Math 212a - Problem set 3

Transcription

1 Mah 212a - Problem se 3 Convexiy, arbirage, and probabiliy Tuesday, epember 23, 2014, Due ep 30 The purpose of his problem se is o describe a formula in he pricing of opions which was involved in he big financial crisis of 1998 (which pales in comparison o he crisis of 2008 bu wih similar governmen bail-ou) I mean he infamous Black-choles formula In he course of doing so we will encouner he free marke based foundaions for probabiliy heory due o Bruno De Finei La prevision: ses lois iques, ses sources subjecives (1937) Annales de l Insiiu Henri Poincaré We will sar wih a beauiful and seemingly harmless heorem of Caraheodory A he end, I will append some hisorical commens by co Kominers o his problem se when he ook 212a in 2008 Horse racing erminoy: A be on a horse o place, means ha you are being ha your horse will finish firs or second If you be on a horse o show, means ha your are being ha your horse will finish firs, second or hird I will no allow such bes Conens 1 Farkas lemma 2 11 Caraheodory s heorem 2 12 The separaion heorem for closed convex bodies 3 13 Farkas lemma 4 2 De Finei s arbirage heorem 4 21 The price of a one period European call opion 5 22 Odds 7 23 A muliperiod sock model 7 3 The Black-choles formula 8 31 Warm up 8 32 The mean and variance of Y = e X, X normal 9 33 Deermining he drif from De Finei Black-choles 10 1

2 1 Farkas lemma 11 Caraheodory s heorem There are several varians of his heorem Here is he one we will use: Theorem 1 [Caraheodory] Le a 1,, a r be vecors in a real vecor space V Le x be a linear combinaion of he a i wih non-negaive coefficiens Then x is a linear combinaion wih non-negaive coefficiens of a linearly independen subse of he a i 1 Prove his heorem [Hin: Wrie x = k λ i x i, λ i > 0 i=1 where he x i range over a subse of cardinaliy k of he a i and we have chosen k minimal wih his propery If k = 0 hen x = 0 and here is nohing o prove Oherwise, we wish o show ha he x i are linearly independen uppose no Then here exis α i R no all zero wih α i x i = 0 i We may assume ha a leas one of he α i is posiive (Oherwise muliply hem all by 1) o α i > 0 for a leas one i, and choose m so ha λ m /α m is minimal among he posiive α i how ha hen we can wrie x as a combinaion wih non-negaive coefficiens of he x i, i m] The se of all linear combinaions of he a i wih non-negaive coefficiens is called he cone generaed by he a i Corollary 1 The cone generaed by a finie number of vecors in a real (opoical) vecor space is closed Proof By Caraheodory s heorem, his cone is he union of he cones generaed by he linearly independen subses, and here are finiely many of hese o i is enough o prove he corollary under he addiional assumpion ha he vecors in quesion are linearly independen The subspace spanned by hese vecors is closed (being finie dimensional) and we may idenify his subspace wih R k wih he vecors idenified wih he sandard basis If we do his, he cone becomes he closed (firs) orhan consising of all vecors wih non-negaive coordinaes ince he cone generaed by a finie number of vecors is convex, he corollary implies ha he cone generaed by a finie number of vecors is a closed convex se o if A is an m n marix, hen he se {Ax} x R n, x 0 is a closed convex subse of R m Here we are using he noaion x 0 for x R n o denoe he asserion ha all he coordinaes of x are non-negaive 2

