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Chapter 3 Arbitrage Pricing 3.1 Binomial Pricing Return to the binomial pricing model Please see: Cox, Ross and Rubinstein, J. Financial Economics, 7(1979), 229–263, and Cox and Rubinstein (1985), Options Markets, Prentice-Hall. Example 3.1 (Pricing a Call Option) Suppose u =2;d =0:5;r = 25 (interest rate), S 0 =50 . (In this and all examples, the interest rate quoted is per unit time, and the stock prices S 0 ;S 1 ;::: are indexed by the same time periods). We know that S 1 ! = 100 if ! 1 = H 25 if ! 1 = T Find the value at time zero of a call option to buy one share of stock at time 1 for $50 (i.e. the strike price is $50). The value of the call at time 1 is V 1 ! =S 1 !,50 + = 50 if ! 1 = H 0 if ! 1 = T Suppose the option sells for $20 at time 0. Let us construct a portfolio: 1. Sell 3 options for $20 each. Cash outlay is ,$60: 2. Buy 2 shares of stock for $50 each. Cash outlay is $100. 3. Borrow $40. Cash outlay is ,$40: 59 60 This portfolio thus requires no initial investment. For this portfolio,the cash outlay at time 1 is: ! 1 = H ! 1 = T Pay off option $150 $0 Sell stock ,$200 ,$50 Pay off debt $50 $50 ,,,, , ,,,,, $0 $0 The arbitrage pricing theory (APT) value of the option at time 0 is V 0 =20 . Assumptions underlying APT: Unlimited short selling of stock. Unlimited borrowing. No transaction costs. Agent is a “small investor”, i.e., his/her trading does not move the market. Important Observation: The APT value of the option does not depend on the probabilities of H and T . 3.2 General one-step APT Suppose a derivative security pays off the amount V 1 at time 1, where V 1 is an F 1 -measurable random variable. (This measurability condition is important; this is why it does not make sense to use some stock unrelated to the derivative security in valuing it, at least in the straightforward method described below). Sell the security for V 0 at time 0. ( V 0 is to be determined later). Buy 0 shares of stock at time 0. ( 0 is also to be determined later) Invest V 0 , 0 S 0 in the money market, at risk-free interest rate r .( V 0 , 0 S 0 might be negative). Then wealth at time 1 is X 1 4 = 0 S 1 +1+rV 0 , 0 S 0 = 1 + rV 0 + 0 S 1 ,1 + rS 0 : We want to choose V 0 and 0 so that X 1 = V 1 regardless of whether the stock goes up or down. CHAPTER 3. Arbitrage Pricing 61 The last condition above can be expressed by two equations (which is fortunate since there are two unknowns): 1 + rV 0 + 0 S 1 H,1 + rS 0 =V 1 H (2.1) 1 + rV 0 + 0 S 1 T,1 + rS 0 =V 1 T (2.2) Note that this is where we use the fact that the derivative security value V k is a function of S k , i.e., when S k is known for a given ! , V k is known (and therefore non-random) at that ! as well. Subtracting the second equation above from the first gives 0 = V 1 H , V 1 T S 1 H , S 1 T : (2.3) Plug the formula (2.3) for 0 into (2.1): 1 + rV 0 = V 1 H , 0 S 1 H , 1 + rS 0 = V 1 H , V 1 H , V 1 T u , dS 0 u , 1 , rS 0 = 1 u , d u , dV 1 H , V 1 H , V 1 T u , 1 , r = 1+r ,d u,d V 1 H+ u, 1, r u ,d V 1 T: We have already assumed ud 0 . We now also assume d 