The Itˆo Integral

Chapter 14 The It ˆ o Integral The following chapters deal with Stochastic Differential Equationsin Finance. References: 1. B. Oksendal, Stochastic Differential Equations, Springer-Verlag,1995 2. J. Hull, Options, Futures and other Derivative Securities, Prentice Hall, 1993. 14.1 Brownian Motion (See Fig. 13.3.) ; F ; P is given, always in the background, even when not explicitly mentioned. Brownian motion, B t; ! :0;1!IR , has the following properties: 1. B 0 = 0; Technically, IP f! ; B 0;!=0g=1 , 2. B t is a continuous function of t , 3. If 0=t 0 t 1 :::  t n , then the increments B t 1  , B t 0 ; ::: ; Bt n  , Bt n,1  are independent,normal, and IE B t k+1  , B t k =0; IEBt k+1  , B t k  2 = t k+1 , t k : 14.2 First Variation Quadratic variation is a measure of volatility. First we will consider first variation, FV f ,ofa function f t . 153 154 t t 1 2 t f(t) T Figure 14.1: Example function f t . For the function pictured in Fig. 14.1, the first variation over the interval 0;T is given by: FV 0;T  f =ft 1 ,f0 , f t 2  , f t 1  + f T  , f t 2  = t 1 Z 0 f 0 t dt + t 2 Z t 1 ,f 0 t dt + T Z t 2 f 0 t dt: = T Z 0 jf 0 tj dt: Thus, first variation measures the total amount of up and down motion of the path. The general definition of first variation is as follows: Definition 14.1 (First Variation) Let =ft 0 ;t 1 ;::: ;t n g be a partition of 0;T , i.e., 0=t 0 t 1 :::  t n = T: The mesh of the partition is defined to be jjjj = max k=0;::: ;n,1 t k+1 , t k : We then define FV 0;T  f = lim jjjj!0 n,1 X k=0 jf t k+1  , f t k j: Suppose f is differentiable. Then the Mean Value Theorem implies that in each subinterval t k ;t k+1  , there is a point t  k such that f t k+1  , f t k =f 0 t  k t k+1 , t k : CHAPTER 14. The ItˆoIntegral 155 Then n,1 X k=0 jf t k+1  , f t k j = n,1 X k=0 jf 0 t  k jt k+1 , t k ; and FV 0;T  f = lim jjjj!0 n,1 X k=0 jf 0 t  k jt k+1 , t k  = T Z 0 jf 0 tj dt: 14.3 Quadratic Variation Definition 14.2 (Quadratic Variation) Thequadraticvariationof a function f on an interval 0;T is hf iT = lim jjjj!0 n,1 X k=0 jf t k+1  , f t k j 2 : Remark 14.1 (Quadratic Variation of Differentiable Functions) If f is differentiable, then hf iT = 0 , because n,1 X k=0 jf t k+1  , f t k j 2 = n,1 X k=0 jf 0 t  k j 2 t k+1 , t k  2 jjjj: n,1 X k=0 jf 0 t  k j 2 t k+1 , t k  and hf iT   lim jjjj!0 jjjj: lim jjjj!0 n,1 X k=0 jf 0 t  k j 2 t k+1 , t k  = lim jjjj!0 jjjj T Z 0 jf 0 tj 2 dt =0: Theorem 3.44 hB iT =T; or more precisely, IP f! 2 ; hB :; !iT = Tg =1: In particular, the paths of Brownian motion are not differentiable. 