Chapter 15 It ˆ o’s Formula 15.1 It ˆ o’s formula for one Brownian motion We want a rule to “differentiate” expressions of the form f B t ,where f x is a differentiable function. If B t were also differentiable, then the ordinary chain rule would give d dt f B t = f 0 B tB 0 t; which could be written in differential notation as df B t = f 0 B tB 0 t dt = f 0 B tdB t However, B t is not differentiable, and in particular has nonzero quadratic variation, so the correct formula has an extra term, namely, df B t = f 0 B t dB t+ 1 2 f 00 Bt dt |z dB t dB t : This is Itˆo’s formula in differential form. Integrating this, we obtain Itˆo’s formula in integral form: f B t , f B 0 | z  f 0 = Z t 0 f 0 B u dB u+ 1 2 Z t 0 f 00 B u du: Remark 15.1 (Differential vs. Integral Forms) The mathematically meaningful form of Itˆo’s for- mula is Itˆo’s formula in integral form: f B t , f B 0 = Z t 0 f 0 B u dB u+ 1 2 Z t 0 f 00 B u du: 167 168 This is because we have solid definitions for both integrals appearing on the right-hand side. The first, Z t 0 f 0 B u dB u is an Itˆointegral, defined in the previous chapter. The second, Z t 0 f 00 B u du; is a Riemann integral, the type used in freshman calculus. For paper and pencil computations, the more convenient form of Itˆo’s rule is Itˆo’s formula in differ- ential form: df B t = f 0 B t dB t+ 1 2 f 00 B t dt: There is an intuitive meaning but no solid definition for the terms df B t;dBt and dt appearing in this formula. This formula becomes mathematically respectable only after we integrate it. 15.2 Derivation of It ˆ o’s formula Consider f x= 1 2 x 2 ,sothat f 0 x=x; f 00 x=1: Let x k ;x k+1 be numbers. Taylor’s formula implies f x k+1  , f x k =x k+1 , x k f 0 x k + 1 2 x k+1 , x k  2 f 00 x k : In this case, Taylor’s formula to second order is exact because f is a quadratic function. In the general case, the above equation is only approximate, and the error is of the order of x k+1 , x k  3 . The total error will have limit zero in the last step of the following argument. Fix T0 and let =ft 0 ;t 1 ;::: ;t n g be a partition of 0;T . Using Taylor’s formula, we write: f B T  , f B 0 = 1 2 B 2 T  , 1 2 B 2 0 = n,1 X k=0 f B t k+1  , f B t k  = n,1 X k=0 B t k+1  , B t k  f 0 B t k  + 1 2 n,1 X k=0 B t k+1  , B t k  2 f 00 B t k  = n,1 X k=0 B t k Bt k+1  , B t k  + 1 2 n,1 X k=0 B t k+1  , B t k  2 : CHAPTER 15. Itˆo’s Formula 169 We let jjjj!0 to obtain f B T  , f B 0 = Z T 0 B u dB u+ 1 2 hBiT | z  T = Z T 0 f 0 B u dB u+ 1 2 Z T 0 f 00 B u | z  1 du: This is Itˆo’s formula in integral form for the special case f x= 1 2 x 2 : 15.3 Geometric Brownian motion Definition 15.1 (Geometric Brownian Motion) Geometric Brownian motion is S t=S0 exp n Bt+  , 1 2  2  t o ; where  and 0 are constant. Define f t; x=S0 exp n x +   , 1 2  2  t o ; so S t=ft; B t: Then f t =   , 1 2  2  f; f x = f; f xx =  2 f: According to Itˆo’s formula, dS t= dft; B t = f t dt + f x dB + 1 2 f xx dB dB | z  dt =, 1 2  2 fdt+f dB + 1 2  2 fdt =S tdt + St dB t Thus, Geometric Brownian motion in differential form is dS t= S tdt + St dB t; and Geometric Brownian motion in integral form is S t=S0 + Z t 0 S u du + Z t 0 Su dB u: 170 15.4 