3 12 The separaion heorem for closed convex bodies Le be a closed convex non-empy subse of a real Banach space V x V, x Then he separaion heorem assers ha Theorem 2 [eparaion] There exiss a coninuous linear funcion l on V and a real number c such ha l(y) c for all y and l(x) < c We will only need his heorem for finie dimensional vecor spaces so we will only prove i in his case We will herefore ake V = R n wih is sandard meric, and we will wrie vecors as x, y ec Lemma 1 For any x V here is a unique poin p(x, ) which is closes o x, ie such ha x p(x, ) x y y Indeed, if B(x, r) denoes he closed ball cenered a x hen B(x, r) is compac and is non-empy for large enough r The funcion y x y is coninuous, and hence aains a minimum on such a non-empy B(x, r), say a y 0, and clearly x y 0 x y y We mus show ha y 0 is unique uppose ha here is also a y 1 wih x y 1 x y y Then x, y 0, y 1 form an isosceles riangle wih verex a x and hence if we ake z = 1 2 (y 0 + y 1 ) o be he midpoin of he base hen z since is convex and x z < x y 0 unless y 1 = y 0 Now suppose ha x so p(x, ) x so ha we may form he uni vecor u(x, ) := 1 (x p(x, )) d(x, ) where d(x, ) denoes he disance from x o so d(x, ) = x p(x, ) Le H denoe he hyperplane hrough p(x, ) orhogonal o u(x, ) o H is defined by H = {z u(x, ) (x z) = u(x, ) (x p(x, )) = d(x, )} The hyperplane H is a suppor hyperplane in he sense ha y = p(x, ) H and lies enirely in one of he half spaces defined by H Indeed, le H denoe he closed halfspace bounded by H which does no conain x We claim ha is compleely conained in his halfspace Indeed, suppose ha here is some y no in H Consider he line segmen [p(x, ), y] and le z be he poin of his line segmen closes o x Then z and x z < x p(x, ) conradicing he definiion of p(x, ) This shows ha H is a suppor hyperplane and complees he proof of Theorem 2 Noice ha a suppor hyperplane for a cone is a codimension one subspace (passing hrough he origin)we will acually need he exisence of hese suppor hyperplanes Le 3

4 13 Farkas lemma This says ha for A an n m marix ( x R m, x 0, Ax = b) ( y R n (y A 0 y b 0)) (1) 2 Prove Farkas lemma [Hin: Use he resul of he preceding secion abou suppor hyperplanes for cones] The Farkas Lemma is someimes formulaed as an alernaive: Theorem 3 [The Farkas Lemma] Eiher x 0 wih Ax = b or y R n wih y A 0 and y b < 0 bu no boh 2 De Finei s arbirage heorem uppose ha here is an experimen having m possible oucomes for which here are n possible wagers Tha is, if you be he amoun y on wager i you win he amoun yr i (j) if he oucome of he experimen is j Here y can be posiive, zero, or negaive A being sraegy is a vecor y = (y 1,, y n ) which means ha you simulaneously be he amoun y i on wager i for i = 1,, n o if he oucome of he experimen is j, your gain (or loss) from he sraegy y is n y i r i (j) i=1 Theorem 4 [De Finei s arbirage heorem] Exacly one of he following is rue: Eiher here exiss a probabiliy vecor p = (p 1,, p m ) so p j 0 j, j p j = 1 such ha m p j r i (j) = 0 i = 1,, m, or j=1 here exiss a being sraegy y such ha n y i r i (j) > 0 j = 1, m i=1 In oher words eiher here exiss a probabiliy disribuion on he oucome under which all bes have expeced gain equal o zero, or else here is a being sraegy which always resuls in a posiive win 4

5 3 Prove De Finei s heorem [Hin: Consider he marix r 1 (1) r 1 (m) A := r n (1) r n (m) 1 1 and vecor Use Farkas] b := I is imporan o observe ha De Finei s heorem does no say ha p is unique Bu here are special circumsances in which uniqueness is obvious 21 The price of a one period European call opion uppose ha m = 2 and ha here is exacly one wager o he marix A is given by r(1) r(2) A = 1 1 If r(1) r(2) his marix is non-singular wih inverse A r(2) = r(2) r(1) 1 r(1) so A 1 b = 1 r(2) r(1) ( 1 r(2) 0 = 1 r(1) 1) 1 r(2) r(2) r(1) r(1) o if r(2) > 0 and r(1) < 0 boh enries are posiive and yield he unique probabiliy vecor 1 p 1 r(2) p = = p r(2) r(1) r(1) Of course, if boh r(1) and r(2) are posiive any wager which assigns a posiive be o boh is a guaraneed win, and if boh are negaive hen any wager which assigns a negaive be o boh is a guaraneed win Now suppose ha anoher wager is allowed uppose his be has he reurn a if 1 occurs and he reurn b if 2 occurs Then according o De Finei s heorem, unless a(1 p) + bp = 0, 5

