Financial Mathematics. Christel Geiss Department of Mathematics University of Innsbruck

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1 Financial Mathematics Christel Geiss Department of Mathematics University of Innsbruck September 11, 212

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3 Contents 1 Introduction Financial markets Types of financial contracts Example: the European call option Basics of Probability theory 13 1 Finite probability spaces CRR model 17 1 Filtration Martingales Non-arbitrage pricing 25 1 The market model Strategies Properties of the conditional expectation Admissible strategies and arbitrage Fundamental Theorem 37 1 Separation of convex sets in R N Martingale transforms The fundamental theorem of asset pricing Complete markets and option pricing American Options 51 1 Stopping Times The Snell Envelope Decomposition of Supermartingales

4 4 CONTENTS 4 Pricing and hedging of American options American options and European options Some stochastic calculus 63 1 Brownian motion Some properties of the Brownian motion Conditional expectation and martingales Itô s integral for simple integrands Itô s integral for general integrands Itô s formula Continuous time market models 75 1 The stock price process Trading strategies Risk neutral pricing 77 1 The Black-Scholes model 81 1 The equivalent martingale measure Q Pricing: The Black-Scholes formula Example of infinitely many EMM s Example of no EMM s Bonds Currency markets Credit risk 93 1 A simple model for credit risk

5 1. Introduction 1.1 Financial markets Financial markets are places where individuals and corporations can buy or sell financial securities and products. These markets are not only a possibility to purchase assets, they also are used for risk transfer. Already centuries ago financial contracts have been made. It is known that in the Antique Greece olives were sold by the farmers using forward contracts (i.e. quality and price was aggreed upon in advance). In the beginning of the 17th century the first official stock exchange was opened in Amsterdam. Especially tulips, originally from Turkey, became extremely popular among rich merchants. Traders purchased bulbs at higher and higher prices planning to re-sell them for profit. Due to the nature of (growing) tulips which only can be moved in a certain time of the year the concept of futures contracts was developped. But suddenly the interest in tulips decreased and prices fell rapidly ( tulip mania, speculative bubble ). In the United States the Chicago Board of Trade (CBOT) was created in 1848 as an exchange market for futures and options. 5

6 6 CHAPTER 1. INTRODUCTION 1.2 Types of financial contracts financial contracts primary securities derivative securities bonds stocks preferred stocks shares options futures a security is a piece of paper representing a promise bonds are certificates issued by a government or a public company promising to repay a fixed interest rate at a specified time a share (or stock) is a security representing partial ownership of a company and/or makes dividend payments according to the profits. Shares are traded on a stock exchange preferred stocks are entitled to a fixed dividend an asset (in Finance) is anything owned, whether in possession or by right to take possession, by a person or a company, the value of which can be expressed in monetary terms stock cash } current assets

7 1.2. TYPES OF FINANCIAL CONTRACTS 7 land building machinery goodwill copyrights education... fixed assets intangible assets a forward contract is an agreement between two parties to buy or sell an asset (which can be of any kind) at a pre-agreed future point in time. a futures contract is a forward contract the has been standardized: - amount to be traded: for example a fixed number of barrels of oil currency (US dollar often) - quality - delivery month - last trading date Futures contracts are traded on a futures exchange an option gives the holder of the option the right to buy (or sell) a security (shares) at a predefined time (or timeperiod) in the future and for a pre-determined amount. Types of options: - stock options - foreign exchange options - interest rate options (=largest derivatives market in the world) - warrants - options on bonds - swaptions -... long position someone agrees to buy the asset short position someone agrees to sell the asset One purpose of derivatives is as a form of insurance to move risk from someone who cannot afford a major loss to someone who could absorb the loss, or is able to hedge against the risk by buying some other derivatives.

8 8 CHAPTER 1. INTRODUCTION The central topic of Financial Mathematics is the fair valuation of derivatives. One key equation used to value derivatives is the Black-Scholes-Equation (published 1973). Fischer Black and Myron Scholes received the Nobel Prize in Economics for this. After 1973 trading with options increased rapidly. 1.3 Example: the European call option Someone buys at time a European call option. Then he can (but does not have to) buy a given number of shares (1 share here) for a fixed price K (= the so called strike price ) at a fixed time T. shareprice K S T T time

9 1.3. EXAMPLE: THE EUROPEAN CALL OPTION 9 If S T > K If S T K then he will buy the shares for the price K and if he sells them immediately his gain = S T K price of the option he will not buy and his loss = price of the option Question: How to determine a fair price for an option? 1. If the price would be : then the option holder (= the one who bought the option) could make a riskless profit: this is against the rules of the market 2. if the option price is too high and if there is no sign that the share price S T will be much higher than the strike price K, nobody will buy this option. Summary: European call-option; C :=option price The gain (outcome) of the option holder (= buyer) { ST K C if S T > K C if S T K writer (=seller of the option) { K ST + C if S T > K C if S T K

10 1 CHAPTER 1. INTRODUCTION seller s gain writer s gain payoff (S T K) + K buyer s gain (S T K) + C S T writers gain K payoff (S T K) + S T The purpose of a European call option: 1. The writer reduces the risk in case S t will go down: he gets C. 2. The buyer hopes that S T > K + C and takes the risk that S t will go down. In this case he loses the price C of the option. purpose: it is a form of insurance (for the writer) A fair price of an European call option f(s T ) = (S T K) + (Example) A fair price of f(s T ) would be a price where both the writer and the buyer could not make riskless profit. We consider the following example: Assume 2 trading dates: and T. at time share { price S = 2 $ 2 $ with probability p at time T S T = 7.5 $ with probability 1 p ( < p < 1). Let the strike price be K = 15 (dollar). { 5 $ if ST = 2 the option writer has to pay nothing if S T = 7.5

11 1.3. EXAMPLE: THE EUROPEAN CALL OPTION 11 We can do the following: hedging (=counterbalancing action to protect oneself from losing). Let us assume here for simplicity that the interest rate r=. That means one can borrow from the bank without paying interest. The writer sells the option, so he gets C. He borrows (-ϕ ) dollar from the bank and can buy ϕ 1 = C ϕ (S = 1) 1 shares at time. The portfolio (ϕ,ϕ 1 ) is correctly chosen if We get and Then ϕ 1 + ϕ 1 2 = 5 ϕ 1 + ϕ = } ϕ = 7.5ϕ ϕ 1 = 5 ϕ 1 = 5 = ϕ = = 3. C = 1ϕ 1 + ϕ = 4 3 = 1 is the fair price for the option. Hence at the time the writer gets 1 dollar for the option and borrows 3 dollars from the bank. With these 4 dollars he can buy.4 shares. Case 1: S T = 2. The option is exercised at a cost of 5$. The writer repays the loan (cost 3$) and sells the shares (gain.4 2 = 8). Balance of trade: =. Case 2: S T = 7.5 The option is not exercised (cost = ). The writer repays the loan (cost 3$) and sells the shares (gain = 3) Balance of trade: 3-3=. If C > 1, then the writer can make (by hedging like above) the riskless profit C 1. If C < 1 the option holder can make a riskless profit by the following procedure: buy the option (cost C ), sell.4 shares (gain: 4) and put 4 C to the bank account. Then, at time T

12 12 CHAPTER 1. INTRODUCTION { 4 C = 1 C for S T = 2 4 C = 1 C for S T = 7.5 Where the 5 is the payoff of the option and 1 C is the riskless profit. Summary In this example we did 1. find the hedging portfolio (ϕ, ϕ 1 ) by solving the equation ϕ + ϕs T = f(s T ) ( here f(s T ) = (S t K) + ). 2. We calculated the fair price namely how much money a trader would need at time to have the amount f(s T ) at time T: Remark fair price = ϕ + ϕ 1 S. One can compute the fair price of an option also by using a martingalemeasure. For this we introduce probability theory.