Quadratic variation of geometric Brownian motion In the integral form of Geometric Brownian motion, S t=S0 + Z t 0 S u du + Z t 0 S u dB u; the Riemann integral F t= Z t 0 S u du is differentiable with F 0 t=S t . This term has zero quadratic variation. The Itˆointegral Gt= Z t 0 Su dB u is not differentiable. It has quadratic variation hGit= Z t 0  2 S 2 udu: Thus the quadratic variation of S is given by the quadratic variation of G . In differential notation, we write dS t dS t= S tdt + StdB t 2 =  2 S 2 t dt 15.5 Volatility of Geometric Brownian motion Fix 0  T 1  T 2 .Let =ft 0 ;::: ;t n g be a partition of T 1 ;T 2  .Thesquared absolute sample volatility of S on T 1 ;T 2  is 1 T 2 , T 1 n,1 X k=0 S t k+1  , S t k  2 ' 1 T 2 , T 1 T 2 Z T 1  2 S 2 u du '  2 S 2 T 1  As T 2  T 1 , the above approximation becomes exact. In other words, the instantaneous relative volatility of S is  2 . This is usually called simply the volatility of S . 15.6 First derivation of the Black-Scholes formula Wealth of an investor. An investor begins with nonrandom initial wealth X 0 and at each time t , holds t shares of stock. Stock is modelled by a geometric Brownian motion: dS t= S tdt + S tdBt: CHAPTER 15. Itˆo’s Formula 171 t can be random, but must be adapted. The investor finances his investing by borrowing or lending at interest rate r . Let X t denote the wealth of the investor at time t .Then dX t= tdS t+rXt, tS t dt =tS tdt + StdB t + r X t , tS t dt = rX tdt +tSt,r | z  Risk premium dt +tStdB t: Value of an option. Consider an European option which pays g S T  at time T .Let v t; x denote the value of this option at time t if the stock price is S t=x . In other words, the value of the option at each time t 2 0;T is v t; S t: The differential of this value is dv t; S t = v t dt + v x dS + 1 2 v xx dS dS = v t dt + v x S dt + S dB+ 1 2 v xx  2 S 2 dt = h v t + S v x + 1 2  2 S 2 v xx i dt + Sv x dB A hedging portfolio starts with some initial wealth X 0 and invests so that the wealth X t at each time tracks v t; S t . We saw above that dX t= rX +,rS dt + SdB: To ensure that X t=vt; S t for all t , we equate coefficients in their differentials. Equating the dB coefficients, we obtain the  -hedging rule: t=v x t; S t: Equating the dt coefficients, we obtain: v t + S v x + 1 2  2 S 2 v xx = rX +,rS: Butwehaveset =v x , and we are seeking to cause X to agree with v . Making these substitutions, we obtain v t + S v x + 1 2  2 S 2 v xx = rv + v x  , rS; (where v = v t; S t and S = S t ) which simplifies to v t + rS v x + 1 2  2 S 2 v xx = rv: In conclusion, we should let v be the solution to the Black-Scholes partial differential equation v t t; x+rxv x t; x+ 1 2  2 x 2 v xx t; x=rv t; x satisfying the terminal condition v T; x= gx: If an investor starts with X 0 = v 0;S0 and uses the hedge t=v x t; S t , then he will have X t=vt; S t for all t , and in particular, X T = gST . 