AN INTRODUCTION TO STOCHASTIC CALCULUS AND BLACK-SCHOLES OPTION PRICING

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1 AN INRODUCION O OCHAIC CALCULU AND BLACK-CHOLE OPION PRICING ZACHRY WANG Abstract. his paper is a expositio of the mathematics behid the Black- choles model of pricig a Europea optio. After we briefly metio the mai defiitios of measure-theoretic probability, we defie Browia motio ad prove its key properties. Next, we itroduce stochastic calculus by costructig the Itō itegral ad provig Itō s formula. Fially, we use these tools to derive the famous Black-choles partial differetial equatio. Cotets 1. Mathematical Probability 1 2. Browia Motio 4 3. he Itō Itegral 8 4. Itō s Formula 1 5. Derivatio of the Black-choles Equatio Assumptios ad Potetial Risks 17 Ackowledgmets 17 Refereces Mathematical Probability Here, we will defie some of the basic machiery of probability. Defiitio 1.1. A σ-algebra F o Ω is a collectio of subsets of Ω such that the followig are true: (1) F. (2) If A F, the A c F. (3) If {A i } i N is a coutable subset of F, the A i F. We also defie a sub-σ-algebra as a subcollectio of F that is also a σ-algebra. Defiitio 1.2. he fuctio P : F [, 1] is a probability measure if the followig hold: (1) If {A i } i N are pairwise disjoit sets i F, the P( A i) = P(A i). (2) P(Ω) = 1. A set Ω, a σ-algebra F, ad a probability measure P make up a probability space, which we deote (Ω, F, P). Date: DEADLINE: Draft AUGU 18 ad Fial versio AUGU 29,

2 2 ZACHRY WANG Defiitio 1.3. he Borel σ-algebra B(R) is the smallest σ-algebra that cotais all ope sets i R. Defiitio 1.4. A real-valued fuctio f : Ω R is F-measurable if for all B B(R), f 1 (B) F. Defiitio 1.5. A radom variable X is a fuctio X : Ω R that is F- measurable. Note that we ca also get radom vectors by replacig R with R. Propositio 1.6. σ(x) := {X 1 (B) : B B(R)} forms a σ-algebra. Proof. We see that = X 1 ( ). If A σ(x), the there exists B A B(R) such that X 1 (B A ) = A. hus, we see that A c = (X 1 (B A )) c = X 1 (BA c ) σ(x). If A 1, A 2,... σ(x), the there exists B 1, B 2,... B(R) such that X 1 (B i ) = A i. We get that A i = [X 1 (B i )] = X 1 ( B i) σ(x). We call σ(x) the σ-algebra geerated by X. Defiitio 1.7. he distributio measure of X is the probability measure µ X that assigs to each B B(R) the measure µ X (B) = P{X B}. If X is a radom vector, we ca get joit distributios by replacig R with R. Defiitio 1.8. he cumulative distributio fuctio of X is the fuctio F : R [, 1] such that F (x) = P[X x]. Defiitio 1.9. A radom variable X has a desity fuctio f : R [, ) if P[a X b] = b a f(x)dx. Defiitio 1.1. A radom variable X is a ormal radom variable with mea µ ad variace σ 2, deoted N(µ, σ 2 ), if its probability desity fuctio is equal to /(2σ 2 ) φ(x) = e (x µ)2 σ. 2π I particular, a stadard ormal radom variable is a ormal radom variable with mea ad variace 1. Defiitio he expectatio of X is defied to be E[X] = X(ω)dP(ω), give that X is itegrable. Defiitio he variace of X is defied to be Ω Var(X) = E[(X E[X]) 2 ]. Defiitio he covariace of X ad Y is defied to be Cov(X, Y ) = E[(X E[X])(Y E[Y ])].