13 2. Basics of Probability theory 1 Finite probability spaces Definition 2.1. Let Ω = {ω 1,, ω N } be a finite set. Assume p i >, i = 1,, N such that N p i = 1. Then P is a probability measure: For A Ω we set P(A) := ω i A P({ω i}). Example 2.2. Rolling a die A := rolling an odd number P(A) =? Ω = {1, 2, 3, 4, 5, 6} P({ω}) = 1 6, ω Ω. It follows from the definition that P(Ω) = N P({ω i }) = N p i = 1. P( ) = ω i P({ω i }) =. We define F := 2 Ω be the power set of Ω = the set of all subsets of Ω. 13

14 14 CHAPTER 2. BASICS OF PROBABILITY THEORY Example 2.3. For Ω = {1, 2} we have 2 Ω = {{1, 2}, {1}, {2}, }. The power set 2 Ω has 2 #Ω elements. Definition 2.4. [ σ - field, σ - algebra ] Let Ω be a non-empty set. A system F of subsets A Ω is a σ-field or σ-algebra on Ω if 1., Ω F, 2. A F A C := Ω\A F, 3. A 1, A 2,... F A i F. Remark 2.5. If Ω is finite, it is enough to check in (3) that A 1, A 2 F implies A 1 A 2 F. Examples 1. 2 Ω is a σ-field. 2. Let Ω be a set and assume A 1,..., A M is a finite partition of Ω i.e. A 1,..., A M are mutually disjoint: A i A j =, i j and M A i = Ω. Then F = { i=j } A i : J {1,..., M} = {, A 1,..., A M, A 1 A 2, A 1 A 3,..., Ω} is a σ-field. We say F is generated by A 1,..., A N and use the notation F := σ(a 1,...A N ). A finite probability space can be thought of in two ways:

15 1. FINITE PROBABILITY SPACES 15 Ω = {ω 1,..., ω N } Ω is finite F = 2 Ω, P({ω i }) = p i >, P(Ω) = 1 Finite probability space (Ω, F, P) Ω arbitrary F = σ(a 1,...A N ) is a σ-field of a finite partition, P(A i ) > i = 1,..., N, A F A = k J A k P(A) = k J P(A k)

16 16 CHAPTER 2. BASICS OF PROBABILITY THEORY

17 3. The Cox-Ross-Rubinstein model (CRR-model, binomial tree model) We want to model the time-development of shares and bonds with a simple model: Assume T = {, 1,..., T } are trading dates (T = trading horizon). S = (S, S1,..., ST ) is a riskless bond (or bank account). S 1 := S = (S 1,..., S T ) is a risky (i.e. random) stock. We assume a constant interest rate r >, i.e. if S = 1, then S 1 = 1 + r, S k = (1 + r)k, k =, 1,..., T. S t riskless bond for r= t 17

18 18 CHAPTER 3. CRR MODEL The random behavior of the stock S will be modeled as follows: < p < 1 fixed. { Sn (1 + a) with probability 1 p S n+1 = S n (1 + b) with probability p If we choose 1 < a < b then Ω := {ω = (ɛ 1,..., ɛ T ) : ɛ i {1 + a, 1 + b}} S t (ω) = S ɛ 1, ɛ 2,..., ɛ t, t T Hence each ω Ω corresponds to one possible case of a stock development. We can also compute the probability of each case: where P ( {(ɛ 1,..., ɛ T )} ) = p k (1 p) T k k := #{i : ɛ i = 1 + b}. is the binomial distribution. The defined P is clearly a probability measure on Ω: P({ω}) > ω Ω. We have to check that P(Ω) = = = P(Ω) = 1. ɛ i {1+a,1+b};,...,T T k= T k= P ( {(ɛ 1,..., ɛ T )} ) ω with k = #{i, ɛ i = 1 + b} ( ) T p k (1 p) T k k = ( p + (1 p) ) T = 1. p k (1 p) T k

19 1. FILTRATION 19 Example 3.1. T = {, 1, 2, 3} ω = (1 + b, 1 + b, 1 + b) S S 3 (ω) p p (1 + b)s p (1 + a)(1 + b)s 1 p (1 + a)s (1 + a) 3 S 1 Filtration The investor does not know at time how the values of S t, t = 1,..., T will be. At time t > he knows all about S, S 1,...S t but nothing about S t+1,..., S T.

20 2 CHAPTER 3. CRR MODEL We model the situation using a filtration. Definition 3.2. A filtration is an increasing sequence of σ-fields: {, Ω} = F F 1... F T Definition 3.3. Assume f : Ω {m 1,...m N }, and G is a σ-field on Ω. Then f is G-measurable f 1 ({m i }) = {ω Ω : f(ω) = m i } G m i If we have functions f 1, f 2,..., f l : Ω {m 1,..., m N } then G = σ(f 1,...f l ) denotes the smallest σ-field, such that all functions f 1,..., f l are G-measurable. Example 3.4. CRR model: We assume F t = σ{s,...s t } is the information which the investor has till time t. T = {, 1, 2, } S := 1. Ω = { ω = (ɛ 1, ɛ 2 ) : ɛ i {1 + a, 1 + b} } F 1 := σ{s, S 1 } S 1 S 1 (ω) = 1 + a ω = (1 + a, 1 + a) or ω = (1 + a, 1 + b) S 1 (ω) = 1 + b ω = (1 + b, 1 + a) or ω = (1 + b, 1 + b) Hence F 1 = {, Ω, {(1 + a, 1 + a), (1 + a, 1 + b)}, {(1 + b, 1 + a), (1 + b, 1 + b)} }. But S 2 (ω) = (1 + a) 2 ω = (1 + a, 1 + a) S 2 (ω) = (1 + a)(1 + b) ω = (1 + a, 1 + b) or ω = (1 + b, 1 + a) S 2 (ω) = (1 + b) 2 ω = (1 + b, 1 + b) Consequently, S 2 is not F 1 -measurable. We say that (f n ) T n= (f n : Ω R) is adapted to (F n ) T n= if it holds that f n is F n -measurable n. If f n is F n 1 -measurable n we say (f n ) T n= is predictable.

21 2. MARTINGALES 21 2 Martingales and conditional expectation We assume we have a finite probability space (Ω, F, P). Hence we can find a partition A 1,..., A N of Ω with F = σ(a 1,..., A N ). If f : Ω R is F-measurable it can always be written as f(ω) = N a i 1 Ai (ω) using indicator functions which are defined by { 1 ω A, 1 A (ω) := ω A c. We define the expectation of f by Ef := N a i P(A i ). Remark 3.5. Let Ω = {ω 1,..., ω N }. Then Ef := N f(ω i )P({ω i }). Example 3.6. Rolling a die: Ω = {1,..., 6}. The expectation is f(i) = i, i = 1,..., 6. with a i R Ef = = = Definition 3.7. Let (Ω, F, P) be a finite probability space and f : Ω R an F-measurable function. Let G F be a sub-σ-field of F. If 1. g : Ω R is G-measurable and 2. Eg1 G = Ef1 G G G. (3.1)

22 22 CHAPTER 3. CRR MODEL We say g is the conditional expectation of f with respect to G and write g := E[f G] Example 3.8. Let Ω = {1, 2,..., 2 N } As sub-σ-field we choose G N = 2 Ω, f(ω) = ω, P({ω}) = 1 2 N G N 1 = σ { {1, 2}{3, 4},..., {2 N 1, 2 N } } We want to compute E[f G N 1 ]. Clearly, if (3.1) holds for all sets G = {2k 1, 2k} k = 1,..., 2 N 1 then (3.1) holds for all sets G G N 1. By definition, if g := E[f G N 1 ], then g(2k 1, 2k) = g(2k), k Eg1 {2k 1,2k} = Ef1 {2k 1,2k} Eg1 {2k 1,2k} On the other hand Iteration: Ef1 {2k 1,2k} = g(2k 1)E1 {2k 1,2k} = g(2k 1)P ( {2k 1, 2k} ) = 2 2 N g(2k 1) = f(2k 1)P ( {2k 1} ) + f(2k)p(2k) = 2k 1+2k 2 N g(2k 1) = g(2k) = 2k 1 + 2k 2 G N 2 := σ { {1, 2, 3, 4},..., {2 N 3, 2 N 2, 2 N 1, 2 N } } G = {, Ω}. We define E[f G N 1 ] =: f N 1 E[f G N 2 ] =: f N 2 Ef =: f.

