of arbitrage, i.e. the opportunity of making a profit with no possibility of incurring a loss. This is also commonly referred to as free lunch. In an ideal market, such possibility should not exist, for otherwise it would lead to violation of the conservation wealth, for example, if all investors use such possibility, then everybody gets rich and nobody suffers.
After the brief introduction 3.1, we consider in 3.2 some formulation of the absence of arbitrage for the hyperfinite time line. There are various ways of doing this, since the concept of infinitely close is involved. We consider one that compared with the riskless interest rate. Under this assumption, the constraints of the price movements of a contingent claim can be established.
In 3.3, we will see how Black-Scholes type equations arise from such formulation of arbitrage free.
In reality, arbitrage possibility may exist for a very brief time before equilibrium takes effect and eliminates such possibility. More realistically, the problem is an subjective one: one needs to formulate risk precisely in order to judge whether there is arbitrage opportunity. For example, in every financial model, the variance of a stock and the so-called riskless interest rate are at best a guess work and hence the difficulty of translating the apparent discrepancy into actual arbitrage.
The fundamental assumption behind most mathematical models of finance...