3 AN INRODUCION O OCHAIC CALCULU AND BLACK-CHOLE OPION PRICING3 Defiitio A radom vector (X 1,..., X ) is joitly ormal if it has the joit desity fuctio 1 f(x) = (2π) det C exp ( 1 2 (x µ)c 1 (x µ) ), x = (x 1,, x ) R, where µ = (µ 1,, µ ) is a row vector with µ i = E[X i ], deoted the mea vector, ad C is a by matrix, with c ij = Cov(X i, X j ), deoted the covariace matrix. Defiitio We call σ-algebras G 1, G 2 F idepedet if P(A 1 A 2 ) = P(A 1 ) P(A 2 ) for all A i G i. We say radom variables are idepedet if the σ-algebras geerated by them are idepedet. Defiitio Let X be a itegrable radom variable with respect to F ad let G be a sub-σ-algebra of F. he coditioal expectatio of X give G, deoted E[X G], is a radom variable o Ω such that (1) E[X G] is G-measurable, (2) For all A G, E[X G](ω)dP(ω) = X(ω)dP(ω). A A he existece ad uiqueess of coditioal expectatio is guarateed by the Rado-Nikodym theorem. he key properties of coditioal expectatios are stated here: Propositio Let X, Y be itegrable radom variables. true about coditioal expectatios: (1) If a, b R, the E[aX + by G] = ae[x G] + be[y G]. (2) If X is G-measurable, the: E[XY G] = XE[Y G]. (3) If H is a sub-σ-algebra of G, the (4) If X is idepedet of G, the E[E[X G] H] = E[X H]. E[X G] = E[X]. he followig are Defiitio A stochastic process is a collectio of radom variables {X t } idexed by t, where is a idex set. I this paper, we cosider = [, ), which represets time. We deote a stochastic process {X t } or {X t } t. here are typically two ways to thik about a stochastic process. For each fixed t, we have a radom variable. o we could thik of a stochastic process as a collectio of radom variables idexed by. For each fixed ω, we have a fuctio from to R. o, we could thik of a stochastic process as a collectio of fuctios idexed by ω. I the latter, we will call these fuctios paths. Defiitio he collectio {F t } of σ-algebras of Ω idexed by t is a filtratio if for all t, s such that s t, we have F s F t. We deote the filtratio {F t } or {F t } t.

4 4 ZACHRY WANG Defiitio 1.2. A stochastic process {X t } is F t -adapted if, for each t, the radom variable X t is F t -measurable. Defiitio Cosider a F t -adapted stochastic process {M t }. If he {M t } is a martigale. E[M t F s ] = M s for all s t, 2. Browia Motio Browia motio will be our model of radom motio ad will be a fudametal buildig block as we costruct our models represetig uderlyig asset prices. We defie the stadard Browia motio as follows: Defiitio 2.1. A stadard Browia motio is a stochastic process {B t } t with the followig properties: (1) With probability 1, B =. (2) For all t 1 t 2... t, the icremets B t2 B t1, B t3 B t2,..., B t B t 1 are idepedet. (3) For t s, B t B s N(, t s). (4) With probability 1, the fuctio t B t is cotiuous. For a rigourous proof of existece of the stadard Browia motio usig liear iterpolatio o the dyadic ratioals, see [1]. We create a filtratio to model iformatio available at each time t i a stadard Browia motio. We typically equip a stadard Browia motio with a augmeted filtratio {F t } t which satisfies the followig properties: (1) For each t, B t is F t -measurable. (2) E[ B t ] < for all t. (3) For all t s, E[B t F s ] = B s. he details of the costructio of {F t } t ca be foud i [5]. For our purposes, we aim to prove three importat properties of stadard Browia motio that will be used later i our costructio of the stochastic itegral. hese properties are that a stadard Browia motio is a martigale, is owhere differetiable with probability 1, ad accumulates quadratic variatio at a rate of oe uit per time. heorem 2.2. tadard Browia motio is a martigale. Proof. Fix s, t such that s t. he E[B t F s ] = E[B t B s + B s F s ] = E[B t B s F s ] + E[B s F s ] = E[B t B s ] + B s = B s. Martigales are ofte thought of as a model of a fair game. I a fiacial cotext, if we model a asset s price as a Browia motio, the the martigale property says that the expected future price based o all iformatio up to this poit i time will be the price today. he martigale is the mathematical formulatio of the Efficiet Market Hypotheisis.