23 2. MARTINGALES 23 y f f N 1 f N x Remark 3.9. G G 1... G N is a filtration. Then (f k ) N k= with f k := E[f G k ] is an adapted sequence. Moreover it holds E[f k+1 G k ] = f k k. Definition 3.1. Let (Ω, F, P) be a finite probability space. An (F n ) T n= adapted process (M n ) T n= is 1. a a martingale if E[M n+1 F n ] = M n, n < T, 2. a a supermartingale if E[M n+1 F n ] M n, n < T, 3. a a submartingale if E[M n+1 F n ] M n, n < T.

24 24 CHAPTER 3. CRR MODEL

25 4. Finite market models and non-arbitrage pricing 1 The market model Let (Ω, F, P) be a finite probability space where we agree on the convention P(A) > A F, A i.e. every event is possible. Trading dates : T = {, 1,..., T } The information available to the investors at time t we model by the σ- field F t where we assume {, Ω} = F F 1... F T = F. The securities (assets) are modelled by a stochastic process in R d+1 : (S t, S 1 t,..., S d t ) t T. Here St denotes the bond (or bank account) and is assumed to be nonrandom while St 1,..., St d models the share prices at time t for d different shares and will be random (=depend on ω). We want that S i is (F t )-adapted for all i = 1,..., d. This can be achieved by setting F t := σ(s 1 u,..., S d u : u t) The tuple ( Ω, F, P, (F t ), (S t,...s d t ) ) is the (securities) market model. 25

26 26 CHAPTER 4. NON-ARBITRAGE PRICING 2 Strategies Example 4.1. A trading strategy is a predictable process ϕ = (ϕ t,...ϕ d t ) T t=1 where ϕ i t denotes the number of shares of asset i the investor owns at time t. For fixed t the vector (ϕ t,..., ϕ d t ) is called the portfolio at time t. The wealth process V t (ϕ) is given by V (ϕ) = ϕ 1 S, the investor s initial wealth, d V t (ϕ) = ϕ t S t = ϕ i tst i t T, t 1. i= The investor trades at time t 1 which leads to the portfolio ϕ t. At time t he will have ϕ t S t = V t (ϕ). If he uses exactly his wealth V t (ϕ) to trade at time t, then it must hold V t (ϕ) = ϕ t S t = ϕ t+1 S t. where ϕ t S t is the wealth which comes out from choosing ϕ t at time t 1 and ϕ t+1 S t the needed wealth to buy the portfolio ϕ t+1 at time t. We call ϕ self-financing if ϕ t S t = ϕ t+1 S t, t = 1,..., T 1 Let us introduce discounted prices: S models the bond, i.e. for example:, S t = (1 + r) t if we assume a constant interest rate r, and it holds Then S t >, t =,..., T. S t = ( ) 1, S1 t,..., Sd t St St is the vector of the discounted prices. (Clearly, in case of interest rate r = the discounted price and the share price are equal). Now

27 2. STRATEGIES 27 is the discounted wealth V (ϕ) at t. Ṽ t (ϕ) = 1 (ϕ St t S t ) = ϕ t S t Proposition 4.2. The following assertions are equivalent. 1. ϕ is self-financing. 2. V t (ϕ) = V (ϕ) + t k=1 ϕ k (S k S k 1 ), 1 t T. 3. Ṽ t (ϕ) = V (ϕ) + t k=1 ϕ k ( S k S k 1 ), 1 t T. Proof (1) (2) : We know that and this gives V t (ϕ) = ϕ t S t = d ϕ i tst i i= V t (ϕ) = ( V t (ϕ) V t 1 (ϕ) ) + + ( V 1 (ϕ) V (ϕ) ) + V (ϕ) = (ϕ t S t ϕ t 1 S t 1 ) + + (ϕ 1 S 1 ϕ 1 S ) + ϕ 1 S = ϕ t (S t S t 1 ) + + ϕ 1 (S 1 S ) + V (ϕ) if and only if ϕ is self-financing, i.e. it holds ϕ t 1 S t 1 = ϕ t S t 1. (1) (3) : ϕ t S t = ϕ t+1 S t ϕ t St = ϕ t+1 St. S St Example 4.3. A self-financing strategy ϕ bank account first share second share time S S 1 S (1 +.5)

28 28 CHAPTER 4. NON-ARBITRAGE PRICING Day : investors money: V (ϕ) = 3$. The portfolio chosen at time ϕ 1 = (ϕ 1, ϕ 1 1, ϕ 2 1) = (1, 5, 2) V (ϕ) = ϕ 1 S = = 3. Day 1: investors value of ϕ 1 : V 1 (ϕ) = ϕ 1 S 1, V 1 (ϕ) = = 31 which is the amount that can be used for the new portfolio ϕ 2. It is self-financing: If ϕ 2 = ( 7 1,5, 8, 1), then ϕ 1 S 1 = 31 =! ϕ 2 S 1. ϕ 2 S 1 = = 31. Day 2: V 2 (ϕ) = 7 1, 5 (1, 5) = 32, 5. Proposition 4.4. For any predictable process (ϕ 1 t,..., ϕ d t ) T t=1 and for any V R, there exists a unique predictable process (ϕ t ) T t=1 such that the strategy ϕ = (ϕ, ϕ 1,..., ϕ d ) is self-financing and V (ϕ) = V. Proof. If ϕ is self-financing we get by Proposition 4.2 (3) Ṽ t (ϕ) = V (ϕ) + On the other hand, = V (ϕ) + t ϕ k ( S k S k 1 ) k=1 t ϕ k( S k S k 1) + ϕ 1 k( S k 1 S k 1) 1 + k=1 +ϕ d k( S d k S d k 1). (4.1)

29 3. PROPERTIES OF THE CONDITIONAL EXPECTATION 29 Ṽ t (ϕ) = ϕ t S t = ϕ t + ϕ 1 t S 1 t + + ϕ d t S d t. (4.2) From (4.1) and (4.2) we conclude ϕ t = V (ϕ) + t = V + t 1 k=1 k=1 d j=1 ϕj k ( S j k S j k 1 ) d j=1 ϕj t S j t d j=1 ϕj k ( S j k S j k 1 ) d j=1 ϕj t S j t 1 From this it follows that (ϕ t ) T t=1 is uniquely defined. Moreover, it is predictable, i.e. ϕ t is F t 1 -measurable because V is a constant F F t 1 measurable, ϕ j t is F t 1 -measurable for j = 1,..., d, S t 1 is F t 1 -measurable, addition and multiplication does keep the measurability. Questions we want to answer 1. How can we get market models ( Ω, F, P, T, (F t ), (S t, S 1 t,..., S d t ) ) where riskless profit is not possible? 2. Is there always a self-financing strategy ϕ to hedge the pay-off V T (ϕ) = f(s T )? 3. Is there a fair price for an option? 3 Properties of the conditional expectation Assume P is a probability measure on (Ω, F). Then with the known properties of P. If P : A P(A) A F := σ(a 1,..., A N ) f = N a i 1 Ai

