Variation Spectrum Suppose ffl S(t) is a continuous function on [0;T], ffl N is a large integer. For n = 1;:::;N, set For p > 0, set vars;n(p) := S n

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1 Lecture 7: Bachelier Glenn Shafer Rutgers Business School April 1, 2002 ffl Variation Spectrum and Variation Exponent ffl Bachelier's Central Limit Theorem ffl Discrete Bachelier Hedging 1

2 Variation Spectrum Suppose ffl S(t) is a continuous function on [0;T], ffl N is a large integer. For n = 1;:::;N, set For p > 0, set vars;n(p) := S n := S NX n=1 n T : N j S n j p = N 1 X n=0 jds n j p : ffl vars;n(p) is the p-variation of S. ffl vars;n is the variation spectrum for S. Variation Exponent For N sufficiently large, j S n j < 1 for all n, and so ffl vars;n(p) decreases continuously in p, ffl lim p!0 vars;n(p) = N, ffl lim p!1 vars;n(p) = 0. Let's pretend that there is a sharply defined value of p where vars;n(p) goes from being very large to being very small. This value of p is S's variation exponent. We write vex S for S's variation exponent. 2

3 Intuitively, S's variation exponent, vex S, is the value of p where vars;n(p) = goes from large to small. NX n=1 j S n j p To make this rigorous, think about the behavior of vars;n(p) for fixed p as N goes to infinity. If there is a value p 0 such that ffl lim N!1 vars;n(p) = 1 for p < p 0, and ffl lim N!1 vars;n(p) = 0 for p > p 0, then p 0 is S's variation exponent. This works for ffl ordinary functions (vex = 1) and ffl paths of diffusion processes (vex = 2). But beware the gap between continuous theory and discrete reality! ffl If we had a continuous function S, then we could (maybe) define vex S by letting N! 1 for each fixed p. ffl When we have only S 0 ;S 1 ;:::;S N for fixed N (always), there is no well-defined point where vars;n(p) transitions from large to small. 3

4 Intuitively, S's variation exponent, vex S, is the value of p where vars;n(p) = goes from large to small. N 1 X n=0 jds n j p Two reasons why the variation exponent is theoretically interesting: ffl It is related to the notion of the order of magnitude of ds relative to dt. ffl It is related to one notion of fractal dimension. Here we venture onto Benoit Mandelbrot's turf. Hölder's Exponent Suppose ds n has the order of magnitude (1=N) H. Then vars;n(p) has the same order of magnitude as N 1 X 1 Hp 1 Hp = N = N 1 Hp : N N n=0 This is large for p < 1=H ffl and small for p > 1=H + ffl. So H, which Mandelbrot calls the Hölder exponent, is the same as 1= vex. Fractal Dimension: See Figure 9.5 on p

5 Mandelbrot's concept of box dimension The box dimension of an object in the plane is the power to which we should raise 1=dt to get, to the right order of magnitude, the number of dt dt boxes required to cover it. ffl In the case of an object of area A, about A=(dt) 2 boxes are required, so the box dimension is 2. ffl In the case of a smooth curve of length T, T=dt boxes are required, so the box dimension is 1. ffl In the case of the graph of a function on [0;T] with Hölder exponent H, we must cover a vertical distance (dt) H above the typical increment dt on the horizontal axis, which requires (dt) H =dt boxes. So the order of magnitude of the number of boxes needed for all T=dt increments is (dt) H 2 ; the box dimension is 2 H. dim = 2 H vex = 1=H 5

6 Typical values for our three related measures of the jaggedness of a continuous real-valued function S. The figures for a substochastic function are reported for price series by Mandelbrot. H vex dim definition 1=H 2 H range [0; 1] [1; 1] [1; 2] smooth function substochastic diffusion A sample path for ordinary Brownian motion A sample path for fractional Brownian motion with H = 0:6 One of Mandelbrot's several alternatives to geometric Brownian motion is fractional Brownian motion with a larger Hölder exponent. This entails long-range correlation. 6