5 AN INRODUCION O OCHAIC CALCULU AND BLACK-CHOLE OPION PRICING5 heorem 2.3. With probability 1, the paths geerated by a stadard Browia motio are owhere differetiable. Proof. It suffices to show that the paths are owhere differetiable o the iterval [,1]. Our first observatio is that if f(t) is a cotiuous fuctio o [,1] ad is differetiable at some s [, 1], the f satisifes a Lipschitz coditio at s. hat is, there exists L N such that f(t) f(s) L t s ( t 1). his is true because f(s + h) f(s) lim = f (s) h h implies δ > s.t. for t s < δ, we have f(t) f(s) < ( f (s) + 1) t s. O the other had, for t s δ, we have f(t) f(s) 2M = 2M δ δ 2M δ t s, where M is a upper boud of f(t) o [,1]. hus, { f(t) f(s) max ( f (s) + 1), 2M } t s. δ For L N, let A L := {ω Ω : s [, 1] s.t. t [, 1], B t (ω) B s (ω) L t s }. Usig our previous observatio, we see that L=1 A L cotais all ω Ω i which the sample path is cotiuous ad differetiable for some s [, 1]. We will show that P [ L=1 A L] =. Fix 3. For each ω A L, there exists k 3 such that for some s [k/, (k + 3)/], we have B t (ω) B s (ω) L t s ( t 1). It follows that for j = k, k + 1, k + 2, B(j+1)/ (ω) B j/ (ω) B(j+1)/ (ω) B s (ω) + Bs (ω) B j/ (ω) 3L + 3L = 6L. However, B (j+1)/ (ω) B j/ (ω) N(, 1/), thus [ P B (j+1)/(ω) B j/(ω) 6L ] = P [ Z 6L ] 12L 2π. By idepedece of Browia icremets, we see that for each k 3, P { B(j+1)/ (ω) B j/ (ω) } ( ) 3 12L < 1728L3. 2π 3/2 k j k+2

6 6 ZACHRY WANG Fially, by takig the uio over these sets for k =, 1,..., 3, we see that 2 P[A L ] P { B(j+1)/ (ω) B j/ (ω) } k= k j k+2 ( 2) 1728L3 3/2 as. Hece, P[A L ] =. he idea of a everywhere cotiuous fuctio that is owhere differetible is pretty bizzare for its ow sake. he first of these fuctios to be explicitly costructed was by Weierstrass i It ca be prove that the Riema-tieltjes itegral is defied oly for a itegrator of bouded variatio. But a fuctio of bouded variatio is almost everywhere differetiable. he owhere differetiability of a stadard Browia motio implies that Browia motio is ot of bouded variatio, ad hece, the Riema-tieltjes itegrals caot be applied. his is oe of the primary reasos for the creatio of the Itō itegral. Defiitio 2.4. Let f : [, ) R be a fuctio. Defie 1 Q Π (f, ) = [f(t j+1 ) f(t j )] 2, j= where Π = {t, t 1,..., t } with t i = i. he quadratic variatio of f up to time is defied to be provided the limit exists. Q(f, ) = lim Q Π(f, ), Π heorem 2.5. For each, Q(B t, ) = with probability 1, where B t is a sample path of Browia motio. Proof. o prove our claim, it suffices to show that Q Π (B t, ), which is a radom variable, has expectatio ad its variace approaches as Π. We start by provig the first part of this by observig that Next, we cosider the variace. 1 E[Q Π (B t, )] = E[(B tj+1 B tj ) 2 ] j= 1 = Var(B tj+1 B tj ) j= 1 = t j+1 t j =. j=