30 3 CHAPTER 4. NON-ARBITRAGE PRICING the expectation of f is defined by N Ef := a i P(A i ). Let us first recall some basic properties of the expectation. Proposition 4.5. Assume Ω and A 1,..., A N F = σ(a 1,..., A N ). Then it holds is a partition of Ω. Set 1. A function f : Ω R is F-measurable f is constant on A 1,...A N, i.e. f can be represented by f(ω) = N a i1 Ai (ω), a i R, ω Ω. 2. If f 1 and f 2 are F-measurable and a, b R then af 1 + bf 2 and f 1 f 2 are F-measurable. 3. E(af 1 + bf 2 ) = aef 1 + bef 2, for f 1, f 2 F-measurable and a, b R. Proof. We only prove 1. and leave 2. and 3. as an exercise. (1) Assume If all the a i s are different, then N f = a i 1 Ai, a i R f 1 ({a i }) = A i F, i = 1,..., N If some a i s are equal, we can arrange that f = n b j 1 Bj, b j s different and B 1,..., B n is a partition on Ω while all B j s are unions of some A i s B j F Since f 1 ({b j }) = B j, f is F-measurable. Assume f is not constant on all A 1,..., A N. We will show that then f is not F-measurable: If there exists A j such that f that is not constant on A j then j.

31 3. PROPERTIES OF THE CONDITIONAL EXPECTATION 31 Because f is a function we have ω 1, ω 2 A i a = f(ω 1 ) f(ω 2 ) = b ω 1 f 1 ({a}) ω 2 f 1 ({b}) f 1 ({a}) f 1 ({b}) = But F consists only of unions of A 1,...A N, that means for any set A F it holds either {ω 1, ω 2 } A or {ω 1, ω 2 } A c. Consequently, f is constant on any A j. Example 4.6. If f 1 = 1 A, f 2 = 1 B then and Notice that f 1 + f 2 = 1 A + 1 B = 1 A B + 1 A B = 1 (A\B) (B\A) + 21 A B + 1 (A B) c f 1 f 2 = 1 A 1 B = 1 A B. A B = (A c B c ) c F and A\B = A B c F. Proposition 4.7. Let F = σ(a 1,..., A N ) like above. Then it holds 1. If G is a σ-field with G F and f is G-measurable, then E[f G] = f. 2. tower-property : f is F-measurable, G 1 and G 2 are σ-fields such that G 1 G 2 F then E[E[f G 1 ] G 2 ] = E[E[f G 2 ] G 1 ] = E[f G 1 ].

32 32 CHAPTER 4. NON-ARBITRAGE PRICING 3. If g is G-measurable and G F then E[fg G] = ge[f G]. Proof: Exercise. Example 4.8. If A 1, A 2, A 3 form a partition of Ω and P(A 1 ) = 1 1, P(A 2) = 7 1, P(A 3) = 2 1, F = σ(a 1, A 2, A 3 ), then Assume that f = a 1 1 A1 + a 2 1 A2 + a 3 1 A3, Ef = a a a 3 1. G = σ(a 1 A 2, A 3 ) σ(a 1, A 2, A 3 ), then h = E[fg G] is by definition G-measurable, i.e. we have h = b 1 1 A1 A 2 + b 2 1 A3. We want to evaluate b 1 and b 2. By definition, E(h1 B ) =! E(f1 B ) B G. As it follows from the Lemma below it is sufficient to test only with B {A 1 A 2, A 3 }. We start with the condition From the LHS we get and the RHS implies E [ (b 1 1 A1 A 2 + b 2 1 A3 )1 A1 A 2 ] = E(f1A1 A 2 ). Eb 1 1 A1 A 2 = b 1 P(A 1 A 2 ) Ea 1 1 A1 + a 2 1 A2 = a 1 P(A 1 ) + a 2 P(A 2 ).

33 4. ADMISSIBLE STRATEGIES AND ARBITRAGE 33 This implies b 1 = a 1 + 7a 2 1 For B = A 3 we get from 1 8 = a 1 + 7a 2. 8 E(b 1 1 A1 A 2 + b 2 1 A3 )1 A3 = E(f1 A3 ) that it should hold which implies b 2 = a 3. Hence Eb 2 1 A3 = Ea 3 1 A3 E[f G] = a 1 + 7a 2 1 A1 A a 3 1 A3. Lemma 4.9. Let F be a σ-field, f an F-measurable function and G = σ(b 1,..., B n ) where B 1,..., B n is a partition of Ω. Assume that h is G- measurable and Then E(h1 Bj = E(f1 Bj ) B j, j = 1,..., n. E(h1 B ) = E(f1 B ) B G. 4 Admissible strategies and arbitrage If ϕ t <, we had borrowed the amount ϕ t from the bank at time t 1. If ϕ i t < for i {1,..., d} we say that we are short a number ϕ i t of assets (shares) i. Borrowing and short-selling is allowed as long as the value of the portfolio V t (ϕ) is always non-negative. Definition 4.1. if 1. A strategy ϕ is admissible if it is self-financing and V t (ϕ) t T. 2. An arbitrage opportunity is an admissible strategy ϕ such that V (ϕ) = and EV T (ϕ) >. (Arbitrage means a possibility of riskless profit: free lunch.)

34 34 CHAPTER 4. NON-ARBITRAGE PRICING 3. The market is viable if it does not contain any arbitrage opportunities, i.e. if it holds V (ϕ) = V T (ϕ) = admissible ϕ. Let us assume in the following that Ω = {ω 1,..., ω N }. We can identify the space of all functions f : Ω R with R N : Define f ( f(ω 1 ),..., f(ω N ) ) R N C := { x = (x 1..., x N ) R N : x i, i = 1,..., N and there exists i : x i > } C is a convex cone. Definition A subset C of a vector space is a convex cone if it holds: x, y C x + y C x C, a > ax C Define Ψ a := set of admissible strategies. Recall that ϕ is self-financing if and only if Ṽ t (ϕ) = V (ϕ) + t ϕ k ( S k S k 1 ). k=1 The discounted gains process will be defined by G t (ϕ) := t ϕ k ( S k S k 1 ). k=1 Lemma If the market (Ω, F, P, (F t ), (S t )) is viable (does not admit any arbitrage opportunities) then it holds G T (ϕ) C predictable (ϕ 1 t,..., ϕ d t ) T t=1.

35 4. ADMISSIBLE STRATEGIES AND ARBITRAGE 35 Proof. Assume G T (ϕ) C. We will show that the market is not viable. First we prove that if G t (ϕ), t =,..., T it follows that the market is not viable. Notice that G t (ϕ) = t k=1 ϕ k ( S k S k 1 ) = t k=1 ϕ k ( S k S k 1 ) + ϕ1 k ( S k 1 S k 1 1 ) + + ϕd k ( S k d S k 1 d ). Hence G T (ϕ) does not depend on ϕ. Proposition 4.4 implies that given (ϕ 1,..., ϕ d ) which is predictable and V = then there is a predictable and self-financing ϕ such that So we conclude that Ṽ t (ϕ) = V + G t (ϕ). Ṽ (ϕ) =, Ṽ t (ϕ) t =,..., T. But G T (ϕ) C means G T (ϕ)(ω i ) i = 1,..., N and there exists i with G T (ϕ)(ω i ) >. Hence G T (ϕ) = N G T (ϕ)(ω i )P({ω i }) G T (ϕ)(ω i ))P({ω i }) >. So there exists an arbitrage opportunity and the market is not viable. Now we consider the general case i.e. Gt (ϕ) can have negative values. Set t = sup { t : P({ω : G t (ϕ) < }) > } Clearly, 1. t T 1 (since G T (ϕ) C), 2. P({ G t (ϕ) < }) >, 3. Gt (ϕ), t = t + 1,..., T

36 36 CHAPTER 4. NON-ARBITRAGE PRICING We define a new strategy as follows. For i = 1,..., d put {, if t ψt(ω) i t, := 1 A (ω)ϕ i t(ω), if t > t where A = {ω : G t (ϕ) < } F t and hence ψ t is predictable. It holds t { G t (ψ) = ψ k ( S k S t t k 1 ) = ( 1 A Gt (ϕ) G t (ϕ) ) t > t k=1 so that by construction we have G t (ϕ) and G t (ϕ) > on A. Thus G t (ψ), t =,..., T, GT (ϕ) > on A. Hence E G N T (ψ) = G T (ψ)(ω i )P({ω i }) ω i A G T (ψ)(ω i )P({ω i }) >, i.e. the market is not viable which means G T (ψ) C. Remark About the assumptions on our market : in contrary to reality we always assume here: a frictionless market: no transaction costs, short sale and borrowing without any limit (ϕ i t R), the securities are perfectly divisible: S i t [, ).