6 here will some combinaion of he wo wagers ha has a guaraneed profi As an illusraion, suppose ha an asse (say a sock) has value (0) a he presen ime, and has only wo possible values a ime 1: Eiher (1) = u(0) or (1) = d(0) u > 1 > d In oher words, eiher he sock can go up by a facor of u or down by a facor of d uppose also ha if money is kep in he bank for his period i increases by a facor of 1 + r o he curren value of M fuure dollars is M(1 + r) 1 o r(2) = u d 1, r(1) = 1 + r 1 + r 1 and 1 p = u 1 r u d, p = 1 + r d u d (2) Noice ha hese wo values, p and 1 p have nohing o do direcly wih any inuiive idea of how probable he sock is o go up or down Of course, he curren marke price will be influenced by wha people believe he sock will do, so here is an indirec relaion beween p and inuiive probabiliy This is De Finei s marke based approach o he foundaions of probabiliy A European call opion is he righ o buy a number of shares of sock a ime 1 a a specified srike price K Le C be he (curren) price of he call opion Le K = k(0) Thus if he sock goes up by a facor of u hen he gain per uni purchased is u k 1 + r C since you can by he sock a ime 1 for a price of k(0) and sell i immediaely a he price u(0) and he opion coss you C dollars oday If he sock goes down by a facor of d you lose C dollars (I am assuming ha d k u) o unless u k 0 = (1 p)c + p 1 + r C = p u k 1 + r C = 1 + r d u k u d 1 + r C De Finei s heorem guaranees he exisence of a mixed sraegy of buying or selling he sock and buying or selling he opion wih a sure profi ince a fundamenal law of economics says ha here is no free lunch we mus have C = 1 + r d u d u k 1 + r (3) This is he fair price of he opion in he sense ha if he opion were priced differenly, an arbirageur could make a guaraneed profi 6

7 22 Odds In some siuaions he only ype of wagers allowed are ones ha choose one of he oucomes i = 1, m and hen be ha i is he oucome An example: a horse race where each be is on a single horse and one can only be ha horse wins or does no win The reurn on such a be is usually quoed in erm of odds If he odds agains oucome i are o i o 1 hen a one uni be will reurn o i if i occurs and 1 if i is no he oucome 4 how ha unless m i= o i = 1 you can make a sure profi a he race rack Remember ha a my race rack you can be a posiive or negaive amoun on any horse o win or any combinaion of such bes, bu here is no be on a horse o place or o show 5 uppose ha here are hree horses and he odds are 1,2, and 3 By he previous problem a sraegy exiss for a sure win Find such a sraegy 23 A muliperiod sock model uppose ha here are n consecuive periods and he bank ineres rae is r per period (0) denoes he iniial price of he sock and (i) is price a ime i uppose ha (i) is eiher u(i 1) or d(i 1) where d < 1 + r < u ock may be purchased or sold a any one of he imes i = 0, 1,, n Le X i = 1 if he price goes up by he facor u from ime i 1 o ime i and X i = 0 if i goes down The succession of sock prices can be regarded as he oucome of an experimen whose possible values are given by he vecor X = (X 1,, X n ) According o De Finei s heorem, here mus be probabiliies on hese oucomes which make all bes fair Oherwise here is a way of making a sure profi 6 how ha o saisfy he De Finei no sure profi condiion, he X i mus be independen and have probabiliy p for X i = 1 and 1 p for X i = 0 where p is given by (2) [Hin: Consider he following be: Choose i and a specific vecor (x 1,, x i 1 ) of zeros and ones Observe he sock marke If X j = x j for every j = 1,, i 1 buy a uni of sock a ime i 1 and sell i a ime i] o W = n i=1 X i 7