7 AN INRODUCION O OCHAIC CALCULU AND BLACK-CHOLE OPION PRICING7 Var[(B tj+1 B tj ) 2 ] = E[((B tj+1 B tj ) 2 (t j+1 t j )) 2 ] = E[(B tj+1 B tj ) 4 ] 2(t j+1 t j )E[(B tj+1 B tj ) 2 ] + (t j+1 t j ) 2 = E[(B tj+1 B tj ) 4 ] 2(t j+1 t j )(t j+1 t j ) + (t j+1 t j ) 2 = E[(B tj+1 B tj ) 4 ] (t j+1 t j ) 2 We kow that B tj+1 B tj (t j+1 t j )Z, where Z N(, 1). ice E[Z 4 ] = 3, we have that Hece, ad E[(B tj+1 B tj ) 4 ] = E[(t j+1 t j ) 2 Z 4 ] = 3(t j+1 t j ) 2. Var[(B tj+1 B tj ) 2 ] = E[(B tj+1 B tj ) 4 ] (t j+1 t j ) 2 = 3(t j+1 t j ) 2 (t j+1 t j ) 2 = 2(t j+1 t j ) 2, 1 Var[Q Π (B(t), )] = Var[B(t j+1 ) B(t j )] 2 j= 1 = 2(t j+1 t j ) 2 j= 1 2 Π (t j+1 t j ) j= = 2 Π. hus, we have lim Var[Q Π(B t, )] =. Π Applyig Chebyshev s iequality gives us that for ay λ, P ( Q Π (B t, ) λ) = P ( Q Π (B t, ) E [Q Π (B t, )] λ) 1 λ 2 E [Q Π(B t, )] as λ. Hece Q Π (B t, ) coverges to i probability. Because of this, there exists a subsequece of Q Π (B t, ) that coverges to with probability 1. hat is, Q(B t, ) = ad we prove the theorem. lim Q Π(B t, ) =, with probability 1, Π he sigificace of this calculatio is that the quadratic variatio is ot zero. It ca be show that the quadratic variatio of fuctios with cotiuous derivatives is zero. Hece, quadratic variatio terms are almost ever see i stadard calculus. Due to this differece, whe we are developig stochastic calculus, the stochastic fudametal theorem of calculus (Itō s formula) will look differet from the

8 8 ZACHRY WANG stadard fudametal theorem of calculus to icorporate the effects of quadratic variatio. 3. he Itō Itegral o why are we tryig to build this stochastic itegral? I the big picture, we will have two pieces. First, we will have a stochastic process with its accompayig filtratio represetig a radom asset price. ecod, we wat to create a stochastic process that will represet a particular portfolio maagemet strategy. he purpose of the itegral will be to show how much moey will be made give the strategy ad a model for asset price movemets. he strategy will be represeted by the itegrad ad the movemet of the asset will be represeted by the itegrator. Our costructio will very closely follow the costructio i [2]. Defiitio 3.1. Let V = V (, ), where, R, be the set of fuctios (stochastic processes) X : [, ) Ω R which satisfies the followig properties: (1) X(t, ω) is B F measurable, where B is the Borel σ-algebra o [, ). (2) X(t, [ ω) is F t -adapted. ] (3) E X2 (t, w)dt <. Just like the Riema itegral is built usig step fuctios, we will first defie stochastic itegrals directly o a very well-behaved subset of V, which we will call elemetary processes. he, for the remaiig processes i V, we will defie the stochastic itegral by approximatio usig elemetary processes Defiitio 3.2. We call H V a elemetary process if for each ω, the path ca be writte i the form H (t, ω) = j e j (ω)i [tj,t j+1)(t), where e j are idexed radom variables, is a fixed positive iteger, I [tj,t j+1) is a idicator fuctio, ad j2 if j2 t j := if j2 < if j2 >. Defiitio 3.3. Give the elemetary process i defiitio 3.2, we defie its Itō itegral as follows: H (t, ω)db t := j e j (ω)[b tj+1 B tj ](ω). Example 3.4. Let H (t, ω) = j B tj I [tj,t j+1)(t). If we go back to our portfolio allocatio aalogy, H (t, ω)db t

