Stochastic Processes and their Applications in Financial Pricing
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1 Stochastic Processes ad their Applicatios i Fiacial Pricig Adrew Shi Jue 3, 1 Cotets 1 Itroductio Termiology.1 Fiacial Stochastics Mathematical Stochastics Browia Motio The Ito Itegral ad the Ito Differetial Ito s Lemma Fiacial Applicatios Black Scholes Equatio Coclusio 6 A Appedix 7 1
2 1 Itroductio Ivestors purchase stocks ad bods i the fiacial market, puttig their fuds at risk for the opportuity to ear a retur. Sice the time of Phoeicias, they have sought to miimize this risk value for each level of expected retur. I order to do so, a whole rage fiacial istrumets have bee developed, kow as derivatives, assets who derive assets from aother fiacial asset. The ature of derivative assets provides a iterestig coduit for the aalysis ad applicatio of Browia motio ad solvig partial derivative equatios, while maitaiig its real world applicatios. Numerous articles have bee writte o modelig movemets i fiacial markets with stochastic calculus. Perhaps the most famous of these described the Nobel Prize wiig Black-Scholes optio pricig model]. I several articles, mathematicias, specifically Robert Almgre s1] ad Aastasios Malliaris5], have attempted to more rigorously bridge the gap betwee radom motio ad optio pricig. Termiology.1 Fiacial Asset: A object that provides a claim to future cash flows. Efficiet Market Hypothesis: There is o opportuity for arbitrage i the market Derivative: A fiacial asset that derives its value from aother asset. Optio: A derivative that provides the opportuity, but ot obligatio to buy or sell a asset at a predetermied price i the future. Strike Price: The predetermied price for executig a optio. For a call optio, if the market price rises above the strike price, the ivestor will be willig to buy. For a put optio, if the market price falls below the strike price, the ivestor will wat to sell the uderlyig asset.. Stochastics Probability Space: A costruct of three compoets, (Ω, F, P ), where 1. Ω is the set of all possible outcomes.. F is the set of all evets, where each evet has zero or more outcomes. 3. P is the assigmet of probabilities to each evet With Probability 1: Also kow as almost surely. The probability of a evet occurig teds to 1 give some limit. Note that this differs from surely i that surely idicates that o other evet is possible, while almost surely idicates that other evets become less ad less likely. A collectio of sets F is called a σ-algebra if for a sequece of sets A k F, 1 A k F ad is closed uder complemetatio. The sets A F are F-measurable. M, T ] deotes the set of fuctios f(t) such that f(t) is defied o, T ], measurable with respect to the σ-algebra F t for all t, ad f(t) ds fiite with probability 1.
3 3 Mathematical Stochastics 3.1 Browia Motio The realm of fiacial asset pricig borrows heavily from the field of stochastic calculus. The price of a stock teds to follow a Browia motio. Defiitio A stochastic process w(t) is said to follow a Browia motio o, T ] if it satisfies the followig: 1. w() =.. w(t) is almost surely cotiuous. 3. For arbitrary t 1, t,..., t, where < t 1 < t <... < t < T, the variables w(), w(t 1 ) w(), w(t ) w(t 1 ),..., w(t ) w(t ) are mutually idepedet. I other words, is a process with idepedet icremets. 4. The mea (or expected) value, Ew(t) is. 5. The process w(t) takes o a ormal distributio desity aroud its mea. More specifcally, Ew(t k1 ) w(t k )] = t k1 t k. 3. The Ito Itegral ad the Ito Differetial A atural respose to a Browia motio w(t) is the desire to itegrate with respect to it. Thus, for a fuctio/process f over a probability space ω, we seek to make sese of a stochastic itegral f(t, x)dw(t) Note that with a differetial fuctio g(t), we ca evaluate the Riema Stieljes itegral of by usig lim j= f(t)dg(t) f(t )g tj1 g tj ]. Where t is a poit o the iterval t j, t j1 ), ad the series coverges to the same limit regardless of our selectio of t. With Browia motio, because of the idepedet icremet quality, w(t) is owhere differetiable, ad as such, we caot evaluate the Riema Stieljes itegral (the limit of the sum is depedet o our selectio of t ). As a result, there are as may stochastic itegrals as there are selectios of t. Defiitio The Ito itegral, takes our selectio of t as the left edpoit, t j. We thus have f(t, x)dw(t) = lim j= f(t j )w tj1 w tj ] With a stochastic itegral, it seems oly suitible to have a stochastic differetial. Is defied as follows. Defiitio Suppose there exist two fuctios u(t) ad v(t) i M, T ] such that for all t f T, X(t f ) X( ) = u(t)dt The the Ito differetial of a process X(t) is defied to be dx(t) = u(t)dt v(t)dw(t) v(t)dw(t) 3
