ONE PERIOD MODELS t = TIME = 0 or 1

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1 BASIC INSTRUMENTS: * S t : STOCK * B t : RISKLESS BOND ONE PERIOD MODELS t = TIME = 0 or 1 B 0 = 1 B 1 = 1 + r or e r * FORWARD CONTRACT: AGREEMENT TO SWAP $$ FOR STOCK - AGREEMENT TIME: t = 0 - AGREEMENT PRICE: F 0 - SWAP TIME t = 1 - SWAP: $$ F 0 FOR 1 STOCK S 1 ACTUAL INSTRUMENT: W t : W 0 = 0 W 1 = S 1 F 0 Autumn Per A. Mykland

2 ABITRAGE ARGUMENT: F 0 = e r S 0 If F 0 < e r S 0 : FORM PORTFOLIO AT t = 0: NET POSITION SELL 1 STOCK BUY S 0 # BONDS ENTER 1 FORWARD CONTR TOTAL PORTFOLIO S t S 0 B t W t V t = S t + S 0 B t + W t VALUE: V 0 = S 0 + S = 0 V 1 = S 1 + e r S 0 + S 1 F 0 = e r S 0 F 0 > 0 V t IS AN ARBITRAGE: MONEY FOR NOTHING NOT SUPPOSED TO OCCUR Autumn Per A. Mykland

3 MORE GENERALLY... * K + 1 ASSETS, PRICE S j t, j = 1,...,K S 0 = B IS BOND: S 0 0 = 1, S 0 1 = e r * PORTFOLIO = ( 0,..., K ): HOLD j # OF SECURITY S j t * VALUE OF PORTFOLIO: V t ( ) = K j S j t j=0 * ARBITRAGE: A PORTFOLIO FOR WHICH V 0 ( ) 0 V 1 ( ) 0 V 1 ( ) > 0 IN SOME SCENARIO (= WITH PROBABILITY > 0) Autumn Per A. Mykland

4 TWO STATE MODEL FOR STOCK S 0.. S 1 = us 0 S 1 = ds 0 SCENARIO H: S 1 (H) SCENARIO T: S 1 (T) Conditions to avoid arbitrage Consider portfolio: buy 1 B 0 # of bonds, 1 S 0 # of stocks at time 0. Properites: V 0 = 1 B 0 B 0 1 S 0 S 0 = 0 V 1 = 1 B 0 B 1 1 S 0 S 1 = { e r u under scenario H e r d under scenario T If e r u: V 1 0 under H V 1 > 0 under T ARBITRAGE NOT ALLOWED It follows that u > e r (unless P(T)=0: e r = u OK) Similarly: portfolio 1 B 0 bonds, 1 S 0 stocks => e r > d CONCLUSION: no arbitrage implies u > e r > d. Autumn Per A. Mykland

5 S 0 CALL OPTIONS V 1 = (S 1 K) + V 0 =???.. S 1 = us 0 S 1 = ds 0 SCENARIO H SCENARIO T Suppose us 0 > K > ds 0 otherwise V 1 = S 1 K (ds 0 K) or V 1 = 0 (us 0 K) REPLICATING PORTFOLIO Buy 0 bonds and 1 stocks at time 0 portfolio: V t ( ) = 0 B t + 1 S t replication: (S 1 K) + = 0 B S 1 FINDING THE s: 2 EQUATIONS, 2 UNKNOWNS: us 0 K = 0 e r + 1 us 0 SCENARIO H 0 = 0 e r + 1 ds 0 SCENARIO T OR: 1 = us 0 K us 0 ds 0 and 0 = e r 1 ds 0 PRICE FOR THIS OPTION: V 0 = 0 B S 0 = 1 e r ( ds 0 + e r S 0 ) Autumn Per A. Mykland

6 ARGUMENT DEPENDS ON bond, stock can be bought or sold in any quantity bond, stock can be short sold (in the case of bond: this means that borrowing rate is same as lending rate) no bid-ask spread binomial model Binomial model is oversimplification Brownian motion is close to binomial model Increasingly realistic models as the course progresses ARGUMENT DOES NOT DEPEND ON Assumption of no arbitrage, except that u > e r > d Autumn Per A. Mykland

7 MORE GENERAL DERIVATIVE SECURITIES IN THE ONE PERIOD BINOMIAL MODEL payoff V 1 (H) or V 1 (T) or V 1 = f(s 1 ) where f(s) = { (s K) + call option (K s) + put option etc REPLICATING PORTFOLIO Buy 0 bonds and 1 stocks at time 0 portfolio: V t ( ) = 0 B t + 1 S t replication: f(s 1 ) = 0 B S 1 FINDING THE s: 2 EQUATIONS, 2 UNKNOWNS: f(us 0 ) = 0 e r + 1 us 0 SCENARIO H f(ds 0 ) = 0 e r + 1 ds 0 SCENARIO T OR: 1 = f(us 0) f(ds 0 ) us 0 ds 0 and 0 = e r uf(ds 0 ) df(us 0 ) u d PRICE FOR THIS OPTION: V 0 = 0 B S 0 Autumn Per A. Mykland