9 AN INRODUCION O OCHAIC CALCULU AND BLACK-CHOLE OPION PRICING9 represets the returs we will get from iitially holdig B of the asset ad at times t j, j = 1, 2,..., we chage to holdig B tj of the asset. We see that [ ] E H (t, ω)db t = E[B tj (B tj+1 B tj )] j = j E[B tj ]E[(B tj+1 B tj )] =. he followig lemma is crucial for extedig the Itō itegral to geeral itegrads. Lemma 3.5. If H (t, w) V is elemetary, the ( ) 2 [ ] E H (t, ω)db t = E (H (t, ω)) 2 dt. Proof. Notice that for i < j, E [ e i (B ti+1 B ti )e j (B tj+1 B tj ) ] = E [ e i (B ti+1 B ti )e j ] E [ (Btj+1 B tj ) ] = because (B tj+1 B tj ) is idepedet of F tj. hus, ( ) 2 E H (t, ω)db t E [ e i (B ti+1 B ti )e j (B tj+1 B tj ) ] = i,j = i E [ e i (B ti+1 B ti )e i (B ti+1 B ti ) ] = i E [ (e i ) 2] E [ (B ti+1 B ti ) 2] = E [ (e i ) 2] (t i+1 t i ) i [ ] = E (H (t, ω)) 2 dt. Lemma 3.6. Let H V. he there exists a sequece of elemetary fuctios H V such that [ ] E (H H ) 2 dt as. Details of the proof ca be foud i [2].

10 1 ZACHRY WANG { } Usig lemmas 3.5 ad 3.6, we ca show that H db t is a Cauchy sequece i L 2 (P ) space. ice L 2 (P ) is complete, we see that lim H (t, ω)db t is a well-defied limit. his allows us to defie the Itō itegral. Defiitio 3.7. Let H V (, ). We defie its Itō itegral as H(t, ω)db t = lim H (t, ω)db t (i L 2 (P )) where {H } is a sequece of elemetary fuctios such that [ ] E (H(t, ω) H (t, ω)) 2 dt as. 4. Itō s Formula I this sectio, we prove the stochastic fudametal theorem of calculus or Itō s formula. heorem 4.1. (Itō s formula for the time-idepedet case with respect to Browia motio) uppose f is a C 2 fuctio ad {B t } t is a stadard Browia motio. he for each t, f(b t ) f(b ) = f (B s )db s his ca also be writte i differetial form as df(b t ) = f (B t )db t f (B t )dt. f (B s )ds. Proof. We defie a partitio of the iterval [, ] by t i = i/ for i. he (4.2) f(b ) f() = ( f(bti ) f(b ) ti 1). We will estimate each term f(b ti ) f(b ti 1 ) usig a two-term aylor expasio f(y) f(x) = (y x)f (x) (y x)2 f (x) + r(x, y). Here, r(x, y) is the remaider term, which is bouded by r(x, y) (y x) 2 h(x, y), where h(x, y) is a oegative, uiformly cotiuous, bouded fuctio with h(x, x) =. We ca rewrite equatio 4.2 usig the aylor expasio, so we have

11 AN INRODUCION O OCHAIC CALCULU AND BLACK-CHOLE OPION PRICING11 (4.3) f(b ) f() = = ( f(bti ) f(b )) ti 1 f (B ti 1 )(B ti B ti ) + 1 f (B ti 1 )(B ti B ti 1 ) r(b ti 1, B ti ), with the remaider terms r(b ti 1, B ti ) bouded by r(b ti 1, B ti ) (B ti B ti 1 ) 2 h(b ti 1, B ti ). We deal with the terms of (4.3) oe by oe. For the first term of (4.3), sice f is cotiuous, f (B ti 1 )(B ti B ti ) f (B t )db t, i probability. We rewrite the secod term of (4.3) as follows: (4.4) 1 2 f (B ti 1 )(B ti B ti 1 ) 2 = [ f (B )(B ti 1 t i B )2 ti 1 (t i t i 1 ) ] f (B ti 1 )(t i t i 1 ). ice f is cotiuous, the limit of the secod term of (4.4) is equal to 1 lim f (B ti 1 )(t i t i 1 ) = 1 f (B t )dt. 2 2 We boud the first term of (4.4) usig that fact that for Z N(, 1), we have that E[Z 4 ] = 3, which implies that Var[Z 2 ] = E[Z 4 ] E[Z 2 ] 2 = 3 1 = 2. Usig the defiitio of Browia motio ad the properties of expectatio ad variace, we get that which implies that ad B ti B ti 1 N(, ) Z, E[(B ti B ti 1 ) 2 ] = E [ ] Z2 = E[Z2 ] = Var((B ti B ti 1 ) 2 ) = Var ( Z2) = 2 2 Var( Z 2) =

