Chapter 7 Jensen’s Inequality 7.1 Jensen’s Inequality for Conditional Expectations Lemma 1.23 If ' : IR!IR is convex and IE j'X j  1 ,then IE 'X jG  'IE X jG: For instance, if G = f; g;'x= x 2 : IEX 2  IEX 2 : Proof: Since ' is convex we can express it as follows (See Fig. 7.1): 'x= max h' h is linear hx: Now let hx=ax + b lie below ' . Then, IE 'X jG  IE aX + bjG  = aIE X jG+b = hIEXjG This implies IE 'X jG  max h' h is linear hIE X jG = 'IE X jG: 91 92 ϕ Figure 7.1: Expressing a convex function as a max over linear functions. Theorem 1.24 If fY k g n k=0 is a martingale and  is convex then f'Y k g n k=0 is a submartingale. Proof: IE 'Y k+1 jF k   'IE Y k+1 jF k  = 'Y k : 7.2 Optimal Exercise of an American Call This follows from Jensen’s inequality. Corollary 2.25 Given a convex function g :0;1!IR where g 0 = 0 . For instance, g x= x,K + is the payoff function for an American call. Assume that r  0 . Consider the American derivative security with payoff g S k  in period k . The value of this security is the same as the value of the simple European derivative security with final payoff g S n  , i.e., f IE 1 + r ,n g S n  = max  f IE 1 + r , g S   ; where the LHS is the European value and the RHS is the American value. In particular  = n is an optimal exercise time. Proof: Because g is convex, for all  2 0; 1 we have (see Fig. 7.2): g x = g x +1,:0  g x+1, :g 0 = g x: CHAPTER 7. Jensen’s Inequality 93 ( x, g(x))λλ ( x, g( x))λλ (x,g(x)) x Figure 7.2: Proof of Cor. 2.25 Therefore, g  1 1+r S k+1   1 1+r gS k+1  and f IE h 1 + r ,k+1 g S k+1 jF k i = 1 + r ,k f IE  1 1+r gS k+1 jF k   1 + r ,k f IE  g  1 1+ r S k+1  jF k   1 + r ,k g  f IE  1 1+ r S k+1 jF k  = 1 + r ,k g S k ; So f1 + r ,k g S k g n k=0 is a submartingale. Let  be a stopping time satisfying 0    n .The optional sampling theorem implies 1 + r , g S    f IE 1 + r ,n g S n jF   : Taking expectations, we obtain f IE 1 + r , g S    f IE  f IE 1 + r ,n g S n jF    = f IE 1 + r ,n g S n  : Therefore, the value of the American derivative security is max  f IE 1 + r , g S    f IE 1 + r ,n g S n  ; and this last expression is the value of the European derivative security. Of course, the LHS cannot be strictly less than the RHS above, since stopping at time n is always allowed, and we conclude that max  f IE 1 + r , g S   = f IE 1 + r ,n g S n  : 94 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 Figure 7.3: A three period binomial model. 7.3 Stopped Martingales Let fY k g n k=0 be a stochastic process and let  be a stopping time. We denote by fY k^ g n k=0 the stopped process Y k^ ! ! ;k=0;1;::: ;n: Example 7.1 (Stopped Process) Figure 7.3 shows our familiar 3-period binomial example. Define  ! =  1 if ! 1 = T; 2 if ! 1 = H: Then S 2^ !  ! = 8      : S 2 HH=16 if ! = HH; S 2 HT =4 if ! = HT; S 1 T =2 if ! = TH; S 1 T=2 if ! = TT: Theorem 3.26 A stopped martingale (or submartingale, or supermartingale) is still a martingale (or submartingale, or supermartingale respectively). Proof: Let fY k g n k=0 be a martingale, and  be a stopping time. Choose some k 2f0;1;::: ;ng . The set f  kg is in F k ,sotheset f  k +1g= f  kg c is also in F k . We compute IE h Y k+1^ jF k i = IE h I f kg Y  + I f k+1g Y k+1 jF k i = I f kg Y  + I f k+1g IE Y k+1 jF k  = I f kg Y  + I f k+1g Y k = Y k^ : CHAPTER 7. Jensen’s Inequality 95 96 . Chapter 7 Jensen’s Inequality 7.1 Jensen’s Inequality for Conditional Expectations Lemma 1.23 If '. 'Y k : 7.2 Optimal Exercise of an American Call This follows from Jensen’s inequality. Corollary 2.25 Given a convex function g :0;1!IR where g