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Finance: a Quantitative Introduction Chapter 7 - Part 2 Option Pricing Foundations The setting Complete Markets Arbitrage free markets Risk neutral valuation Finance: A Quantitative Introduction Chapter 7 - part 2 Option Pricing Foundations Nico van der Wijst 1 Finance: A Quantitative Introduction c Cambridge University Press The setting Complete Markets Arbitrage free markets Risk neutral valuation 1 The setting 2 Complete Markets 3 Arbitrage free markets 4 Risk neutral valuation 2 Finance: A Quantitative Introduction c Cambridge University Press The setting Complete Markets Arbitrage free markets Risk neutral valuation Recall general valuation formula for investments: t X Exp [Cash flowst ] Value = t (1 + discount ratet ) 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 3 Finance: A Quantitative Introduction c Cambridge University Press The setting Complete Markets Arbitrage free markets Risk neutral valuation Introduce pricing principles in state preference theory old, tested modelling framework excellent framework to show completeness and arbitrage more general than binomial option pricing Also introduce some more general concepts equivalent martingale measure state prices, pricing kernel, few more Gives you easy entry to literature (+ pinch of matrix algebra, just for fun, can easily be omitted) 4 Finance: A Quantitative Introduction c Cambridge University Press The setting Time and states Complete Markets An example Arbitrage free markets Present values Risk neutral valuation 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' 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. 5 Finance: A Quantitative Introduction c Cambridge University Press The setting Time and states Complete Markets An example Arbitrage free markets Present values Risk neutral valuation The states of the world can be defined in different ways Examples: general economic circumstances: 'recession' with return on stock portfolio of -5% 'expansion' with return on stock portfolio of +16% result of a specific action, such as drilling for oil: large well medium-sized well dry well each with a cash flow attached to it 6 Finance: A Quantitative Introduction c Cambridge University Press The setting Time and states Complete Markets An example Arbitrage free markets Present values Risk neutral valuation Elaborate a simple example with: 1 period - 2 points in time 3 future states of the world: 2 3 2 3 w1 bust W = 4 w2 5 = 4 normal 5 w3 boom The states have a given probability of occurring: 2 0:3 3 prob(wi ) = 4 0:45 5 0:25 7 Finance: A Quantitative Introduction c Cambridge University Press The setting Time and states Complete Markets An example Arbitrage free markets Present values Risk neutral valuation There are 3 investment opportunities, Y1; Y2; Y3 Y1 pays off: 4 in state 1 5 in state 2 6 in state 3, or in matrix notation: 2 4 3 Y1(W ) = 4 5 5 6 For simplicity, the addition (W ) will be omitted Payoffs of all investments in different states in payoff matrix Ψ: 2 4 1 2 3 Ψ = 4 5 7 4 5 6 10 16 8 Finance: A Quantitative Introduction c Cambridge University Press The setting Time and states Complete Markets An example Arbitrage free markets Present values Risk neutral valuation Present value of investments Y found by: calculating expected value of payoffs discounting them with an appropriate rate The expected payoffs are e.g.: E(Y1) = 0:3 × 4 + 0:45 × 5 + 0:25 × 6 = 4: 95 or in matrix notation probT Ψ: 2 0:3 3T 2 4 1 2 3 4 0:45 5 4 5 7 4 5 = 4:95 5:95 6:40 0:25 6 10 16 9 Finance: A Quantitative Introduction c Cambridge University Press The setting Time and states Complete Markets An example Arbitrage free markets Present values Risk neutral valuation Let the required returns on the investments be: 10% 13% 16% Gives following present values of Y1; Y2; Y3: (e.g. 4:95=1:1 = 4:5; etc.) v = 4:5 5:25 5:5 Don't look at prices or returns as such, but: at mutual relations between them what these relations mean for capital market 10 Finance: A Quantitative Introduction c Cambridge University Press The setting Risk free and state securities Complete Markets Definition of completeness Arbitrage free markets Geometric representation of completeness Risk neutral valuation Risk free and state securities On perfect capital markets (assumed here): 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. 