12 12 ZACHRY WANG Next, we show that the summads i the first term of (4.4) have mea zero. his is true because [ E f (B ti 1 ) ( (B ti B ti 1 ) 2 ] ) Fti 1 [ = f ((Bti (B ti 1 )E B ti 1) 2 [ = f ((Bti (B ti 1 )E B ti 1 )2 ) ]. ) Fti 1 ] Now, we check that the summads are ucorrelated. If i < j, the [ E f (B ti 1 ) ( (B ti B ti 1 ) 2 ) f (B ( tj 1) (B tj B tj 1) 2 ] ) Ftj 1 ] = f (B ti 1 ) ( (B ti B ti 1 ) 2 [ ) f (B tj 1 )E ((Btj B )2 tj 1 ) F tj 1 =, a.s. hus we have that ( [ Var f (B ti 1 )(B ti B ti 1 ) 2 ] ) ( = Var f (B ti 1 )(B ti B ti 1 ) 2 ) ( f 2 L Var (B ti B ti 1 ) 2 ) = f 2 L Var ( (B ti B ti 1 ) 2) = f 2 L 2 2 2, as hus, we see that the limit of the first term of (4.4) becomes [ f (B ti 1 )(B ti B ti 1 ) 2 ], as.

13 AN INRODUCION O OCHAIC CALCULU AND BLACK-CHOLE OPION PRICING13 o prove that the third term of (4.3) goes to zero as, we use the Cauchy- chwartz iequality ad get [ ] [ ] E r(b ti 1, B ti ) E (B ti B ti 1 ) 2 h(b ti 1, B ti ) = E[(B ti B ti 1 ) 4 ]E[h(B ti 1, B ti ) 2 ] = = = [ 2 E Z4 ] E[h(B ti 1, B ti ) 2 ] 2 E[h(B t i 1, B ti ) 2 ] E[h(B ti 1, B ti ) 2 ]. By costructio, we ow that for ay ɛ >, there exists δ > such that h(x, y) < ɛ, give that x y < δ. hus, we kow that E[h(B ti 1, B ti ) 2 ] h 2 L P ( B ti B ti 1 > δ ) + ɛ 2 = h 2 L P( Z > δ) + ɛ2 2ɛ 2 for sufficietly large. herefore, [ ] E (B ti B ti 1 ) 2 h(b ti 1, B ti ) 6 ɛ for sufficietly large, which implies that (B ti B ti 1 ) 2 h(b ti 1, B ti ) i probability as. We thus fiish the proof of the theorem. Example 4.5. Let f(x) = x 3. he f (x) = 3x 2, f (x) = 6x. hus, f(b t ) = 3BudB 2 u + 1 6B u du = 3B 2 udb 2 u + 3B u du We ow defie a stadard Itō process to expad the class of stochastic processes that we ca use as itegrators for our stochastic itegral. I our cotext, it allows us to use other stochastic processes for uderlyig price movemet besides a stadard Browia motio. Defiitio 4.6. We say that a process {X t } is a stadard Itō process if X t = X + a(ω, s)ds + b(ω, s)db s, where t [, ], ad a ad b are adapted, measurable processes that satisfy the itegrability coditios ( ) ( ) P a(ω, s) ds < = 1 ad P (b(ω, s)) 2 ds < = 1