8 is binomial wih parameers n and p and so (n) = u W d n W (0) (4) Now suppose ha you could also buy or sell an opion o buy sock a ime n a a price equal o k(0) The presen value of he opion is given by he random variable (1 + r) n (0)(u W d n W k) + per uni sock and so he fair price of he opion per uni sock is C = (1 + r) n E [ (u W d n W k) +] (0) (5) [I am using he noaion f + o denoe he funcion x max(f(x), 0)] 3 The Black-choles formula 31 Warm up To give an illusraion of equaions (2), (4), and (5), suppose ha each period is an inerval of lengh /n and for each period we have u = e σ /n and d = e σ /n for some posiive number σ Also, he ineres rae r per period is now aken o be r n (so ha he (uncompounded) rae over he inerval of lengh is r) Then (2) says ha p = 1 + r/n e σ /n e σ /n e σ /n Using he Taylor series approximaions o second order Then we ge e σ /n = 1 + σ /n + σ 2 2n e σ /n = 1 σ /n + σ 2 2n p = σ /n σ 2 /2n + r/n 2σ /n = 1 2 ( o () = d n u W d (0) or () (0) 1 + r σ2 /2 σ = /n 2σ /n ) ( u ) = n d + W d σ /n 4 8

9 = nσ /n + 2σ /nw Recall ha W = n 1 X i and he X i are independen random variables each aking on he value 1 wih probabiliy p and 0 wih probabiliy 1 p o Thus E(W ) = np and Var (W ) = np(1 p) [ ] () E = nσ /n + 2σnp /n (0) In he above expression for p le us wrie µ := r σ 2 so we have p = 1 ( 1 + µ ) /n 2 σ and we obain We have [ ] () E = µ (0) p(1 p) = µ2 = 1 σ 2 n 4 for large n Using 1 4 for he variance of he X i we ge [ ] () = Var σ 2 (0) [ )] The cenral limi heorem says ha is (approximaely) normally ( () (0) disribued wih mean µ and variance σ 2 The compounded ineres for he enire period is ( 1 + r n) n which, according o Euler, ends o e r To summarize: Under he above assumpions and approximaions he price () of a sock a ime is a random variable of he form e X (0) where X is a Gaussian wih mean µ and variance σ 2 Furhermore, he compounded ineres rae r is relaed o µ and σ by r = µ σ2 Of course he cenral limi heorem holds under much more general hypoheses, so his model of he sock price raio is more general han jus aking he limi of (4) In he lieraure, a random variable of he form e X where X is normal is called normal meaning ha is arihm has a normal disribuion We begin wih some facs abou he exponenial of a normal disribuion 32 The mean and variance of Y = e X, X normal Le X be a normal (=Gaussian) random variable wih mean µ and variance σ 2 Le Y be he random variable Y = e X We claim ha and E(Y ) = e µ+ 1 2 σ2, (6) Var(Y ) = e 2µ+σ2 e σ2 1 (7) 7 Prove hese formulas, firs for µ = 0 and hen for he general case 9