172 15.7 Mean and variance of the Cox-Ingersoll-Ross process The Cox-Ingersoll-Ross model for interest rates is drt= ab,crtdt +  q rt dB t; where a; b; c;  and r0 are positive constants. In integral form, this equation is rt= r0 + a Z t 0 b , cru du +  Z t 0 q ru dB u: We apply Itˆo’s formula to compute dr 2 t .Thisis df rt ,where f x=x 2 . We obtain dr 2 t=dfrt = f 0 rt drt+ 1 2 f 00 rt drt drt =2rt  ab,crt dt +  q rt dB t  +  ab , crt dt +  q rt dB t  2 =2abrt dt , 2acr 2 t dt +2r 3 2 t dB t+ 2 rt dt =2ab +  2 rt dt , 2acr 2 t dt +2r 3 2 t dB t The mean of rt . The integral form of the CIR equation is rt= r0 + a Z t 0 b , cru du +  Z t 0 q ru dB u: Taking expectations and remembering that the expectation of an Itˆo integral is zero, we obtain IErt= r0 + a Z t 0 b , cIEru du: Differentiation yields d dt IErt=ab,cIErt = ab , acIErt; which implies that d dt h e act IErt i = e act  acIErt+ d dt IErt  = e act ab: Integration yields e act IErt , r0 = ab Z t 0 e acu du = b c e act , 1: We solve for IErt : IErt= b c +e ,act  r0 , b c  : If r0 = b c ,then IErt= b c for every t .If r0 6= b c ,then rt exhibits mean reversion: lim t!1 IErt= b c : CHAPTER 15. Itˆo’s Formula 173 Variance of rt . The integral form of the equation derived earlier for dr 2 t is r 2 t=r 2 0+2ab +  2  Z t 0 ru du , 2ac Z t 0 r 2 u du +2 Z t 0 r 3 2 udB u: Taking expectations, we obtain IEr 2 t=r 2 0 + 2ab +  2  Z t 0 IEru du , 2ac Z t 0 IEr 2 u du: Differentiation yields d dt IEr 2 t=2ab +  2 IErt , 2acIEr 2 t; which implies that d dt e 2act IEr 2 t=e 2act  2acIEr 2 t+ d dt IEr 2 t  = e 2act 2ab +  2 IErt: Using the formula already derived for IErt and integrating the last equation, after considerable algebra we obtain IEr 2 t= b 2 2ac 2 + b 2 c 2 +  r0 , b c    2 ac + 2b c ! e ,act +  r0 , b c  2  2 ac e ,2act +  2 ac  b 2c , r0  e ,2act : var rt= IEr 2 t , IErt 2 = b 2 2ac 2 +  r0 , b c   2 ac e ,act +  2 ac  b 2c , r0  e ,2act : 15.8 Multidimensional Brownian Motion Definition 15.2 ( d -dimensional Brownian Motion) A d -dimensional Brownian Motion is a pro- cess B t=B 1 t;::: ;B d t with the following properties:  Each B k t is a one-dimensional Brownian motion;  If i 6= j , then the processes B i t and B j t are independent. Associated with a d -dimensional Brownian motion, we have a filtration fF tg such that  For each t , the random vector B t is F t -measurable;  For each t  t 1  :::  t n , the vector increments B t 1  , B t;::: ;Bt n  , Bt n,1  are independent of F t . 174 15.9 Cross-variations of Brownian motions Because each component B i is a one-dimensional Brownian motion, we have the informal equation dB i t dB i t=dt: However, we have: Theorem 9.49 If i 6= j , dB i t dB j t=0 Proof: Let =ft 0 ;::: ;t n g be a partition of 0;T .For i 6= j ,definethesample cross variation of B i and B j on 0;T to be C  = n,1 X k=0 B i t k+1  , B i t k  B j t k+1  , B j t k  : The increments appearing on the right-hand side of the above equation are all independent of one another and all have mean zero. Therefore, IEC  =0: We compute varC   . First note that C 2  = n,1 X k=0  B i t k+1  , B i t k   2  B j t k+1  , B j t k   2 +2 n,1 X `k B i t `+1  , B i t `  B j t `+1  , B j t `  : B i t k+1  , B i t k  B j t k+1  , B j t k  All the increments appearing in the sum of cross terms are independent of one another and have mean zero. Therefore, varC  =IEC 2  = IE n,1 X k=0 B i t k+1  , B i t k  2 B j t k+1  , B j t k  2 : But B i t k+1  , B i t k  2 