14 14 ZACHRY WANG We ow state the geeral form of the Itō formula. he proof is similar, ad will be omitted here. heorem 4.7. (Itō formula for a geeral Itō process) uppose f(t, x) is a fuctio where f t (t, x), f x (t, x) ad f xx (t, x) are defied ad cotiuous ad {X t } is a Itō process with the format above. he, f(t, X t ) = f(, ) + f t (s, X s)ds f 2 x 2 (s, X s)b 2 (ω, s)ds. his equatio ca also be writte i differetial form as df(t, X t ) = f t (t, X t)dt + f x (t, X t)dx t f 2 x 2 (t, X t)b 2 (ω, t)dt. f x (s, X s)dx s 5. Derivatio of the Black-choles Equatio Defiitio 5.1. A derivative is a fiacial security i which the value is depedet o the state of aother fiacial security (the uderlyig security). Defiitio 5.2. A call optio is a derivative that gives the ower the right (but ot obligatio) to buy a uderlyig asset for a predetermied price (the strike price). imilarly, a put optio gives the ower the right to sell for a predetermied price. Europea optios predetermie the date that the ower ca exercise the optio, while America optios give the user freedom to exercise at aytime before a specified date. he geeral approach to pricig a derivative security is to (1) Outlie how much the security pays at each time give each possible state of the uderlyig security. (2) Create a replicatig portfolio. hat is, create a portfolio allocatio strategy that has the same value at each time ad at each possible state as the derivative. (3) Calculate the price of the replicatig portfolio. his will be the price of the derivative security. he philosophy for pricig derivatives this way is rooted i the o-arbitrage assumptio. A arbitrage opportuity is a way to make riskless profit. More precisely, it is a asset allocatio strategy that is guarateed to have o egative et cash flows at ay time ad positive et cash flow i at least oe state. his assumptio is theoretically soud ad for the most part empirically true. Arbitrage opportuities are ofte quickly discovered, ad quickly exploited util prices move to elimiate the opportuity. If the price of the derivative is ay other price tha the price of the replicatig portfolio, the there will be a arbitrage opportuity. If the derivative is overpriced, sell it ad buy the replicatig portfolio. If the derivative is uderpriced, doig the opposite. Either way, this will violate the o-arbitrage assumptio. hus,

15 AN INRODUCION O OCHAIC CALCULU AND BLACK-CHOLE OPION PRICING15 the derivative will have the same price as the replicatig portfolio, assumig o arbitrage. Here we will cotruct a very simple example to illustrate these ideas. Example 5.3. uppose we live i a two-period world cosistig of time 1 ad time 2. Let be a stock that is worth $2 at time 1, ad could be worth either $4 or $1 at time 2. Let V be a derivative security that, at time 2, is worth $3 if the price of moves to $4 ad is worth $ if the price of moves to $1 (V is a Europea call optio with strike price $1). Assume also that we led ad borrow moey with o iterest (usig a bod, β). o the questio is, what should the price of the derivative security be at time 1? o fid the replicatig portfolio, we eed to fid weights a ad b such that a+bβ gives the same payoff as the derivative V. We kow that: (1) if is worth $4 i period 1, the derivative is worth $3 ad the replicatig portfolio is worth 4a + b. o we eed 3 = 4a + b. (2) if is worth $1 i period 1, the derivative is worth $ ad the replicatig portfolio is worth a + b. o we eed = a + b. hus, we get that a = 1 ad b = 1 ad the price of V at time 1 should be ( 1) 1 = 1. Remark 5.4. A importat observatio is that the price does ot deped o the probability of the stock goig up or dow. As a cosequece, the price does ot deped o the expectatio. ake for example, if i the above example had a 1/2 probability of goig up ad 1/2 probability of goig dow. he E[V ] = For the Europea call optio, o the exercise date, if the price of the uderlyig security is less tha the strike price, the the call optio is worthless. I example 5.3, whe the price of is $1 at time 2, V is worthless. Otherwise, the cotract is worth the price of the uderlyig security mius the strike price. I example 5.3, whe the price of is $4 at time 2, V is worth $3. he value of the call optio as a fuctio of the stock price o the expiratio date is h(x) = max{, x (strike price)} Now we begi our derivatio of the Black-choles equatio. Let t model the price of the uderlyig security, which is assumed to follow a geometric Browia motio (5.5) d t = µ t dt + σ t db t. Let β t model the price of a bod, which is assumed to be a determiistic process that icreases i value expoetially (5.6) dβ t = rβ t dt. Now we wat to build our replicatig portfolio. Let a t ad b t be adapted processes that represet the combiatio of stock ad bods we hold at time t, respectively. At ay give time, the value of the portfolio is V t = a t t + b t β t ice this is a replicatig portfolio, we have V = h( ).