11 Finance: A Quantitative Introduction c Cambridge University Press The setting Risk free and state securities Complete Markets Definition of completeness Arbitrage free markets Geometric representation of completeness Risk neutral valuation Risk free security created by: combining the investments Y1−3 in a portfolio choosing the portfolio weights xn such that payoffs are equal: 4x1 + 1x2 + 2x3 = 1 5x1 + 7x2 + 4x3 = 1 6x1 + 10x2 + 16x3 = 1 3 equations with 3 unknowns, system can be solved: x1 = 33=124 x2 = −5=124 x3 = −3=248 12 Finance: A Quantitative Introduction c Cambridge University Press The setting Risk free and state securities Complete Markets Definition of completeness Arbitrage free markets Geometric representation of completeness Risk neutral valuation These weights define: the riskless security but also the risk free interest rate: PV(investments) × weights = PV(riskless security) (4:5 × 33=124) + (5:25 × −5=124) + (5:5 × −3=248) = 0:9194: Given PV, and payoff of 1, risk free interest rate is 1 = 1:088 or 8; 8%: 0:9194 Notice: we use small and negative fractions of investments (short selling), use perfect market assumption 13 Finance: A Quantitative Introduction c Cambridge University Press The setting Risk free and state securities Complete Markets Definition of completeness Arbitrage free markets Geometric representation of completeness Risk neutral valuation Use same procedure to create a portfolio that: pays off 1 if state of the world 1 occurs zero in all other states: 4x10 + 1x20 + 2x30 = 1 5x10 + 7x20 + 4x30 = 0 6x10 + 10x20 + 16x30 = 0 System is solvable too: x1 = 9=31 x2 = −7=31 x3 = 1=31 Can repeat procedure, find portfolios that pay off 1 in state 2 or 3 14 Finance: A Quantitative Introduction c Cambridge University Press The setting Risk free and state securities Complete Markets Definition of completeness Arbitrage free markets Geometric representation of completeness Risk neutral valuation Use a little matrix algebra instead, to find matrix of weights X that satisfies : ΨX = I Ψ is the payoff matrix X is 3×3 matrix of 3 weights in 3 equations I is the identity matrix: 2 3 2 3 2 3 4 1 2 x1;1 x2;1 x3;1 1 0 0 4 5 7 4 5 4 x1;2 x2;2 x3;2 5 = 4 0 1 0 5 6 10 16 x1;3 x2;3 x3;3 0 0 1 15 Finance: A Quantitative Introduction c Cambridge University Press The setting Risk free and state securities Complete Markets Definition of completeness Arbitrage free markets Geometric representation of completeness Risk neutral valuation This system is solved by X = Ψ−1I = Ψ−1 i.e. by taking the inverse of the payoff matrix : 2 9 1 5 3 31 62 − 124 Ψ−1 = 6 − 7 13 − 3 7 4 31 62 124 5 1 17 23 31 − 124 248 These weights give 3 portfolios, each of which: pays off 1 in only one state of the world and zero in all other states 16 Finance: A Quantitative Introduction c Cambridge University Press The setting Risk free and state securities Complete Markets Definition of completeness Arbitrage free markets Geometric representation of completeness Risk neutral valuation Such securities are called: state securities, or pure securities, or primitive securities, or Arrow-Debreu securities Prices of state securities found by multiplying: the weights matrix with present value vector (of the existing securities): vΨ−1 = 0:298 0:419 0:202 Prices of state securities also known as state prices 17 Finance: A Quantitative Introduction c Cambridge University Press The setting Risk free and state securities Complete Markets Definition of completeness Arbitrage free markets Geometric representation of completeness Risk neutral valuation In matrix algebra: v = 4:5 5:25 5:5 and 2 4 1 2 3 Ψ = 4 5 7 4 5 6 10 16 so that the state prices are: 2 4 1 2 3−1 4:5 5:25 5:5 4 5 7 4 5 = 0:298 0:419 0:202 6 10 16 18 Finance: A Quantitative Introduction c Cambridge University Press The setting Risk free and state securities Complete Markets Definition of completeness Arbitrage free markets Geometric representation of completeness Risk neutral valuation Market completeness defined State securities allow construction of any payoff pattern simply as combination of state securities State securities could be constructed because: the existing securities span all states i.e. there are no states without a payoff. A market where that is the case is said to be complete It is complete because: no "new" securities can be constructed new = payoff patterns cannot be duplicated with existing securities 19 Finance: A Quantitative Introduction c Cambridge University Press The setting Risk free and state securities Complete Markets Definition of completeness Arbitrage free markets Geometric representation of completeness Risk neutral valuation Can also be stated the other way around: If state securities can be constructed for all states, the market has to be complete.
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