Pricing in terms ofMarket Probabilities_ The Radon-Nikodym Theorem

Chapter 9 Pricing in terms of Market Probabilities: The Radon-Nikodym Theorem. 9.1 Radon-Nikodym Theorem Theorem 1.27 (Radon-Nikodym) Let IP and f IP be two probability measures on a space ; F  . Assume that for every A 2F satisfying IP A=0 , we also have f IP A=0 . Then we say that f IP is absolutely continuous with respect to IP. Under this assumption, there is a nonegative random variable Z such that f IP A= Z A ZdIP; 8A 2F; (1.1) and Z is called the Radon-Nikodym derivative of f IP with respect to IP. Remark 9.1 Equation (1.1) implies the apparently stronger condition f IEX = IEXZ for every random variable X for which IE jXZj  1 . Remark 9.2 If f IP is absolutely continuous with respect to IP, and IP is absolutely continuous with respect to f IP , we say that IPand f IP are equivalent. IPand f IP are equivalent if and only if IP A=0 exactly when f IP A=0; 8A2F: If IPand f IP are equivalent and Z is the Radon-Nikodym derivative of f IP w.r.t. IP, then 1 Z is the Radon-Nikodym derivative of IP w.r.t. f IP , i.e., f IEX = IEXZ 8X; (1.2) IEY = f IEY: 1 Z  8Y: (1.3) (Let X and Y be related by the equation Y = XZ to see that (1.2) and (1.3) are the same.) 111 112 Example 9.1 (Radon-Nikodym Theorem) Let =fHH; HT; T H; T T g , the set of coin toss sequences of length 2. Let P correspond to probability 1 3 for H and 2 3 for T ,andlet e IP correspond to probability 1 2 for H and 1 2 for T .Then Z ! = e IP! IP! ,so Z HH= 9 4 ;ZHT = 9 8 ;ZTH= 9 8 ;ZTT= 9 16 : 9.2 Radon-Nikodym Martingales Let  be the set of all sequences of n coin tosses. Let IP be the market probability measure and let f IP be the risk-neutral probability measure. Assume IP !   0; f IP !  0; 8! 2 ; so that IPand f IP are equivalent. The Radon-Nikodym derivative of f IP with respect to IPis Z ! = f IP! IP! : Define the IP-martingale Z k 4 = IE Z jF k ;k=0;1;::: ;n: We can check that Z k is indeed a martingale: IE Z k+1 jF k  = IE IE Z jF k+1 jF k  = IE Z jF k  = Z k : Lemma 2.28 If X is F k -measurable, then f IEX = IEXZ k  . Proof: f IEX = IEXZ = IE IEXZjF k  = IE X:IE ZjF k  = IE XZ k : Note that Lemma 2.28 implies that if X is F k -measurable, then for any A 2F k , f IEI A X=IEZ k I A X; or equivalently, Z A Xd f IP = Z A XZ k dIP: CHAPTER 9. Pricing in terms of Market Probabilities 113 0 1 1 2 2 2 2 Z = 1 Z (H) = 3/2 Z (T) = 3/4 Z (HH) = 9/4 Z (HT) = 9/8 Z (TH) = 9/8 Z (TT) = 9/16 2/3 1/3 1/3 2/3 1/3 2/3 Figure 9.1: Showing the Z k values in the2-periodbinomialmodel example. The probabilitiesshown are for IP, not f IP . Lemma 2.29 If X is F k -measurable and 0  j  k ,then f IE X jF j = 1 Z j IEXZ k jF j : Proof: Note first that 1 Z j IE XZ k jF j  is F j -measurable. So for any A 2F j ,wehave Z A 1 Z j IE XZ k jF j d f IP = Z A IE XZ k jF j dIP (Lemma 2.28) = Z A XZ k dIP (Partial averaging) = Z A Xd f IP (Lemma 2.28) Example 9.2 (Radon-Nikodym Theorem, continued) We show in Fig. 9.1 the values of the martingale Z k . We always have Z 0 =1 ,since Z 0 = IEZ = Z  ZdIP = e IP =1: 9.3 The State