7 ffl Does it make sense to model stock prices, a la Mandelbrot, using a stochastic process with Hölder exponent slightly larger than 1 (variation exponent 2 slightly smaller than 2)? ffl Does it make sense, with finite N, to try to identify so precisely the value p 0 where var(p) goes from large to small? NO. It is completely arbitrary to say where a decreasing function of p (going from N to 0 as p goes from 0 to 1) becomes small". Microsoft and the S&P 500 for 600 trading days starting January 1, Microsoft unit var(2) var(2:5) dollar 742 1; 100 median ($48:91) 0:310 0:066 initial ($22:44) 1:473 0:463 final ($86:06) 0:100 0:016 S&P 500 unit var(2) var(2:5) index point 37; ; 000 median (775) initial (621) 0:063 0:098 0:010 0:017 final (1124) 0:030 0:004 7

8 So what sense can we make of a variation exponent (or Hölder exponent or box dimension) for finite N? Remember the rigorous definition for continuous functions: If there is a value p 0 such that ffl lim N!1 vars;n(p) = 1 for p < p 0, and ffl lim N!1 vars;n(p) = 0 for p > p 0, then p 0 is S's variation exponent. This suggests a practical meaning for theoretical statements such as vex» 2. ffl Not var(p) is small for p larger than 2." ffl Rather for a fixed positive number ffl, we can make var(2 + ffl) as small as we want by making N sufficiently large (i.e., making dt sufficiently small)." This restricts the wildness of a price series without claiming that it is a realization of a stochastic process. 8

9 In order to understand our book's ideas about pricing derivatives, the reader must abandon the notion that progress can be made by finding a better" stochastic model. The issue is not what the variation exponent really is in the limit, because there is no limit. Perhaps at most daily rebalancing (or less) is feasible. vex» 2 means for a fixed positive number ffl, we can make var(2 + ffl) as small as we want by making dt sufficiently small." Maybe dt = 1 day is small enough for the ffl we need. Maybe not. This needs to be examined empirically. Asking for a true stochastic process (an exact variation exponent) is like asking what would happen in a different game, one in which rebalancing can be performed every millisecond rather than every day. What players would do in a different game may have little relevance to what the players in our game do. 9

10 Bachelier's Central Limit Theorem Parameters: N, C 1 Protocol: Market announces S n 2 R and D 0 > 0. K 0 = R 1 1 U(z) N S0 ;D 0 (dz). FOR n = 1;:::;N: Investor announces M n 2 R and V n 2 R. Market announces S n 2 R and D n 0. K n := K n 1 + M n x n + V n (( S n ) 2 + D n ). Additional Constraints on Market: ffl D n > 0 for n = 1;:::;N 1 ffl D N = 0 ffl j D n j» N C for all n ffl j S n j» p C N for all n Investor's Goal: K N U(S N ) ffl ffl j D n j» C N for all n implies var D;N(1)» C. ffl j S n j» C p N for all n implies vars;n(2)» C 2. 10

11 Review: Proof of the game-theoretic form of De Moivre's theorem R We want a martingale V starting at U(z) N0;1(dz) and ending up with U(S N )» V N + ffl. On round n + 1, Skeptic knows x 1 ;:::;x n. So he buys (@U=@s)(S n ;D n ) tickets, where with S n := S n = U(s; D) := nx x i ; D n := 1 n N ; i=1 Z 1 U(z) Ns;D(dz): 1 du(s n ;D n ) + 2 U (S n;d n )ds (S n;d n )dd 2 (S0 n ;D0 n)(ds n ) 2 2 (S0 n ;D0 n)ds n dd n + 1 2(S0 n ;D0 n)(dd n ) 2 ; for n = 0;:::;N 1, where (Sn;D 0 n) 0 is a point strictly between (S n ;D n ) and (S n+1 ;D n+1 ). Notation: A n := A n A n 1 and da n := A n+1 A n. 11