37 5. The fundamental theorem of asset pricing 1 Separation of convex sets in R N Theorem 5.1. Let C R N be a closed convex set and (,..., ) C. Then there exists a real linear functional ξ : R N R and α > such that ξ(x) α x C. Proof Let B(, r) = {x R N : x := (x x 2 N ) 1 2 r} which equals to a closed ball of radius r and center at the origin. Choose r > such that C B(, r). The map x x is a continuous function and C B(, r) is closed and bounded. Let m := min x C B(,r) x. Then there exists an x C B(, r) with x = m. Indeed, take (x n ) n=1 with x n m as n. Then there exists a subsequence (x nk ) such that x nk x for k. The claim follows from x x x nk + x nk because x x nk and x nk m. Hence, x x x C (x B(, r) x > r) Notice that x C λx + (1 λ)x C for λ [, 1] since C is convex. This implies 37

38 38 CHAPTER 5. FUNDAMENTAL THEOREM and therefore λx + (1 λ)x x λ 2 x x + 2λ(1 λ)x x + (1 λ) 2 x x x x. This gives λx x + 2(1 λ)x x 2x x + λx x and 2x x + λ(x x 2x x + x x ) 2x x λ [, 1] For λ this inequality is only true if x x x x = x 2 >. If we define ξ(x) := x x we get a linear functional ξ(x) m 2 = α for x C. Theorem 5.2. Let K be a compact convex subset in R N and V a linear subspace of R N. If V K =, then there exists a linear functional such that 1. ξ(x) >, x K, 2. ξ(x) =, x V. ξ : R N R Therefore, the subspace V is included in a hyperplane that does not intersect K. Proof. The set C := K V = {x R N : (k, v) K V, x = k v} is convex since for x 1, x 2 C we have λx 1 + (1 λ)x 2 = λ(k 1 v 1 ) + (1 λ)(k 2 v 2 )

39 2. MARTINGALE TRANSFORMS 39 = λk 1 + (1 λ)k 2 (λv 1 + (1 λ)v 2 ). Now λk 1 + (1 λ)k 2 K and λv 1 + (1 λ)v 2 V and its difference is in C. The set C is closed because V is closed and K is compact. We have (,..., ) C since V K =. Hence we can apply Theorem 5.1 and find a linear functional ξ : R N R and a constant α > with This implies ξ(x) α x C. ξ(k v) = ξ(k) ξ(v) α k K, v V. Especially, it holds for fixed k K and and v V and all λ R that and because ξ is linear also ξ(k ) ξ(λv ) α, ξ(k ) λξ(v ) α. Consequently, ξ(v) = for all v V and ξ(k) α for all k K. 2 Martingale transforms Let (Ω, F, P) be a finite probability space. Lemma 5.3. Let (F n ) T n= be a filtration (ϕ n ) T n=1 a predictable sequence and (M n ) T n= a martingale. Then the process X := X n := ϕ 1 (M 1 M ) + ϕ(m 2 M 1 ) ϕ n (M n M n 1 ), n = 1,..., T is a martingale with respect to (F n ) T n=. The sequence (X n ) T n= is called a martingale transform of (M n ) by (ϕ n ). Proof. We have that X n is F n -measurable for all n =,..., T. We check the martingale property: [ n+1 E[X n+1 F n ] = E ϕ t (M t M t 1 ) ] Fn t=1

40 4 CHAPTER 5. FUNDAMENTAL THEOREM n = ϕ t (M t M t 1 ) + E[ϕ n+1 (M n+1 M n ) F n ] t=1 = X n where we used that E[ϕ n+1 (M n+1 M n ) F n ] = ϕ n+1 E[(M n+1 M n ) F n ] = ϕ n+1 E[M n+1 F n ] ϕ n+1 M n because (ϕ n ) is predictable and (M n ) is adapted. Theorem 5.4. An adapted real-value process (M n ) T n= is a martingale if and only if E t ϕ n (M n M n 1 ) = t = 1,..., T (5.1) n=1 for all predictable processes (ϕ n ) T n=1. Proof. If (M n ) T n= is a martingale, X t = t n=1 ϕ n(m n M n 1 ) is a martingale transform. Hence by the previous Lemma EX t = t = 1,..., T. Assume (5.1) holds. Let A F n and define Then and consequently ϕ n (ω) := { n n A (ω) n = n + 1. EX T = E1 A (M n +1 M n ) = A F n E[M n +1 F n ] = M n n =,..., T. Definition 5.5. (Independence)

41 2. MARTINGALE TRANSFORMS The sets A, B F are called independent : P(A B) = P(A)P(B). 2. The σ-fields G 1, G 2 F are called independent : P(A B) = P(A)P(B) A G 1, B G 2 (every set of G 1 is independent of every set of G 2 ). 3. If f 1,..., f n : Ω {a 1,..., a M } (a i R) are F-measurable then f 1,..., f n are called independent (random variables) : P({ω : f 1 (ω) = x 1,..., f N (ω) = x n })) = Π n k=1p({ω : f i (ω) = x i ) x i {a 1,..., a M }. In other words all the pre-images of f 1,..., f n are independent sets. 4. An F-measurable function f is called independent from a σ-field G (G F) : f and 1 G are independent G G. Remark to (3) {ω : f 1 (ω) = x 1,..., f N (ω) = x n } = {ω : f 1 (ω) = x 1 and... and f n (ω) = x n } = n ({x k}). k=1 f 1 k Example Tossing a coin 2 times: P(1st toss = heads and 2nd toss = tails ) = P(1st toss = heads )P( 2nd toss = tails ) 2. CRR model Ω = { ω = (ɛ 1,..., ɛ T ), ɛ i {(1 + a), (1 + b)} } P({ω}) = p k (1 p) T k if ω contains k times 1 + b and T k times 1 + a Tossing a coin T-times Write { for each toss 1 + a if tails 1 + b if heads P(tossing heads ) = p P(tossing tails )=1 p

42 42 CHAPTER 5. FUNDAMENTAL THEOREM If then The functions S t (ω) = S ε 1 ε 2 ε t S t+1 (ω) S t (ω) = ε t+1. S 1 S, S 2 S 1,..., are independent: for x i {(1 + a), (1 + b)} S T S T 1 P ( {ω : S 1(ω) S (ω) = x S T (ω) 1,..., S T 1 (ω) = x T } ) = P ( {ω = (ε 1,..., ε T ) : ε 1 = x 1,..., ε T = x T } ) = p k (1 p) T k = Π T t=1p ( {ω : S t S t 1 = ε t = x t }) if k of the x i s are 1+b and T k are equal to 1+a. We have for t = 1,..., T P ( {ω = (ε 1,..., ε T ) : ε t = 1 + b} ) = p where p is the probability that one tosses heads the t-th time if one tosses T-times altogether. The outcome of the other times does not influence that of time t. Theorem 5.7. Let f, g be F-measurable. Proof 1. If f and g are independent then Efg = EfEg 2. If f is independent from the σ-field G(G F) then E[f G] = Ef 1. Let f = n x i 1 Fi, g = m y j 1 Gj j=1