1+r u (otherwise there would be an arbitrage opportunity). Define ~p 4 = 1+r,d u,d ; ~q 4 = u,1,r u,d : Then ~p 0 and ~q0 .Since ~p +~q =1 ,wehave 0 ~p1 and ~q =1,~p . Thus, ~p; ~q are like probabilities. We will return to this later. Thus the price of the call at time 0 is given by V 0 = 1 1+ r ~pV 1 H+ ~qV 1 T : (2.4) 3.3 Risk-Neutral Probability Measure Let be the set of possible outcomes from n coin tosses. Construct a probability measure f IP on by the formula f IP ! 1 ;! 2 ;::: ;! n 4 =~p fj;! j =Hg ~q fj;! j =Tg f IP is called the risk-neutral probabilitymeasure. We denote by f IE the expectation under f IP . Equa- tion 2.4 says V 0 = f IE 1 1+r V 1 : 62 Theorem 3.11 Under f IP , the discountedstock price process f1 + r ,k S k ; F k g n k=0 is a martingale. Proof: f IE 1 + r ,k+1 S k+1 jF k = 1 + r ,k+1 ~pu +~qdS k = 1 + r ,k+1 u1 + r , d u , d + du , 1 , r u , d S k = 1 + r ,k+1 u + ur , ud + du , d , dr u , d S k = 1 + r ,k+1 u , d1 + r u , d S k = 1 + r ,k S k : 3.3.1 Portfolio Process The portfolio process is = 0 ; 1 ;::: ; n,1 ,where k is the number of shares of stock held between times k and k +1 . Each k is F k -measurable. (No insider trading). 3.3.2 Self-financing Value of a Portfolio Process Start with nonrandom initial wealth X 0 , which need not be 0. Define recursively X k+1 = k S k+1 +1+rX k , k S k (3.1) = 1 + rX k + k S k+1 , 1 + rS k : (3.2) Then each X k is F k -measurable. Theorem 3.12 Under f IP , the discountedself-financingportfolioprocessvalue f1 + r ,k X k ; F k g n k=0 is a martingale. Proof: We have 1 + r ,k+1 X k+1 =1+r ,k X k + k 1 + r ,k+1 S k+1 , 1 + r ,k S k : CHAPTER 3. Arbitrage Pricing 63 Therefore, f IE 1 + r ,k+1 X k+1 jF k = f IE 1 + r ,k X k jF k + f IE 1 + r ,k+1 k S k+1 jF k , f IE 1 + r ,k k S k jF k = 1 + r ,k X k (requirement (b) of conditional exp.) + k f IE 1 + r ,k+1 S k+1 jF k (taking out what is known) ,1 + r ,k k S k (property (b)) = 1 + r ,k X k (Theorem 3.11) 3.4 Simple European Derivative Securities Definition 3.1 () A simpleEuropeanderivativesecuritywith expirationtime m is an F m -measurable random variable V m .(Here, m is less than or equal to n , the number of periods/coin-tosses in the model). Definition 3.2 () A simple European derivative security V m is said to be hedgeable if there exists a constant X 0 and a portfolio process = 0 ;::: ; m,1 such that the self-financing value process X 0 ;X 1 ;::: ;X m given by (3.2) satisfies X m ! =V m !; 8! 2: In this case, for k =0;1;::: ;m , we call X k the APT value at time k of V m . Theorem 4.13 (Corollary to Theorem 3.12) If a simple European security V m is hedgeable, then for each k =0;1;::: ;m , the APT value at time k of V m is V k 4 =1+r k f IE1 + r ,m V m jF k : (4.1) Proof: We first observe that if fM k ; F k ; k =0;1;::: ;mg is a martingale, i.e., satisfies the martingale property f IE M k+1 jF k =M k for each k =0;1;::: ;m, 1 ,thenwealsohave f IE M m jF k =M k ;k =0;1;::: ;m, 1: (4.2) When k = m , 1 , the equation (4.2) follows directly from the martingale property. For k = m , 2 , we use the tower property to write f IE M m jF m,2 = f IE f IE M