12 Now 2 U=@s (S0 n;d 0 n) 2 (S n;d n ) 3 (S00 n;d 00 n)ds 0 n 2(S00 n;dn)dd 00 n; 0 where (Sn;D 00 n) 00 is a point strictly between (S n ;D n ) and (Sn 0 ;D0 n), and dsn 0 and ddn 0 satisfy jdsnj 0» jds n j, jddnj 0» jdd n j. Plugging this and the heat equation, into our expansion of du(s n ;D n ), we obtain du(s n ;D n (S n;d n )ds n + 3 U 3 (S00 n;dn)ds 00 n(ds 0 n ) (S n;d n ) (ds n ) 2 + dd 3 2(S00 n;dn)dd 00 n(ds 0 n ) 2 2(S0 n;dn)(dd 0 n ) 2 (S0 n;dn)ds 0 n dd n The first term on the right-hand side is the increment of our martingale V: the gain (S n;d n ) tickets on round n+1. So we only need to show that the other terms are negligible when N is sufficiently large. 12

13 du(s n ;D n (S n;d n )ds n + 3 U 3 (S00 n ;D00 n)dsn(ds 0 n ) (S n;d n ) (ds n ) 2 + dd 3 2(S00 n ;D00 n)dd 0 n(ds n ) 2 2(S0 n ;D0 n)(dd n ) 2 (S0 n ;D0 n)ds n dd n The second term is identically zero, because (ds n ) 2 = x 2 = n+1 1=N = dd n). All the partial derivatives involved in the last four terms are bounded: the heat equation 3 2 = ; 3 2 = = 1 2 = U=@s 3 4 U=@s 4, being averages of U (3) and U (4), are bounded. So the four terms will have at most the order of magnitude O(N 3=2 ), and their total cumulative contribution be at most O(N 1=2 ). 13

14 Proof that Investor has a winning strategy in the Bachelier game when N is sufficiently large For some c, ju (3) (s)j» c and ju (4) (s)j» c for all s 2 R. Notice that K 0 = U(S 0 ;D 0 ). As in our proof of De Moivre's theorem, du(s n ;D n (S n;d n )ds n + 3 U 3 (S00 n ;D00 n)dsn(ds 0 n ) (S0 n ;D0 n)ds n dd n + 1 (S n;d n ) (ds n ) 2 + dd 3 2(S00 n ;D00 n)dd 0 n(ds n ) 2 2(S0 n ;D0 n)(dd n ) 2 : To make du(s n ;D n ) approximate di n, Investor sets So M (S n;d n ) and V (S n;d n ): jdu(s n ;D n ) di n j 2» c jds n j 3 + jdd n jjds n j 2 + jdd n jjds n j + jdd 2 n j» c C 1:5 N 1:5 + C 3 N 2 + C 2 N 1:5 + C 2 N = O N 3=2 : Summing over n = 0;:::;N 1, we obtain fi fi U(S N ;D N ) U(S 0 ;D 0 ) fi (I N I 0 ) fi = O N 1=2 : 14

15 ffl j D n j» CN 1 vex D < 2, and implies vard;n(1)» C and hence ffl j S n j» CN 1=2 implies vars;n(2)» C 2 and hence that vex S» 2. While still remaining within the unrealistic Bachelier setting, we now take a step towards more useful bounds. Discrete Bachelier Hedging Parameters: N, I 0, ffi 2 (0; 1) Players: Market, Investor Protocol: Market announces S 0 2 R and D 0 > 0. FOR n = 1;:::;N: Investor announces M n 2 R and V n 2 R. Market announces S n 2 R and D n 0. I n := I n 1 + M n S n + V n (( S n ) 2 + D n ). Additional Constraints on Market: Market must set D N = 0, D n > 0 for n < N, and must make S 0 ;:::;S N, D 0 ;:::;D N and vars(2+ ffl) < ffi and vard(2 ffl) < ffi: I hope to have time to say a little about Volodya's proof that this weaker constraint suffices (pp ). 15