Introduction

Introduction to Pricing and Hedging of Derivative Securities. By Lars Tyge Nielsen. Textbook in continuous-time finance theory. Oxford University Press, 1999.

The principle behind the modern theory of pricing and hedging of derivatives or contingent claims is as follows. The future payoff to a claim, such as an option on a stock, will in general depend on the price paths of one or more basic securities. Using those basic securities, one can try to construct a trading strategy which replicates the payoff in the sense that in every possible future scenario, its value at the time of maturity of the claim will equal the payoff to the claim. The trading strategy should be self-financing and satisfy an admissibility condition which is imposed in order to rule out the possibility of arbitrage. If such a replicating trading strategy can be found, then the current price of the claim will be equal to the initial amount of money needed to start off the trading strategy.

It turns out that if the claim can be replicated like this, then its
discounted value is a martingale under the so-called risk adjusted probabilities. Hence, its present value can be calculated as the conditional expectation of the future payoff discounted back to the present. This valuation procedure is called the martingale method or the martingale valuation principle. In many cases, the conditional expectation can be calculated fairly explicitly, because we know the probability distributions that are involved. For example, the payoff is often a function of a random variable which is known to be normally distributed under the risk adjusted probabilities.

A variant of the martingale method uses the so-called state prices or
pricing kernel. If the claim can be replicated, then its value multiplied
by the state prices is a martingale, not under the risk-adjusted but under the original probabilities. Again, the claim can be valued by calculating a conditional expectation.

The surprising and powerful Complete Markets Theorem says that in a
wide range of situations, every contingent claim can be replicated by a
trading strategy, and therefore it can be priced by the martingale method.

Once the value of the claim has been calculated by the martingale method, it remains to find the replicating trading strategy. Usually, the value will be expressed as a function of the current prices of the basic securities. If so, then this function carries the information necessary to construct the replicating trading strategy. The derivatives of the function with respect to the basic securities prices are called the deltas of the claim, and they tell us how many shares of each security to hold in the replicating trading strategy.

To flesh out this story, we need to model the securities prices as Ito
processes, define what it means for a trading strategy to be self-financing, and introduce the concepts of state price processes and risk adjusted probabilities. All of this requires stochastic processes and Ito calculus, and that is where we shall begin.