156 Proof: (Outline) Let =ft 0 ;t 1 ;::: ;t n g be a partition of 0;T . To simplify notation, set D k = B t k+1  , B t k  .Definethesample quadratic variation Q  = n,1 X k=0 D 2 k : Then Q  , T = n,1 X k=0 D 2 k , t k+1 , t k : We want to show that lim jjjj!0 Q  , T =0: Consider an individual summand D 2 k , t k+1 , t k =Bt k+1  , B t k  2 , t k+1 , t k : This has expectation 0, so IE Q  , T =IE n,1 X k=0 D 2 k , t k+1 , t k  = 0: For j 6= k ,theterms D 2 j , t j +1 , t j  and D 2 k , t k+1 , t k  are independent, so varQ  , T = n,1 X k=0 varD 2 k , t k+1 , t k  = n,1 X k=0 IE D 4 k , 2t k+1 , t k D 2 k +t k+1 , t k  2  = n,1 X k=0 3t k+1 , t k  2 , 2t k+1 , t k  2 +t k+1 , t k  2  (if X is normal with mean 0 and variance  2 ,then IE X 4 =3 4 ) =2 n,1 X k=0 t k+1 , t k  2  2jjjj n,1 X k=0 t k+1 , t k  =2jjjj T: Thus we have IE Q  , T =0; varQ  , T   2jjjj:T : CHAPTER 14. The ItˆoIntegral 157 As jjjj!0 , varQ  , T !0 ,so lim jjjj!0 Q  , T =0: Remark 14.2 (Differential Representation) We know that IE B t k+1  , B t k  2 , t k+1 , t k  = 0: We showed above that varB t k+1  , B t k  2 , t k+1 , t k  = 2t k+1 , t k  2 : When t k+1 , t k  is small, t k+1 , t k  2 is very small, and we have the approximate equation B t k+1  , B t k  2 ' t k+1 , t k ; which we can write informally as dB t dB t=dt: 14.4 Quadratic Variation as Absolute Volatility On any time interval T 1 ;T 2  , we can sample the Brownian motion at times T 1 = t 0  t 1  :::  t n = T 2 and compute the squared sample absolute volatility 1 T 2 , T 1 n,1 X k=0 B t k+1  , B t k  2 : This is approximately equal to 1 T 2 , T 1 hB iT 2  ,hBiT 1  = T 2 , T 1 T 2 , T 1 =1: As we increase the number of sample points, this approximation becomes exact. In other words, Brownian motion has absolute volatility 1. Furthermore, consider the equation hB iT =T = T Z 0 1dt; 8T  0: This says that quadratic variation for Brownian motion accumulates at rate 1 at all times along almost every path. 158 14.5 Construction of the It ˆ o Integral The integrator is Brownian motion B t;t  0 , with associated filtration F t;t  0 ,andthe following properties: 1. s  t= every set in F s is also in F t , 2. B t is F t -measurable, 8t , 3. For t  t 1  :::  t n , the increments B t 1  , B t;Bt 2  , Bt 1 ;::: ;Bt n  , Bt n,1  are independent of F t . The integrand is  t;t  0 ,where 1.  t is F t -measurable 8t (i.e.,  is adapted) 2.  is square-integrable: IE T Z 0  2 t dt  1; 8T: We want to define the It ˆ o Integral: I t= t Z 0 udB u; t  0: Remark 14.3 (Integral w.r.t. a differentiable function) If f t is a differentiable function, then we can define t Z 0  u df u= Z t 0 uf 0 udu: This won’t work when the integrator is Brownian motion, because the paths of Brownian motion are not differentiable. 14.6 It ˆ o integral of an elementary integrand Let =ft 0 ;t 1 ;::: ;t n g be a partition of 0;T , i.e., 0=t 0 t 1 :::  t n = T: Assume that  t is constant on each subinterval t k ;t k+1  (see Fig. 14.2). We call such a  an elementary process. The functions B t and  t k  can be interpreted as follows:  Think of B t as the price per unit share of an asset at time t . CHAPTER 14. The ItˆoIntegral 159 t )δ( t )δ( δ( ) δ( t ) = t ) tδ( 0=t 0 t t t = T2 34 t 1 0 δ( t ) = 1 δ( t ) = 2 δ( t ) = 3 Figure 14.2: An elementary function  .  Think of t 0 ;t 1 ;::: ;t n as the trading dates for the asset.  