16 16 ZACHRY WANG Next, we assume that our replicatig portfolio is self-fiacig. hat is, ay chage i value i the portfolio must come from either a chage i value of the stock or a chage i value of the bod. (5.7) dv t = a t d t + b t dβ t. Now, substitute (5.5) ad (5.6) ito (5.7) to get (5.8) a t d t + b t dβ t = dv t = a t [µ t dt + σ t db t ] + b t [rβ t dt] = [a t µ t + b t rβ t ] dt + [a t σ t ] db t. O the other had, we assume that V t = f(t, t ) for some well-behaved, smooth fuctio f. hus we apply the Itō formula for a geeral Itō process (theorem 4.7) to get that (5.9) dv t = = f t (t, t )dt f xx(t, t )d t d t + f x (t, t )d t [f t (t, t ) + 12 ] f xx(t, t )σ 2 2t + f x (t, t )µ t dt + f x (t, t )σ t db t. ice (5.8) ad (5.9) must be equal, the coefficiets of the dt terms must be equal ad the coefficiets of the db t terms also must be equal. Whe we set the coefficiets of the db t terms equal, we get a t = f x (t, t ). imilarly, settig the coefficiets of the dt terms equal gives us (5.1) b t = 1 [ f t (t, t ) + 1 ] rβ t 2 f xx(t, t )σ 2 t 2. hus, we substitute (5.1) for b t to get f(t, t ) = V t = a t t + b t β t = f x (t, t ) t + 1 rβ t If we substitute x for t, we get Addig the origial coditio [ f t (t, t ) + 1 ] 2 f xx(t, t )σ 2 t 2 β t. f t (t, x) = 1 2 σ2 x 2 f xx (t, x) rxf x (t, x) + rf(t, x). f(, x) = h(x) for all x R. gives us the Black-choles model for Europea call optios. olvig the partial differetial equatio gives us that the price of the Europea call with a stock price of, strike price of K, ad time remaiig of τ = t is equal to ( log(/k) + (r + 1 Φ σ τ 2 σ2 )τ ) ( log(/k) + (r 1 Ke rτ Φ σ τ 2 σ2 )τ where Φ is the cummulative distributio fuctio of a stadard ormal radom variable. ),

17 AN INRODUCION O OCHAIC CALCULU AND BLACK-CHOLE OPION PRICING17 6. Assumptios ad Potetial Risks Up util this poit, we have carefully built up the mathematical theory. However, it is of paramout importace to uderstad the assumptios that were used to get the formula. ome of these assumptios iclude: (1) Uderlyig securities move accordig to a geometric Browia motio. (2) radig is istat ad cost-free. (3) here is a sigle, costat iterest rate ad costat volatility. (4) ecurity prices move cotiuously. (5) Optio caot be exercised util expiratio. (6) No divideds are paid o the security. Uderstadig the assumptios allows users to be aware of ad appropriately hedge risks that were assumed away i the derivatio, such as tail risk, liquidity risk, or volatility risk. I additio, a uderstadig of the assumptios allows users to seek out the may extesios that exist ad are beig researched that icorporate real-world features like divideds ad America optios ad broade the scope of the model by icorporatig bells ad whistles like geometric fractioal Browia motio for uderlyig security movemets ad stochastic iterest rates ad volatility. Ackowledgmets. I would like to thak my metor Jacob Perlma for his exceptioal guidace ad help i both learig the material ad the paper-writig process. I additio, I wat to thak Professor Peter May ad all of the istructors i the math REU for this woderful program that helped me to both broade my kowledge of mathematics ad icrease my mathematical maturity. Refereces [1] G.F. Lawler. tochastic Calculus: A Itroductio with Applicatios. America Mathematical ociety. 21. [2] B. Oksedal. tochastic Differetial Equatios: A Itroductio with Applicatios. priger- Verlag. 23. [3] W. Rudi. Priciples of Mathematical Aalysis. McGraw-Hill, Ic [4].E. hreve. tochastic Calculus for Fiace. priger-verlag. 24. [5] J.M. teele. tochastic Calculus ad Fiacial Applicatios. priger-verlag. 21.