Price Density Process In order to express the value of a derivative security in terms of the market probabilities, it will be useful to introduce the following state price density process:  k =1+r ,k Z k ;k=0;::: ;n: 114 We then have the following pricing formulas: For a Simple European derivative security with payoff C k at time k , V 0 = f IE h 1 + r ,k C k i = IE h 1 + r ,k Z k C k i (Lemma 2.28) = IE  k C k : More generally for 0  j  k , V j = 1 + r j f IE h 1 + r ,k C k jF j i = 1 + r j Z j IE h 1 + r ,k Z k C k jF j i (Lemma 2.29) = 1  j IE  k C k jF j  Remark 9.3 f j V j g k j =0 is a martingale under IP, as we can check below: IE  j +1 V j +1 jF j  = IE IE  k C k jF j +1 jF j  = IE  k C k jF j  =  j V j : Now for an American derivative security fG k g n k=0 : V 0 = sup  2T 0 f IE 1 + r , G   = sup  2T 0 IE 1 + r , Z  G   = sup  2T 0 IE   G  : More generally for 0  j  n , V j = 1 + r j sup  2T j f IE 1 + r , G  jF j  = 1 + r j sup  2T j 1 Z j IE 1 + r , Z  G  jF j  = 1  j sup  2T j IE   G  jF j : Remark 9.4 Note that (a) f j V j g n j =0 is a supermartingale under IP, (b)  j V j   j G j 8j; CHAPTER 9. Pricing in terms of Market Probabilities 115 S = 4 0 S (H) = 8 S (T) = 2 S (HH) = 16 S (TT) = 1 S (HT) = 4 S (TH) = 4 1 1 2 2 2 2 ζ = 1.00 ζ (Η) = 1.20 ζ (Τ) = 0.6 ζ (ΗΗ) = 1.44 ζ (ΗΤ) = 0.72 ζ (ΤΗ) = 0.72 ζ (ΤΤ) = 0.36 0 1 1 2 2 2 2 1/3 2/3 1/3 2/3 1/3 2/3 Figure 9.2: Showing the state price values  k . The probabilities shown are for IP, not f IP . (c) f j V j g n j =0 is the smallest process having properties (a) and (b). We interpret  k by observing that  k ! IP !  is the value at time zero of a contract which pays $1 at time k if ! occurs. Example 9.3 (Radon-NikodymTheorem, continued) We illustrate the use of the valuation formulas for European and American derivative securities in terms of market probabilities. Recall that p = 1 3 , q = 2 3 .The state price values  k are shown in Fig. 9.2. For a European Call with strike price 5, expiration time 2, we have V 2 HH=11; 2 HHV 2 HH=1:44  11 = 15:84: V 2 HT =V 2 TH=V 2 TT=0: V 0 = 1 3  1 3 15:84 = 1:76:  2 HH  1 HH V 2 HH= 1:44 1:20  11 = 1:20  11 = 13:20 V 1 H = 1 3 13:20 = 4:40 Compare with the risk-neutral pricing formulas: V 1 H = 2 5 V 1 HH+ 2 5 V 1 HT = 2 5 11 = 4:40; V 1 T = 2 5 V 1 TH+ 2 5 V 1 TT=0; V 0 = 2 5 V 1 H+ 2 5 V 1 T= 2 5 4:40 = 1:76: Now consider an American put with strike price 5 and expiration time 2. Fig. 9.3 shows the values of  k 5 , S k  + . We compute the value of the put under various stopping times  : (0) Stop immediately: value is 1. (1) If  HH=HT =2;TH=TT=1 ,thevalueis 1 3  2 3  0:72 + 2 3  1:80 = 1:36: 116 (5-S 0 ) + =1ζ 0 (5-S 0 ) + =1 (5 - S 1 (H)) + = 0 (H)ζ 1 (5 - S + (HH)) = 0 2 (5 - S + (HH)) = 0 2 ζ 2 (HH) 1/3 2/3 1/3 2/3 1/3 2/3 ζ 1 (5 - S 1 + (5 - S 1 + (T)) (T)) (T) = 3 = 1.80 (5 - S 1 (H)) + = 0 (5 - S + 2 (5 - S + 2 ζ 2 (5 - S + 2 (5 - S + 2 ζ 2 (5 - S + 2 (5 - S + 2 ζ 2 (HT)) (HT) (HT)) = 1 = 0.72 (TH)) (TH) (TH)) = 1 = 0.72 (TT)) (TT) (TT)) = 4 = 1.44 Figure 9.3: Showing the values  k 5 , S k  + for an American put. The