8 DISCOUNTING Discounted stock: S t = S t /B t Discounted bond: B t = B t /B t = 1 Discounted portfolio value: V t = V t /B t NUMERAIRE INVARIANCE Portfolio in original numeraire: V t ( ) = 0 B t + 1 S t Portfolio in discounted numeraire: V t ( ) = 0B t + 1 S t B t = 0 B t + 1 S t The number 0, 1 of bonds, stocks is the same in original and discounted numeraire Exit interest. This is often convenient TWO EQUATIONS, TWO UNKNOWNS ON DISCOUNTED SCALE V 1 (H) = us 0 SCENARIO H V 1 (T) = ds 0 SCENARIO T Autumn Per A. Mykland

9 PROBABILISTIC INTERPRETATION Let π(h), π(t) be two numbers From V t = 0 B t + 1 S t : provided π(h)v 1 (H) + π(t)v 1 (T) =π(h) ( 0 B 1(H) + 1 S 1(H)) + π(t) ( 0 B 1(T) + 1 S 1(T)) = 0 (π(h)b 1(H) + π(t)b 1(T)) + 1 (π(h)s 1(H) + π(t)s 1(T)) ( ) = 0 B0 + 1 S0 = V0 { π(h)b 1 (H) + π(t)b1(t) = B0 ( ) π(h)s1(h) + π(t)s1(t) = S0 ( ) B t = 1: (**) <=> π(h) + π(t) = 1 π is a probability measure, provided π(h), π(t) 0 This is the case since, by solving (**)-(***): π(t) = u er u d and π(h) = er d u d If E π is expectation under π: (E π X = X(H)π(H) + X(T)π(T)) ( ) <=> E π V 1 = V 0 ( ) <=> E π S 1 = S 0 π IS A RISK NEUTRAL PROBABILITY MEASURE Autumn Per A. Mykland

10 A RISK NEUTRAL PROBABILITY DISTRIBUTION: π : S j 0 = e r E π S j 1 = E πs j 1 for all j FUNDAMENTAL THEOREM OF ARBITRAGE PRIC- ING: THERE EXISTS A RISK NEUTRAL MEASURE IF AND ONLY IF ARBITRAGE DOES NOT OC- CUR. EVALUATING PRICES USING π: K V 0 = j S j 0 j=1 K = e r j E π S j 1 j=1 = e r E π V 1 BUT WHAT IS π? BRUNO DE FINETTI (1937): FORESIGHT: ITS LOGICAL LAWS, ITS SUBJECTIVE SOURCES: PROBABILITY DOES NOT EXIST. Autumn Per A. Mykland

11 HORSE RACING (from Baxter and Rennie: Financial Calculus. An introduction to derivative pricing.) TWO HORSES: H1, H2 ACTUAL CHANCE BETS PLACED OF WINNING ON HORSE H1 25% $ 5000 H2 75% $10000 TOTAL FOR BOOKIE $15000 PRICE OF BETS ACTUAL PROBABILITIES: π 1 = 1 4 π 2 = 3 4 BETTING $1 ON HORSE H1 : WIN $4 BETTING $1 ON HORSE H2 : WIN $ 4 3 OUTCOME FOR BOOKIE: WINNING HORSE $$ H = 5000 H = 1666 A RISKY BUSINESS Autumn Per A. Mykland

12 HORSE RACING: RISK NEUTRAL PROBABILITY ACTUAL CHANCE OF WINNING BETS PLACED ON HORSE H1 IRRELEVANT $ 5000 H2 $10000 PRICE OF BETS: $ 1 ON H1 GIVES 1 π 1 $ IF WIN $ 1 ON H2 GIVES 1 π 2 $ IF WIN OUTCOME FOR BOOKIE: WINNER $$ H1 H π π OUTCOME = 0 IF π 1 = 1 3, π 2 = 2 3 NESS. A SAFE BUSI- JUST LIKE SELLING OPTIONS... Autumn Per A. Mykland

13 EQUIVALENT: COMPLETE MARKETS 1) ALL RANDOM VARIABLES V 1 CAN BE REP- RESENTED V 1 = 0 B S K S K 1 2) π IS UNIQUE IN CALL OPTION TWO STATE MODEL: DID NOT NEED TO VERIFY EXISTENCE OF PORTFO- LIO Autumn Per A. Mykland