43 2. MARTINGALE TRANSFORMS 43 where we assume that all x i s are different and all y j s are different. Efg = n m E x i y j 1 Fi 1 Gj = E = n n j=1 j=1 m x i y j 1 Fi G j j=1 m x i y j P(F i G j ) = ( n x i P(F i ) )( m y i P(G j ) ) = EfEg j=1 where we used P(F i G j ) = P ( {ω : f(ω) = x i } {g(ω) = y j } ) = P ( {ω : f(ω) = x i } ) P ( {ω : g(ω) = y j } ) = P(F i )P(G j ). 2. Exercise. Proposition 5.8. Assume f 1,..., f n are independent and F-measurable. Let F k = σ(f 1,..., f k ). 1. Then f l, l > k is independent from F k. 2. If Ef k =, for k = 1,..., n then (M t ) with M t := t k=1 f k for t 1 and M = is a martingale with respect to (F t ). 3. If Ef k = 1, for k = 1,..., n then (N t ) with N t := Π t k=1 f k for t 1 and N = 1 is a martingale with respect to (F t ). Proof 1. The idea is to use G F k which can be represented by G = {f 1 = x 1,..., f k = x k } and to show that 2. and 3. are Exercises. P(f l = x l, 1 G = x) = P(f l = x l )P(1 G = x).

44 44 CHAPTER 5. FUNDAMENTAL THEOREM Remark 5.9. One can show that σ ( S1 S,..., S t S t 1 ) = σ(s 1,...S t ) = F t. From (3) it follows now that (S t ) T t= with S t = S S 1 S S 2 S 1 S t S t 1 is a martingale E S t S t 1 = 1 t. 3 The fundamental theorem of asset pricing With the results of the previous sections we will get a characterization of the no arbitrage condition. Definition 5.1. Let P, Q : (Ω, F) [, 1] be probability measures. Then P is said to be equivalent to Q (notation P Q) if and only if P(A) = Q(A) = for any A F. If Ω := {ω 1,..., ω N }, F := 2 Ω and P({ω i }) >, i = 1,..., N then Q P iff Q({ω i }) >, i = 1,..., N. Theorem [Fundamental Theorem of Asset pricing] The market (Ω, F, P, (F t ), (S t )) is viable if and only if there exists a probability measure Q P such that S t i = 1 S S t, i t T are Q-martingales for t i = 1,..., d (Q is called the equivalent martingale measure: EMM). Proof. The proof for (our case, namely #Ω < ) was done by Harrison, Kreps and Pliska between 1979 and For general Ω this theorem was proved by Dalang, Morton and Willinger in 199. Assume Q P and S i t, i = 1,..., d are Q-martingales. By Proposition 4.2(3), if ϕ is self-financing then Ṽ t (ϕ) = V (ϕ) + t ϕ k ( S k S k 1 ). k=1

45 3. THE FUNDAMENTAL THEOREM OF ASSET PRICING 45 We denote by E Q the expectation with respect to Q. Hence by Theorem 5.4 T E QVT (ϕ) = E Q V (ϕ) + E Q ϕ k ( S k S k 1 ) k=1 = E Q V (ϕ) + E Q d = E Q V (ϕ) + d E Q T ϕ i k( S k i S i k 1 ) k=1 T ϕ i k( S k i S i k 1 ) k=1 = E Q V (ϕ). (5.2) If V (ϕ) = then E Q VT (ϕ) =. Now assume that E Q VT (ϕ) = N V T (ϕ)(ω i )Q({ω i }) =. (5.3) If ϕ is admissible, then V T (ϕ)(ω i ), i = 1,..., N. So (5.3) implies that V T (ϕ)(ω i ) =, i = 1,..., N. Consequently, V (ϕ)(ω i ) =, i = 1,..., N V T (ϕ)(ω i ) =, i = 1,..., N for all admissible ϕ. Hence the market does not admit arbitrage opportunities and is not viable. By Lemma 4.12 we have : If the market is viable then where G T (ϕ) C = {x = (x 1,..., x N ) R N, x i, i = 1,..., N, i x i > } G t (ϕ) = t k=1 is the discounted gains process. We define (ϕ 1,..., ϕ d ) predictable ( ϕ 1 k ( S k 1 S 1 k 1 ) + + ϕd k( S k d S d k 1 )) V := { G T (ϕ) : (ϕ 1,..., ϕ d ) predictable.}

46 46 CHAPTER 5. FUNDAMENTAL THEOREM Then V is a linear subspace of R N and ( GT (ϕ)(ω 1 ),..., G T (ϕ)(ω N ) ) R N. By the Lemma we have V C =. We define K := {f = (f(ω 1 ),..., f(ω N )) C : We have that V K = and K is convex: If f, g K then N f(ω i ) = 1}. 1. λf + (1 λ)g C (since C is convex) 2. N ( λf(ωi ) + (1 λ)g(ω i ) ) = λ + (1 λ) = 1 K is compact because it is bounded ( f = N f(ω i) = 1) and closed. Therefore, by Theorem 5.2 there exists a linear functional ξ(x) = ξ 1 x ξ N x N with 1. N ξ if(ω i ) > f K 2. N ξ i G T (ϕ)(ω i ) = (ϕ 1,..., ϕ d ) predictable. Now, if f := (,...,, 1,,..., ), then f K and 1. implies ξ i > for all i = 1,..., N. We define Q({ω i }) := ξ i N ξ. i Then Q is a probability measure, Q P and by 2. E Q GT (ϕ) = In other words, N G N (ϕ)(ω i )Q({ω i }) = E Q T t=1 d ϕ i t( S t i S t 1) i = (ϕ i,..., ϕ d ) predictable.

47 4. COMPLETE MARKETS AND OPTION PRICING 47 or in short form E Q N ϕ t ( S t S t 1 ) = i = 1,..., d (ϕ i t) predictable. Hence by Theorem 5.4 we have that ( S 1,..., S d ) are Q-martingales. Remark The scalar product (or inner product) on a vector space V (= R N ) is a function and v 1, v 2, v V (, ) : V V R such that α, β R 1. (αv 1 + βv 2, v) = α(v 1, v) + β(v 2, v) linearity 2. (v1, v) = (v, v 1 ) symmetry 3. (v, v) and (v, v) = v = positive definite. For (x 1,..., x N ) R N, v, w V the expression (v, w) := N v iw i x i defines a scalar product iff x i > i = 1,..., N. (To see this assume x 1 =. Then v = (1,,..., ) implies (v, v) = which is a contadiction. In the same way it follows that x i < is not possible.) Orthogonality: We define V W orthogonal (v, w) = for all v V and w W. 4 Complete markets and option pricing Let us assume the market model (Ω, F, P, (F t ), (S t )). We already know the European call-option H = (S 1 T K)+ and the European put-option H = (K S 1 T )+. Options can also depend on the whole path of the underlying security. For example, H = ( ) ST 1 S1 1 + S ST 1 + T