10 33 Deermining he drif from De Finei Le us give he following formulaion of De Finei s idea in more absrac form, and moivaed by he preceding discussion: According o De Finei, if A is any asse, whose value a ime = 0 is he number V (A, 0) and whose value a ime is he random variable V (A, ), and if r is he coninuously compounded ineres rae, hen V (A, 0) = e r E(V (A, )) (8) A universe where his holds is called a risk neural universe (Perhaps i should be called a De Finei universe bu I am following sandard usage) Now suppose ha V (A, ) = V (A, 0)e σ 1 2 N+ν (9) where N is he uni normal Lemma 2 If V saisfies (8) and (9) hen 8 Prove he lemma 34 Black-choles Le E denoe he Gauss error funcion so r = ν σ2 (10) E() := 1 2π e x2 /2 dx Le s() = V (A, ) as above and le = s(0) Le C be he value of a European call opion wih srike price K and ime o expiraion (European means ha he call can only be exercised a he expiraion dae) Le r be he coninuously compounded bank (risk free) rae The Black-choles equaion assers ha ( K C = E + (r + ) ( 1 2 σ2 ) σ e r K KE + (r ) 1 2 σ2 ) σ (11) Proof We know ha and ha he value a ime is s() = e σ 1 2 N+ν max(s() K, 0) o C = e r E(max(s() K, 0)) 10

11 Now E(max(s() K, 0)) = 1 2π max(e σ 1 2 y+ν K, 0)e y2 /2 dy The maximum in he inegrand equals 0 if ie if o he expecaion in quesion is 1 2π K ν σ σ 1 2 y + ν K y K ν σ e σ 1 2 y+ν e y2 /2 dy 1 2π K ν σ Ke y2 /2 dy (12) The second inegral is ( ( K K 1 E ν )) ( σ = KE K ν ) ( σ K = KE + ν ) σ This accouns for he second erm in (11) We can wrie he exponenial in he firs inegral as σ 1 2 y + ν y 2 /2 = (y σ 1 2 ) 2 /2 + (ν σ2 ) so he firs inegral in (12) becomes 1 2π K ν σ e σ 1 2 y+ν e y2 /2 dy = e (ν+ 1 2 σ2 ) 1 2π K ν σ σ e z2 /2 dz as we make he change of variables z = y σ 1 2 The inegral on he righ of his equaion evaluaes as ( E K ν σ σ ) ( K = E + ν σ + σ ) E ( K + (ν σ σ2 ) σ We mus muliply his las expression by ) = E ( e r e (ν+ 1 2 σ2 ) = K + (r σ2 ) σ ) This accouns for he firs erm in (11) 11

12 Ediorial commens by co Kominers As wih any economic model, he Black-choles formula makes a few simplifying assumpions which do no hold up in pracice Asse values do no acually follow a sric, saionary -normal process 1 imilarly, he Black-choles model assumes ha reurns in differen markes are uncorrelaed 2 However, hese assumpions are approximaely accurae [mos of he ime] ince Black-choles pricing is clearly and simply calculaed from model parameers, he Black-choles model was revoluionary when i was announced I appeared in he one of he op economics journals, he Journal of Poliical Economy 3 I also led (wih relaed work) o Rober Meron s and Myron choles s 1997 Economics Nobel Prize award 4 In a somewha parallel series of evens, Meron, choles and ohers founded Long-Term Capial Managemen (LTCM), a quaniaive hedge fund based upon he heory underlying he Black-choles model The fund was successful for several years However, afer Russia defauled on is deb in 1998, several basic marke regulariies failed This emporarily rendered he funcional form assumpions of he Black-choles model exraordinarily inaccurae Consequenly, LTCM nearly wen bankrup The Federal Reserve Bank of New York was forced o orchesrae a bailou, agains fears ha he collapse would precipiae a global financial crisis For a fascinaing non-echnical accoun of how he Black choles equaion, or raher he use of his equaion ogeher wih a cerain amoun of hubris by is praciioners almos led o he collapse of he American financial sysem in epember 1998 see he book When Genius Failed by Roger Lowensein 1 Indeed, if asse values were ha well-conrolled, he marke would be far less volaile 2 A look a he curren financial crisis should convince you ha his is definiely no he case in realiy 3 Fischer Black and Myron choles, The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy 81(3)L adly, Fischer Black died in 1995 and was herefore ineligible o receive he award 12