probabilities shown are for IP, not f IP . (2) If we stop at time 2, the value is 1 3  2 3  0:72 + 2 3  1 3  0:72 + 2 3  2 3  1:44 = 0:96 We see that (1) is optimal stopping rule. 9.4 Stochastic Volatility Binomial Model Let  be the set of sequences of n tosses, and let 0 d k 1+r k u k , where for each k , d k ;u k ;r k are F k -measurable. Also let ~p k = 1+r k ,d k u k ,d k ; ~q k = u k ,1 + r k  u k , d k : Let f IP be the risk-neutral probability measure: f IP f! 1 = H g =~p 0 ; f IPf! 1 =Tg=~q 0 ; and for 2  k  n , f IP ! k+1 = H jF k = ~p k ; f IP! k+1 = T jF k = ~q k : Let IP be the market probability measure, and assume IP f! g  0 8! 2  .ThenIPand f IP are equivalent. Define Z ! = f IP! IP! 8! 2; CHAPTER 9. Pricing in terms of Market Probabilities 117 Z k = IE Z jF k ;k=0;1;::: ;n: We define the money market price process as follows: M 0 =1; M k =1+r k,1 M k,1 ;k=1;::: ;n: Note that M k is F k,1 -measurable. We then define the state price process to be  k = 1 M k Z k ;k=0;::: ;n: As before the portfolio process is f k g n,1 k=0 . The self-financing value process (wealth process) consists of X 0 , the non-random initial wealth, and X k+1 = k S k+1 +1+r k X k ,  k S k ;k=0;::: ;n , 1: Then the following processes are martingales under f IP :  1 M k S k  n k=0 and  1 M k X k  n k=0 ; and the following processes are martingales under IP: f k S k g n k=0 and f k X k g n k=0 : We thus have the following pricing formulas: Simple European derivative security with payoff C k at time k : V j = M j f IE  C k M k     F j  = 1  j IE  k C k jF j  American derivative security fG k g n k=0 : V j = M j sup  2T j f IE  G  M      F j  = 1  j sup  2T j IE   G  jF j  : The usual hedging portfolio formulas still work. 118 9.5 Another Applicaton of the Radon-Nikodym Theorem Let ; F ;Q be a probability space. Let G be a sub-  -algebra of F ,andlet X be a non-negative random variable with R  XdQ=1 . We construct the conditional expectation (under Q )of X given G .On G , define two probability measures IP A=QA 8A2G; f IPA= Z A XdQ 8A 2G: Whenever Y is a G -measurable random variable, we have Z  YdIP= Z  YdQ; if Y = 1 A for some A 2G , this is just the definition of IP , and the rest follows from the “standard machine”. If A 2G and IP A=0 ,then QA=0 ,so f IP A=0 . In other words, the measure f IP is absolutely continuous with respect to the measure f IP . The Radon-Nikodym theorem implies that there exists a G -measurable random variable Z such that f IP A 4 = Z A ZdIP 8A2G; i.e., Z A XdQ= Z A ZdIP 8A2G: This shows that Z has the “partial averaging” property, and since Z is G -measurable, it is the con- ditional expectation (under the probability measure Q )of X given G . The existence of conditional expectations is a consequence of the Radon-Nikodym theorem. . Chapter 9 Pricing in terms of Market Probabilities: The Radon-Nikodym Theorem. 9.1 Radon-Nikodym Theorem Theorem 1.27 (Radon-Nikodym) Let IP. (Partial averaging) = Z A Xd f IP (Lemma 2.28) Example 9.2 (Radon-Nikodym Theorem, continued) We show in Fig. 9.1 the values of the martingale Z k . We