48 48 CHAPTER 5. FUNDAMENTAL THEOREM would be one type of a so-called Asian option. In general we define a European option (or a contingent claim) to be a non-negative function H : Ω [, ) which is F-measurable. We say the contingent claim H is attainable if there exists an admissible strategy ϕ with H = V T (ϕ). If the market is viable, then there exists a Q P such that ( S t ) T t= is a Q-martingale and if we find a self-financing strategy ϕ such that It follows H = V T (ϕ) ( resp. H S T E Q Ṽ T (ϕ)) = V (ϕ) H is a no-arbitrage price. This implies that E Q ST is attainable and Q P. In general [ ] H Ṽ t (ϕ) = E Q ST F t is the discounted no-arbitrage price at time t. ) = ṼT (ϕ). is a no-arbitrage price if H Definition The market is complete if every contingent claim is attainable; i.e. for any F T -measurable H there exists an admissible strategy ϕ such that H = V T (ϕ). Remark Completeness is a restrictive assumption: a lot of market models are not complete. But there is a nice mathematical characterization of completeness. Theorem A viable market is complete if and only if there exists a unique Q P such that ( S 1,..., S d ) are Q-martingales. Proof Assume the market is viable and complete. Let H be F T -measurable and H. Completeness implies that there exists an admissible strategy such that H = V T (ϕ) (5.4)

49 4. COMPLETE MARKETS AND OPTION PRICING 49 where ϕ is self-financing so that Ṽ T (ϕ) = V (ϕ) + T ϕ t ( S t S t 1 ). A viable market implies that there exists a Q P such that the ( S t ) t are Q- martingales. We have to show that Q is unique. Let ˆQ be another probability measure such that ˆQ P and ( S t ) t are ˆQ-martingales. Then t=1 Hence E Q Ṽ T (ϕ) = V (ϕ) = EˆQṼ T (ϕ). H H E Q = V ST (ϕ) = EˆQ. ST By assumption H is attainable, so the no arbitrage price is the same for any Q. Choose H = 1 A ST for A F T. Then Q(A) = ˆQ(A) A F T and Q = ˆQ. Assume the market is viable and incomplete. Then there exists H,and H is F T -measurable and not attainable. Defining implies Let V := Hence we get { V + T ϕ t ( S t S t 1 ), t=1 H S T } V R, (ϕ 1,..., ϕ d ) predictable V. W = { f = ( f(ω 1 ),..., f(ω N ) ) : f : Ω R } = R N. We introduce the scalar product V W.

50 5 CHAPTER 5. FUNDAMENTAL THEOREM (f, g) := E Q fg = We take a basis v 1,..., v M V, ˆx := x N f(ω i )g(ω i )Q({ω i }) x := H S T V then M (x, v i )v i V. Indeed, for any v = M k=1 α kv k V it holds Define (ˆx, v) = Obviously ˆQ({ω}) > M α k (x, v k ) k=1 ˆQ({ω}) := M α n (x, v n ) =. n=1 ( ) ˆx(ω) 1 + Q({ω}). 2 supˆω ˆx( ω) ω Ω and N ˆQ(Ω) = Q(ω i ) + Indeed, since 1 V we have N ˆx(ω i ) 2 supˆω ˆx(ˆω) Q({ω i}) = 1. (ˆx, 1) = E Qˆx = N ˆx(ω i )Q({ω i }) =. Hence ˆQ is a probability measure and ˆQ P and ˆQ Q by definition. Finally, we show that ( S t ) is also a ˆQ-martingale. Setting v = N t=1 ϕ t( S t S t 1 ) we have EˆQ N t=1 ϕ t( S t S t 1 ) = N v(ω i) ˆQ({ω i }) = N v(ω i) ( 1 + ˆx(ω i) 2 sup ω ˆx(ω) ) Q({ωi }) 1 = E Q v + E 2 sup ω ˆx(ω) Qvˆx = (ϕ 1,..., ϕ d ) predictable. Hence by the Theorem 5.4 ( S t ) T t= is a ˆQ-martingale.

51 6. American Options 1 Stopping Times Let (Ω, F, P) be a finite probability space, (Ω, F, P, (F t ) T t=, (S t ) T t=) a market model as before. An American option can be exercised at any time t {, 1,..., T } =: T. For example, the American call option with strike price K: Z t = (S 1 t K) +, t =,1,..., T is then a sequence adapted (F T ). The random variable Z t stands for the profit made by exercising the option at time t. For the decision to exercise or not at time t the trader can only use the information available until time t, i.e. the information is given by F t. We describe this using stopping times. Definition 6.1. A random variable τ: Ω T is a stopping time if ({τ = t} = {ω Ω : τ(ω) = t}) Remark 6.2. It holds {τ = t} F t t =,..., T. {τ = t} = {τ t} \ {τ t 1} F t t = 1,..., T {τ t} = {τ = } {τ = 1}... {τ = t} F t t =,..., T Definition 6.3. Let (X t ) T t=o be an adapted sequence and τ a stopping time. We define X τ t (ω) := X t τ(ω) (ω) where (a b := min{a, b}). This means on the set {ω : τ(ω) = k} it holds { Xt τ Xk if t k = X t if t < k. 51

52 52 CHAPTER 6. AMERICAN OPTIONS Theorem 6.4. Let τ be a stopping time. (1) (X t ) is (F t ) adapted (X τ t ) is (F t ) adapted. (2) (X t ) is a martingale (X τ t ) is a martingale. (3) (X t ) is a supermartingale (X τ t ) is a supermartingale. Proof (1) t X t τ = X + 1I {k τ} (X k X k 1 ) k=1 It holds {k τ} = {k > τ} c. But {τ < k} = {τ k 1} F k 1. Hence ϕ(k):=1i {k τ} is a predictable sequence. Clearly, (X t τ ) T t= is adapted. (2) Let (X t ) be a martingale. Since (X τ t x ) is a martingale transform of (X t ) by (ϕ(t)) it follows by Lemma 5.3 that (X τ t ) is a martingale. (3) Can be shown similarly. 2 The Snell Envelope We want to define the price of an American option, for example, for Z t = (S t K) +, t =,..., T. We use a backward in induction. Let t = T. Then for the option price U T it should hold U T = Z T. For t = T 1 the option holder has 2 possibilities: (1) Trading at once (t = T 1) implies that the writer must pay Z T 1 (2) Trading at time t = T. The writer must be able to pay Z T which means that he needs an admissible strategy with the price S T 1E Q [ ZT S T ] F T 1 = ST 1ṼT 1 = V T 1.

53 2. THE SNELL ENVELOPE 53 Here Q=EMM and we assume that the market is complete. Then for the option price it should hold { [ ]} U T 1 = max Z T 1, ST ZT 1E Q ST F T 1 By induction we have { [ ]} U t 1 = max Z t 1, St 1E Ut Q St F t 1. Theorem 6.5. Let (Ω, F, Q) be a finite probability space, (X t ) T t= an (F t ) adapted sequence with X t, t =,..., T. Then the process (U t ) with U T := X T U t 1 := max{x t 1, E Q [U t F t 1 ]} is a supermartingale. It is the smallest supermartingale dominating (X t ), i.e. it holds U t X t, t =,..., T. Remark 6.6. The process (U t ) is called the Snell envelope of (X t ). t =,..., T. So (U t ) is dominat- Proof (of the Theorem) Clearly, U t = max {X t, E Q [U t+1 F t ]} X t, ing (X t ). Moreover, (U t ) is adapted. From E[U t F t 1 ] max{x t 1, E Q [U t F t 1 ]} = U t 1 we conclude that (U t ) is a supermartingale. We have to show that (U t ) is the smallest one. Suppose (Y t ) is a supermartingale dominating (X t ). Then Y T X T = U T. Backward induction: Assume for some t T that Y t U t. Then it follows by the supermartingale property of (Y t ) that Y t 1 E Q [Y t F t 1 ] E Q [U t F t 1 ]. But Y t 1 X t 1 holds also. Consequently, Y t 1 max{x t 1, E Q [U t F t 1 ]} = U t 1.