and B j t k+1  , B j t k  2 are independent of one another, and each has expectation t k+1 , t k  . It follows that varC  = n,1 X k=0 t k+1 , t k  2 jjjj n,1 X k=0 t k+1 , t k =jjjj:T : As jjjj!0 ,wehave varC  !0 ,so C  converges to the constant IEC  =0 . CHAPTER 15. Itˆo’s Formula 175 15.10 Multi-dimensional It ˆ o formula To keep the notation as simple as possible, we write the Itˆoformulafortwo processes driven by a two-dimensional Brownian motion. The formula generalizes to any number of processes driven by a Brownian motion of any number (not necessarily the same number) of dimensions. Let X and Y be processes of the form X t=X0 + Z t 0 u du + Z t 0  11 u dB 1 u+ Z t 0  12 u dB 2 u; Y t=Y0 + Z t 0  u du + Z t 0  21 u dB 1 u+ Z t 0  22 u dB 2 u: Such processes, consisting of a nonrandom initial condition, plus a Riemann integral, plus one or more Itˆo integrals, are called semimartingales. The integrands u;u; and  ij u can be any adapted processes. The adaptedness of the integrands guarantees that X and Y are also adapted. In differential notation, we write dX = dt+ 11 dB 1 +  12 dB 2 ; dY = dt+ 21 dB 1 +  22 dB 2 : Given these two semimartingales X and Y , the quadratic and cross variations are: dX dX =dt+ 11 dB 1 +  12 dB 2  2 ; =  2 11 dB 1 dB 1 | z  dt +2 11  12 dB 1 dB 2 | z  0 + 2 12 dB 2 dB 2 | z  dt = 2 11 +  2 12  2 dt; dY dY =dt+ 21 dB 1 +  22 dB 2  2 = 2 21 +  2 22  2 dt; dX d Y =dt+ 11 dB 1 +  12 dB 2 dt+ 21 dB 1 +  22 dB 2  = 11  21 +  12  22  dt Let f t; x; y  be a function of three variables, and let X t and Y t be semimartingales. Then we have the corresponding Itˆoformula: df t; x; y  = f t dt + f x dX + f y dY + 1 2 f xx dX dX +2f xy dX d Y + f yy dY dY  : In integral form, with X and Y as decribed earlier and with all the variables filled in, this equation is f t; X t;Yt , f 0;X0;Y0 = Z t 0 f t + f x + f y + 1 2  2 11 +  2 12 f xx + 11  21 +  12  22 f xy + 1 2  2 21 +  2 22 f yy  du + Z t 0  11 f x +  21 f y  dB 1 + Z t 0  12 f x +  22 f y  dB 2 ; where f = f u; X u;Yu ,for i; j 2f1;2g ,  ij =  ij u ,and B i = B i u . 176 [...]... tk  = jjjj:T: As jjjj!0, we have varC !0, so C converges to the constant IEC = 0 CHAPTER 15 Itˆ ’s Formula o 175 15.10 Multi-dimensional Itˆ formula o To keep the notation as simple as possible, we write the Itˆ formula for two processes driven by a o two-dimensional Brownian motion The formula generalizes to any number of processes driven by a Brownian motion of any number (not necessarily... dt; dt + 11 dB1 + 12 dB2 11 21 + 12 22  dt dt dt + 21 dB1 + 22 dB2  Let f t; x; y  be a function of three variables, and let X t and Y t be semimartingales Then we have the corresponding Itˆ formula: o 1 df t; x; y = ft dt + fx dX + fy dY + 2 fxx dX dX + 2fxy dX dY + fyy dY dY : In integral form, with X and Y as decribed earlier and with all the variables filled in, this equation is f t; . =0 . CHAPTER 15. Itˆo’s Formula 175 15.10 Multi-dimensional It ˆ o formula To keep the notation as simple as possible, we write the Itˆoformulafortwo processes. (Differential vs. Integral Forms) The mathematically meaningful form of Itˆo’s for- mula is Itˆo’s formula in integral form: f B t , f B 0 = Z t 0 f 0 B