Derivatives Pricing - Complete Frictionless Markets

Overview


The basic insight that leads to derivative pricing in a perfect market follows directly from the law of one price. Essentially, the reasoning is that if one can find a portfolio of assets (with well establised market prices) that has the same payoff as the derivative in question, then the price of the derivative must have the same price as the portfolio.

Note that the replicating portfolio (replicating in the sense that it replicates the derivative payout) need not have a fixed set of assets. The portfolio may require active trading in order to maintain the link to the derivative. That is, it is probably more accurate to say that if one can find a portfolio of assets with a well defined, zero cost, trading strategy that replicates the derivative payoff, then by the law of one price, the cost of the derivate should equal the initial cost to set up the portfolio.

Assumptions


  • Statistical Assumptions The first assumption essentially grounds the modeling within a statistical framework. That is, it assumes that asset prices can be modeled using the axioms of probability.
    • Complete Market - Complete markets refers to a model of the market where there is a price for every asset in every possible state of the world.
  • Frictionless Market - no transaction costs when trading, and no bid-ask spread.
  • No Arbitrage The market does not allow arbitrage. Arbitrage exists when a portfolio can be constructed with zero cost which has a positive probability of a profit and a zero probability of loss. That is, one has the possibility of making money without having to put up any money or take any risk.

    The no arbitrage condition is then used to establish the law of one price.

Model Structures


The axioms of probability require that possible states of the world are modeled as elements in the set of all possible states. Events are given as a set of subsets of the set of all possible world sates, and there is an assignment of probabilities to each event.

When modeling derivatives then, one must create a model of the asset prices that the derivative depends upon. There are two popular methods for modeling asset prices which generally map to whether you model asset prices in a discrete framework or a continuous framework.

  • Tree Structures: market prices (or interest rates) are and a continuous time framework that models prices in a continuous framework
  • Stochastic Calculus : asset prices (or interest rates) are structured as a continuous process and can take on a continuum of values, making them amenable to calculus techniques.
  • Arrow Debreu Pricing

Topics


  • Black Scholes - describes the Black Scholes formula, the celebrated formula for pricing plain vanilla options.
  • Derivative Risk Greeks - describes the standard sensitivty based methods used to measure and manage risks associated with financial derivatives.
  • Risk Neutral Pricing is a powerful and simple framework that encompasses the above frameworks.

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