54 54 CHAPTER 6. AMERICAN OPTIONS Theorem τ = min{t : U t = X t } is a stopping time. 2. The stopped process (U τ t ) is a martingale. Proof 1. {τ = } = {U = X } F since U and X are F -measurable. t 1 {τ = t} = {U s > X s } {U t = X t } s= Since {U s > X s } F s F t and {U t = X t } F t we have {τ = t} F t. 2. Define ϕ(t) := 1I {τ t}. We know that ϕ(t) is predictable. It holds and U τ t U τ t = U + t ϕ(s)(u s U s 1 ) s=1 U τ t 1 = ϕ(t)(u t U t 1 ) = 1I {τ t}(u t U t 1 ) = 1I {τ t}(u t E Q [U t F t 1 ]) since on the set {τ t} = {τ > t 1} it holds U t 1 > X t 1 and hence So it follows E Q [U τ t U t 1 = max{x t 1, E Q [U t F t 1 ]} = E Q [U t F t 1 ]. Ut 1 F τ t 1 ] = E Q [1I {τ t}(u t E Q [U t F t 1 ]) F t 1 ] = 1I {τ t}(e Q [U t F t 1 ] E Q [U t F t 1 ] = i.e. (U τ t ) is a martingale. Definition 6.8. A stopping time σ : Ω T is optimal for (X t ) if E Q X σ = sup E Q X τ τ T where T denotes the set of stopping times τ : Ω T.

55 2. THE SNELL ENVELOPE 55 Interpretation: If we think of X n as the total winnings at time n, then stopping at time σ would maximize the expected gain. Corollary 6.9. τ = min{t : U t = X t } is an optimal stopping time for (X t ) and U = E Q X τ = sup EX τ. τ T Proof The process (U τ t ) is a martingale. It holds U = U τ = E Q U τ T = E Q U T τ = E Q X τ because T τ = τ and U τ = X τ by definition. On the other hand (Ut τ ) is a supermartingale for any τ T by Theorem 6.4. So it follows because (Ut τ ) is a supermartingale and (U t ) dominates (X t ) that U = U τ E Q U τ E Q X τ. Which implies U sup τ T E Q X τ, and since τ T and E Q X τ get U = sup τ T E Q X τ. = U we There is the following characterization for optimal stopping times: Theorem 6.1. A stopping time σ is optimal for (X t ) iff 1. U σ = X σ, 2. U σ is a ((F t ), Q) - martingale (U denotes the Snell envelope of (X t )). Proof If U σ is a martingale, it holds U = E Q U σ T = E Q U σ = E Q X σ. On the other hand (U τ t ) is a supermartingale for any τ T (since (U t ) is a supermartingale, see Theorem 6.4.) Hence U = U τ E Q U τ T = E Q U τ E Q X τ

56 56 CHAPTER 6. AMERICAN OPTIONS because (U t ) dominates (X t ). From σ T we get E Q X σ = sup E Q X τ, i.e. σ is optimal. τ T Assume σ is optimal; i.e. E Q X σ = sup τ T E Q X τ. By Collary 6.9 we have that U = sup τ T E Q X τ. Hence U = E Q X σ E Q U σ because (U t ) dominates (X t ). The process (U t ) is a supermartingale, therefore (U σ t ) is a supermartingale so that also U = U σ E Q U σ and therefore E Q X σ = U = E Q U σ. (6.1) Hence U t X t t, ω X σ = U σ. Since (Ut σ ) is a supermartingale, E Q [UT σ F t ] Ut σ (6.2) and U = U σ E Q Ut σ E Q E Q [UT σ F t ] = E Q U σ = U because of the relations (6.1) and (6.2). Since Ut σ E Q [UT σ F t ] and we conclude that i.e. (U σ t ) is a martingale. E Q U σ t = E Q [U σ T F t ] E Q [U σ T F t ] = U σ t, Remark τ from the Corollary is the smallest optimal stopping time for (X t ).

57 3. DECOMPOSITION OF SUPERMARTINGALES 57 3 Decomposition of Supermartingales We will consider the so called Doob decomposition which we will use to find trading strategies for American options. Doob decomposition is also used to model trading strategies with consumption. Theorem Every supermartingale (U t ) T t= has the following unique decomposition U t = M t A t, where (M t ) is a martingale an (A t ) is a non-decreasing predictable process with A =. Proof Induction t = : From A = we conclude that M = U is uniquely determined. t t + 1: Consider U t+1 U t = M t+1 M t (A t+1 A t ). (6.3) We take the conditional expectation on both sides and assume (M t ) is a martingale and that (A t ) is predictable. Then implies E[U t+1 F t ] U t = E[M t+1 F t ] M t (A t+1 A t ) (A t+1 A t ) = E[U t+1 F t ] U t (6.4) and therefore i.e. (A t ) is non-decreasing. A t A t+1 Remark From (6.3) and (6.4) one gets M t+1 M t = U t+1 E[U t+1 F t ] and A t+1 A t = U t E[U t+1 F t ]. One can find also the largest optimal stopping time for (X t ):

58 58 CHAPTER 6. AMERICAN OPTIONS Definition Define σ : Ω {, 1,..., T } by setting { T if AT (ω) =, σ(ω) := min{t ; A t+1 > } if A T (ω) >. if (U t ) is the Snell envelope of (X t ) and U t = M t A t (Doob decomposition). Theorem σ is the largest optimal stopping time for (X t ) Proof (1) σ is a stopping time: {σ = T } = {A T = } F T and for t T 1 {σ = t} = s t {A s = } {A t+1 > } is F t measurable because {A s = } F s 1 F t 1 for 1 s t and {A t+1 > } F t since A is predictable. (2) σ is optimal: We conlude from U t = M t A t and U σ t = M σ t that (U σ t ) is a martingale. This gives us property (2) of Theorem 6.1 i.e. σ is optimal. We still have to show U σ = X σ. U σ = Σs= T 1 1I {σ=s} U s + 1I {σ=t } U T = Σs= T 1 1I {σ=s} max{x s, E[U s+1 F s ]} + 1I {σ=s} U T We have E[U s+1 F s ] = E[M s+1 A s+1 F s ] = M s A s+1 and A s+1 > on {σ = s}. On the other hand This gives and therefore We get U s = M s A s and A s = on {σ = s}. E[U s+1 F s ] < U s U s = max{x s, E[U s+1 F s ]} = X s. U σ = Σ T 1 s= 1I {σ=s} X s + 1I {σ=t } U T = X σ, because U T = X T by construction, i.e. σ is optimal.

59 4. PRICING AND HEDGING OF AMERICAN OPTIONS 59 (3) σ is the largest: Assume τ σ and Q(τ > σ) >. Then EU τ = EM τ EA τ = EM EA τ < EU = U which means τ is not optimal. 4 Pricing and hedging of American options We assume the market (Ω, F, P, (F t ), (S t )) is viable and complete ((Ω, F, P) finite probability space) and Q is the EMM. An American option is an adapted sequence (Z t ) T t= with Z t. In Section 2 of this chapter we saw: Given an American option (Z t ) T t= its value process (U t ) T t= can be described by { UT = Z T, U t = max{z t, St E Q [ U t+1 S t+1 F t ]}, t T 1. That means, the discounted price of the option Ũt := Ut, t=,...,t is St the Snell envelope under Q of ( Z t ) T t=. Like in Section 2 one can show Ũ t = sup E Q [ Z τ F τ T t,t Sτ t ], where T t,t denotes the set of all stopping time τ : Ω {t,..., T }. Consequently, the price U t of the option (Z t ) T t= is U t = S t Now we use the Doob decomposition: sup E Q [ Z τ F τ T t,t Sτ t ]. Ũ t = M t Ãt where M t is a Q-martingale and Ãt is non-decreasing, predictable and A =. By assumption, the market is complete. This implies the existence of a selffinancing strategy ϕ, such that for H = ST M T it holds T (ϕ) = S T M T.