m jF m,1 jF m,2 = f IE M m,1 jF m,2 = M m,2 : 64 We can continue by induction to obtain (4.2). If the simple European security V m is hedgeable, then there is a portfolio process whose self- financing value process X 0 ;X 1 ;::: ;X m satisfies X m = V m . By definition, X k is the APT value at time k of V m . Theorem 3.12 says that X 0 ; 1 + r ,1 X 1 ;::: ;1 + r ,m X m is a martingale, and so for each k , 1 + r ,k X k = f IE 1 + r ,m X m jF k = f IE1 + r ,m V m jF k : Therefore, X k =1+r k f IE1 + r ,m V m jF k : 3.5 The Binomial Model is Complete Can a simple European derivative securityalwaysbe hedged? It depends on themodel. If the answer is “yes”, the model is said to be complete. If the answer is “no”, the model is called incomplete. Theorem 5.14 The binomial model is complete. In particular, let V m be a simple European deriva- tive security, and set V k ! 1 ;::: ;! k = 1 + r k f IE 1 + r ,m V m jF k ! 1 ;::: ;! k ; (5.1) k ! 1 ;::: ;! k = V k+1 ! 1 ;::: ;! k ;H, V k+1 ! 1 ;::: ;! k ;T S k+1 ! 1 ;::: ;! k ;H, S k+1 ! 1 ;::: ;! k ;T : (5.2) Starting with initial wealth V 0 = f IE 1 + r ,m V m , the self-financing value of the portfolioprocess 0 ; 1 ;::: ; m,1 is the process V 0 ;V 1 ;::: ;V m . Proof: Let V 0 ;::: ;V m,1 and 0 ;::: ; m,1 be defined by (5.1) and (5.2). Set X 0 = V 0 and define the self-financing value of the portfolio process 0 ;::: ; m,1 by the recursive formula 3.2: X k+1 = k S k+1 +1+rX k , k S k : We need to show that X k = V k ; 8k 2f0;1;::: ;mg: (5.3) We proceed by induction. For k =0 , (5.3) holds by definition of X 0 . Assume that (5.3) holds for some value of k , i.e., for each fixed ! 1 ;::: ;! k ,wehave X k ! 1 ;::: ;! k =V k ! 1 ;::: ;! k : CHAPTER 3. Arbitrage Pricing 65 We need to show that X k+1 ! 1 ;::: ;! k ;H= V k+1 ! 1 ;::: ;! k ;H; X k+1 ! 1 ;::: ;! k ;T=V k+1 ! 1 ;::: ;! k ;T: We prove the first equality; the second can be shown similarly. Note first that f IE 1 + r ,k+1 V k+1 jF k = f IE f IE 1 + r ,m V m jF k+1 jF k = f IE 1 + r ,m V m jF k = 1 + r ,k V k In other words, f1 + r ,k V k g n k=0 is a martingale under f IP . In particular, V k ! 1 ;::: ;! k = f IE1 + r ,1 V k+1 jF k ! 1 ;::: ;! k = 1 1+r ~pV k+1 ! 1 ;::: ;! k ;H+ ~qV k+1 ! 1 ;::: ;! k ;T : Since ! 1 ;::: ;! k will be fixed for the rest of the proof, we simplify notationby suppressing these symbols. For example, we write the last equation as V k = 1 1+r ~pV k+1 H + ~qV k+1 T : We compute X k+1 H = k S k+1 H + 1 + rX k , k S k = k S k+1 H , 1 + rS k + 1 + rV k = V k+1 H , V k+1 T S k+1 H , S k+1 T S k+1 H , 1 + rS k +~pV k+1 H + ~qV k+1 T = V k+1 H , V k+1 T uS k , dS k uS k , 1 + rS k +~pV k+1 H + ~qV k+1 T = V k+1 H , V k+1 T u , 1 , r u , d +~pV k+1 H + ~qV k+1 T = V k+1 H , V k+1 T ~q +~pV k+1 H + ~qV k+1 T = V k+1 H : 66 . Chapter 3 Arbitrage Pricing 3.1 Binomial Pricing Return to the binomial pricing model Please see: Cox, Ross and Rubinstein,. $150 $0 Sell stock ,$200 ,$50 Pay off debt $50 $50 ,,,, , ,,,,, $0 $0 The arbitrage pricing theory (APT) value of the option at time 0 is V 0 =20 . Assumptions
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