Eco 525 Financial Economics I Lecture 01 One Period Model Prof Markus K Brunnermeier 21 57 Lecture 01 One Period Model Slide 1 1 Eco 525 Financial Economics I Overview 1 Securities Structure Arrow Debreu securities structure Redundant securities Market completeness Completing markets with options 2 Pricing no arbitrage state prices SDF EMM 3 Optimization and Representative Agent Pareto efficiency Welfare Theorems 21 57 Lecture 01 One Period Model Slide 1 2 Eco 525 Financial Economics I The Economy s 1 State space Evolution of states Two dates t 0 1 S states of the world at time t 1 Preferences U c0 c1 cS 0 s 2 s S slope of indifference curve Security structure Arrow Debreu economy General security structure 21 57 Lecture 01 One Period Model Slide 1 3 Eco 525 Financial Economics I Security Structure Security j is represented by a payoff vector Security structure is represented by payoff matrix NB Most other books use the transpose of X as payoff matrix 21 57 Lecture 01 One Period Model Slide 1 4 Eco 525 Financial Economics I Arrow Debreu Security Structure in R2 One A D asset e1 1 0 c2 This payoff cannot be replicated Payoff Space X c1 Markets are incomplete 21 57 Lecture 01 One Period Model Slide 1 5 Eco 525 Financial Economics I Arrow Debreu Security Structure in R2 Add second A D asset e2 0 1 to e1 1 0 c2 c1 21 57 Lecture 01 One Period Model Slide 1 6 Eco 525 Financial Economics I Arrow Debreu Security Structure in R2 Add second A D asset e2 0 1 to e1 1 0 c2 Payoff space X c1 Any payoff can be replicated with two A D securities 21 57 Lecture 01 One Period Model Slide 1 7 Eco 525 Financial Economics I Arrow Debreu Security Structure in R2 Add second asset 1 2 to c2 Payoff space X c1 New asset is redundant it does not enlarge the payoff space 21 57 Lecture 01 One Period Model Slide 1 8 Eco 525 Financial Economics I Arrow Debreu Security Structure S Arrow Debreu securities each state s can be insured individually All payoffs are linearly independent Rank of X S Markets are complete 21 57 Lecture 01 One Period Model Slide 1 9 Eco 525 Financial Economics I General Security Structure Only bond Payoff space X c2 c1 21 57 Lecture 01 One Period Model Slide 1 10 Eco 525 Financial Economics I General Security Structure Only bond xbond 1 1 Payoff space X can t be reached c2 c1 21 57 Lecture 01 One Period Model Slide 1 11 Eco 525 Financial Economics I General Security Structure Add security 2 1 to bond 1 1 c2 c1 21 57 Lecture 01 One Period Model Slide 1 12 Eco 525 Financial Economics I General Security Structure Add security 2 1 to bond 1 1 c2 Portfolio of buy 3 bonds sell short 1 risky asset c1 21 57 Lecture 01 One Period Model Slide 1 13 Eco 525 Financial Economics I General Security Structure c2 Payoff space X c1 Two assets span the payoff space Market are complete with security structure Payoff space coincides with payoff space of 21 57 Lecture 01 One Period Model Slide 1 14 Eco 525 Financial Economics I General Security Structure Portfolio vector h RJ quantity for each asset Payoff of Portfolio h is j hj xj h X Asset span X is a linear subspace of RS Complete markets X RS Complete markets if and only if rank X S Incomplete markets rank X S Security j is redundant if xj h X with hj 0 21 57 Lecture 01 One Period Model Slide 1 15 Eco 525 Financial Economics I Introducing derivatives Securities property rights contracts Payoffs of derivatives derive from payoff of underlying securities Examples forwards futures call put options Question Are derivatives necessarily redundant assets 21 57 Lecture 01 One Period Model Slide 1 16 Eco 525 Financial Economics I Forward contracts Definition A binding agreement obligation to buy sell an underlying asset in the future at a price set today Futures contracts are same as forwards in principle except for some institutional and pricing differences A forward contract specifies The features and quantity of the asset to be delivered The delivery logistics such as time date and place The price the buyer will pay at the time of delivery Expiration date Today 21 57 Lecture 01 One Period Model Slide 1 17 Eco 525 Financial Economics I Settlement price last transaction of the day Low of the day High of the day The open price Reading price quotes Index futures Daily change Lifetime low Open interest Expiration month 21 57 Lecture 01 Lifetime high One Period Model Slide 1 18 Eco 525 Financial Economics I Payoff diagram for forwards Long and short forward positions on the S R 500 index 21 57 Lecture 01 One Period Model Slide 1 19 Eco 525 Financial Economics I Forward vs outright purchase Forward payoff Bond payoff Forward bond Spot price at expiration 1 020 1 020 Spot price at expiration 21 57 Lecture 01 One Period Model Slide 1 20 Eco 525 Financial Economics I Additional considerations ignored Type of settlement Cash settlement less costly and more practical Physical delivery often avoided due to significant costs Credit risk of the counter party Major issue for over the counter contracts Credit check collateral bank letter of credit Less severe for exchange traded contracts Exchange guarantees transactions requires collateral 21 57 Lecture 01 One Period Model Slide 1 21 Eco 525 Financial Economics I Call options A non binding agreement right but not an obligation to buy an asset in the future at a price set today Preserves the upside potential while at the same time eliminating the unpleasant downside for the buyer The seller of a call option is obligated to deliver if asked Expiration date Today or at buyer s choosing 21 57 Lecture 01 One Period Model Slide 1 22 Eco 525 Financial Economics I Definition and Terminology A call option gives the owner the right but not the obligation to buy the underlying asset at a predetermined price during a predetermined time period Strike or exercise price The amount paid by the option buyer for the asset if he she decides to exercise Exercise The act of paying the strike price to buy the asset Expiration The date by which the option must be exercised or become worthless Exercise style Specifies when the option can be exercised European style can be exercised only at expiration date American style can be exercised at any time before expiration Bermudan style can be exercised during specified periods 21 57 Lecture 01 One Period Model Slide 1 23 Eco 525 Financial Economics I Reading price quotes S P500 Index options Strike price 21 57 Lecture 01 One Period Model Slide 1 24 Eco 525 Financial Economics I Diagrams for purchased