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Derivative Securities (Subject) / Complete Markets (Lesson)
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- Recall general valuation formula for investments: Value = ∑ Exp(Cash Flow) / (1+discount rate)T
- Uncertainty can be accounted for in 3 different ways: 1. Adjust discount rate to risk adjusted discount rate 2. Adjust cash flows to certainty equivalent cash flows 3. Adjust probabilities (expectations operator) from normal to risk neutral or equivalent martingale probabilities
- Introduce pricing principles in state preference theory I old, tested modelling framework I excellent framework to show completeness and arbitrage more general than binomial option pricing
- State-preference theory Developed in 1950’s and 1960’s by Nobel prize winners Arrow and Debreu: Time modelled as discrete points in time at which: uncertainty over the previous period is resolved new decisions are made in periods between points ’nothing happens’ 1.Uncertainty in variables modelled as: discrete number of states of the world can occur on the future points in time each state associated with different numerical value of variables under consideration 2. The states of the world can be defined in different ways general economic corcumstances (recession, expansion) result of a specific action, e.g. drilling for oil
- Risk free and state securities On perfect capital markets: investments are costlessly and infinitely divisible means they can be combined in all possible ways to create the payoff pattern we want An obvious candidate for a wanted pattern: the same payoff in all states of the world i.e. creating a riskless security.
- Calculate the State securities: Given: State W Π, Investment Yi in Matrix ¥ Π, Probability wi Π Search: Search for the Present Value to get Price Calculate E(¥) = prob wT x ¥ Calculate present value PV(¥) = E(¥) / required return ri (Create the risk-free security and weights xi) + maximize ( all equal 1) PV(riskless security) = PV(¥) x xi) Risk-free rate = 1/ PV(riskless security) State securities: PV(¥) x (¥)-1
- Definition state securities: state securities, or pure securities, or primitive securities Arrow-Debreu securities These weights give 3 portfolios, each of which: pays off 1 in only one state of the world and zero in all other states
- Market completeness defined State securities allow construction of any payoff pattern I simply as combination of state securities I State securities could be constructed because: I the existing securities span all states I i.e. there are no states without a payo↵. A market where that is the case is said to be complete It is complete because: I no ”new” securities can be constructed I new = payo↵ patterns cannot be duplicated with existing securities
- A market is complete because: It is complete because no ”new” securities can be constructed (new = payoff patterns cannot be duplicated with existing securities) If state securities can be constructed for all states, the market has to be complete.
- On complete markets all additional securities are linear combinations of original ones additional securities are called redundant securities in examples so far risk free security and state securities are redundant they are formed as combinations of the existing securities A market can only be complete if: number of different (i.e. not redundant) securities = number of states
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- Completeness can be represented geometrically: A and B linearly independent: combinations of A and B span whole 2-dimensional space a+b b-a a-b a -a b -b
- Arbitrage free markets Complete markets imply: any payoff pattern can be constructed Arbitrage free markets imply: patterns are properly priced What is properly priced? - Proper prices offer no arbitrage opportunities
- When do arbitrage opportunities exist? arbitrage opportunities exist if there is investment strategy that either requires investment ≤ 0 today, while all future payoffs ≥ 0 and at least one payoff > 0 or requires investment < 0 today (=profit) and all future payoffs ≥ 0 Less formally: either costs nothing today + payoff later payoff today without obligations later Absence of arbitrage implies a characteristic of state prices : State prices have to be positive Negative state price would mean: buying state security with negative price = receive money and possibly (if state occurs), a payoff of 1 later. A negative net investment now + a non-negative profit later