Think of  t k  as the number of shares of the asset acquired at trading date t k and held until trading date t k+1 . Then the Itˆointegral I t can be interpreted as the gain from trading at time t ; this gain is given by: I t= 8          : t 0 B t , B t 0  | z  =B 0=0 ; 0  t  t 1  t 0 B t 1  , B t 0  +  t 1 B t , B t 1 ; t 1  t  t 2  t 0 B t 1  , B t 0  +  t 1 B t 2  , B t 1  +  t 2 B t , B t 2 ; t 2  t  t 3 : In general, if t k  t  t k+1 , I t= k,1 X j=0  t j B t j +1  , B t j  +  t k B t , B t k : 14.7 Properties of the It ˆ o integral of an elementary process Adaptedness For each t; I t is F t -measurable. Linearity If I t= t Z 0 udB u; J t= t Z 0 udB u then I t  J t= Z t 0 uu dB u 160 t tt t l+1 l k k+1 s t . . . . . Figure 14.3: Showing s and t in different partitions. and cI t= Z t 0 c udB u: Martingale I t is a martingale. We prove the martingale property for the elementary process case. Theorem 7.45 (Martingale Property) I t= k,1 X j=0  t j B t j +1  , B t j  +  t k B t , B t k ; t k  t  t k+1 is a martingale. Proof: Let 0  s  t be given. We treat the more difficult case that s and t are in different subintervals, i.e., there are partition points t ` and t k such that s 2 t ` ;t `+1  and t 2 t k ;t k+1  (See Fig. 14.3). Write I t= `,1 X j=0  t j B t j +1  , B t j  +  t ` B t `+1  , B t `  + k,1 X j =`+1  t j B t j +1  , B t j  +  t k B t , B t k  We compute conditional expectations: IE 2 4 `,1 X j =0  t j B t j +1  , B t j      F s 3 5 = `,1 X j =0  t j B t j +1  , B t j : IE   t ` B t `+1  , B t `      F s  =  t ` IEBt `+1 jF s , B t `  =  t ` B s , B t `  CHAPTER 14. The ItˆoIntegral 161 These first two terms add up to I s . We show that the third and fourth terms are zero. IE 2 4 k,1 X j =`+1  t j B t j +1  , B t j      F s 3 5 = k,1 X j =`+1 IE  IE   t j B t j +1  , B t j      F t j       F s  = k,1 X j =`+1 IE 2 6 4  t j IEBt j+1 jF t j  , B t j  | z  =0     F s 3 7 5 IE   t k B t , B t k      F s  = IE 2 6 4  t k IEBtjF t k  , B t k  | z  =0     F s 3 7 5 Theorem 7.46 (It ˆ o Isometry) IEI 2 t=IE Z t 0  2 udu: Proof: To simplify notation, assume t = t k ,so I t= k X j=0  t j B t j +1  , B t j  | z  D j  Each D j has expectation 0, and different D j are independent. I 2 t= 0 @ k X j=0  t j D j 1 A 2 = k X j =0  2 t j D 2 j +2 X ij  t i  t j D i D j : Since the cross terms have expectation zero, IEI 2 t= k X j=0 IE  2 t j D 2 j  = k X j =0 IE   2 t j IE  B t j +1  , B t j  2     F t j   = k X j =0 IE 2 t j t j +1 , t j  = IE k X j =0 t j+1 Z t j  2 u du = IE Z t 0  2 u du 162 0=t 0 t t t = T2 34 t 1 path of path of δ δ 4 Figure 14.4: Approximating a general process by an elementary process  4 , over 0;T . 14.8 It ˆ o integral of a general integrand Fix T0 .Let  be a process (not necessarily an elementary process) such that   t is F t -measurable, 8t 2 0;T ,  IE R T 0  2 t dt  1: Theorem 8.47 There is a sequence of elementary processes f n g 1 n=1 such that lim n!1 IE Z T 0 j n t ,  tj 2 dt =0: Proof: Fig. 14.4 shows the main idea. In the last section we have defined I n T = Z T 0  n tdB t for every n .Wenowdefine Z T 0  t dB t= lim n!1 Z T 0  n t dB t: [...]... have ZT 0 1 B udB u = 1 B 2T  , 2 T: 2 The extra term 1 T comes from the nonzero quadratic variation of Brownian motion It has to be 2 there, because Z IE but T 0 B u dB u = 0 (Itˆ integral is a martingale) o 1 IE 1 B 2 T  = 2 T: 2 14.10 Quadratic variation of an Itˆ integral o Theorem 10.48 (Quadratic variation of Itˆ integral) Let o I t = Then Zt u dB u: 0 hI it = Zt 0 2u du:... it = Zt 0 2u du: 166 This holds even if is not an elementary process The quadratic variation formula says that at each time u, the instantaneous absolute volatility of I is 2 u This is the absolute volatility of the Brownian motion scaled by the size of the position (i.e t) in the Brownian motion Informally, we can write the quadratic variation formula in differential form as follows: dI t... 14 The Itˆ Integral o Therefore, X n,1 k=0 165 2 Bk Bk+1 , Bk  = 1 Bn , 1 2 2 X n,1 Bk+1 , Bk 2 ; k=0 or equivalently n,  X kT  k + 1T kT  X k + 1T k  B n B , B n = B T , B n n T : n,1 1 1 2 k=0 2 1 2 2 k=0 Let n!1 and use the definition of quadratic variation to get ZT 0 Bu dBu = 1 B 2 T , 1 T: 2 2 Remark 14.4 (Reason for the 1 T term) If f is differentiable with f 0 = 0, then... udB u: I t is a martingale Continuity I t is a continuous function of the upper limit of integration t Rt Itˆ Isometry IEI 2t = IE 0 2 u du o Martingale Example 14.1 () Consider the Itˆ integral o ZT 0 Bu dBu: We approximate the integrand as shown in Fig 14.5 164 2T/4 T/4 T 3T/4 Figure 14.5: Approximating the integrand B u with 4 , over 8 B0 = 0 BT=n n u = : : : :B n,1T ...CHAPTER 14 The Itˆ Integral o 163 The only difficulty with this approach is that we need to make sure the above limit exists Suppose n and m are large positive integers Then varIn T  , Im T  = IE (Itˆ Isometry:) = IE o Z T 0 T Z n t , m t n t , m t 0 ZT 2 !2 dB t dt j nt... t , tj2 dt + 2IE j m t , tj2 dt; = IE 0 0 0 which is small This guarantees that the sequence fIn T g1 has a limit n=1 14.9 Properties of the (general) Itˆ integral o I t = Zt u dB u: 0 Here is any adapted, square-integrable process Adaptedness For each t, I t is F t-measurable Linearity If I t = then Zt 0 u dB u; I t  J t = and Zt cI t = 0 J t = Zt 0 u dB u  u... dt: Proof: (For an elementary process ) Let  = ft0 ; t1; : : : ; tn g be the partition for , i.e., tk  for tk  t  tk+1 To simplify notation, assume t = tn We have hI it = n,1 X hI itk+1 , hI itk : k=0 Let us compute hI itk+1 , hI itk  Let  = fs0; s1 ; : : : ; sm g be a partition tk = s0  s1  : : :  sm = tk+1 : Then I sj+1  , I sj  = sZ+1 j sj tk  dB u = tk  B sj +1  , . trading date t k and held until trading date t k+1 . Then the Itˆointegral I t can be interpreted as the gain from trading at time t ; this gain is given.  t ` B s , B t `  CHAPTER 14. The ItˆoIntegral 161 These first two terms add up to I s . We show that the third and fourth terms are zero. IE