4 3.3 Ito s Lemma Stochastic calculus cotais a aalogue to the chai rule i ordiary calculus. geometric Browia motio, we ca apply Ito s Lemma, which states4]: If a process follows Theorem 3.1 Suppose that the process X(t) has a stochastic differetial dx(t) = u(t)dt v(t)dw(t) ad that the fuctio f(t, x) is oradom ad defied for all t ad x. Additioally, suppose f is cotiuous ad has cotiuous derivatives f t (t, x), f x (t, x), f xx (t, x). The the stochastic process Y (t) = f(t, X(t)) also has a stochastic differetial, ad dy (t) = f t (t, X(t)) f x (t, X(t))u(t) 1 ] f xx(t, X(t))v (t) dt f x (t, X(t))v(t)dw(t) or i itegral form, Y (t f ) Y ( ) = f t (t, X(t)) f x (t, X(t))u(t) v(t) f xx(t, X(t))dt f x (t, X(t))v(t)dw(t) This proof is borrowed largely from Gikhma ad Skorokhod3]. Proof First, let us assume that u(t) ad v(t) are idepedet of t. Let = t < t 1 <... < t = t f. The Y (t f ) Y ( ) = f(t f, X(t f )) f(, X( )) = f(t k1, X(t k1 )) f(t k, X(t k ))] However, ote that the Taylor expasio of the summad is f(t k1, X(t k1 )) f(t k, X(t k )) = f t (t k, X(t k )) (t k1 t k ) f x (t k, X(t k )) X(t k1 ) X(t k )] 1 f xx(t k, X(t k )) X(t k1 ) X(t k )] O(tX, t, X 3 ) The O(tX, t, X 3 ) will become isigificat with probability 1 as max(t k1 t k ), by reasoig similar to tha Lemma A.1. As it turs out, we will show that the X(t k1 ) X(t k )] term retais a O(t) term, so we keep it for ow. Because X(t k1 ) X(t k ) = u(t)(t k1 t k ) v(t)(w(t k1 ) w(t k )), we ca substitute to get f(t k1, X(t k1 )) f(t k, X(t k )) = f t (t k, X(t k ))(t k1 t k ) f x (t k, X(t k ))u(t k )(t k1 t k ) f x (t k, X(t k ))v(t k )w(t k1 ) w(t k )] u(t k) f xx (t k, X(t k ))(t k1 t k ) u(t)v(t)f xx (t k, X(t k ))(t k1 t k )(w(t k1 ) w(t k )) v(t k) f xx (t k, X(t k ))(w(t k1 ) w(t k )) As show i Lemma A.1 (i the appedix), the terms that sum over the (t k1 t k ) term ad the (t k1 t k )(w(t k1 ) w(t k )) term will ted to with probability 1 as max(t k1 t k ). If we ow sum over the remaiig terms, we have Y (t f ) Y ( ) = lim max(t k1 t k ) f t (t k, X(t k ))(t k1 t k ) f x (t k, X(t k ))u(t k )(t k1 t k ) ] f x (t k, X(t k ))v(t k )w(t k1 ) w(t k )] v(t k) f xx (t k, X(t k ))(w(t k1 ) w(t k )) 4
5 Recall that for a Browia process, Ew(t k1 ) w(t k )] = t k1 t k. As we take the limit as t k1 t k, we ca thus replace the w(t k1 ) w(t k )] term. Doig so ad recogizig that the summatios are by defiitio the Ito itegral, we are left with Y (t f ) Y ( ) = f t (t, X(t))dt f x (t, X(t))u(t)dt f x (t, X(t))v(t)dw(t) v(t k) f xx (t, X(t))dt We have ow show Ito s Lemma for costat u ad v. It follows that for step fuctios, the same applies, as they ca be partitioed ito fiitely may costat fuctios over a iterval. As show i Lemma A. (i the appedix), is possible to choose a sequece of step fuctios u ad v so that u(t) u (t) dt v(t) v (t) dt ad X (t) = X( ) t a ds t b dw(s) coverges uiformly to X(t) with probability 1. As f is cotiuously smooth, we ca also say that Y (t) = f(t, X (t)) Y (t) uiformly with probability 1. As Y (t) is piecewise costat, we ca say Y (t f ) Y ( ) = f t (t, X (t)) f x (t, X (t))u (t) 1 ] f xx(t, X (t))v(t) dt f x (t, X (t))v (t)dw(t) As we let, we obtai Ito s Lemma i itegral form, Y (t f ) Y ( ) = f t (t, X(t)) f x (t, X(t))u(t) 1 ] f xx(t, X(t))v (t) dt f x (t, X(t))v(t)dw(t) The differetial form of this is the theorem we set out to prove. dy (t) = f t (t, X(t)) f x (t, X(t))u(t) 1 ] f xx(t, X(t))v (t) dt f x (t, X(t))v(t)dw(t) 4 Fiacial Applicatios 4.1 Black-Scholes Equatio This brigs us to the Black-Scholes equatio for optio pricig. Cosider a sigle stock, with price S(t), which varies with time. Almgre argues that the value of the optio derivig from that stock should have a market value thas a fuctio of S ad t. Let us call this D(t) = V (t, S(t)). I the world of fiace, the most sigificat descriptor of the profitability of a asses its rate of retur. I order to describe the pertrubatios of the retur o a share of stock, we will model it a geometric Browia motio. 5
6 Defiitio A process takes o geometric (also kow as expoetial) Browia motio if its logarithm follows a Browia motio. I other words, oly fractioal chages take place as radom variatio. Its differetial takes o the form ds = as(t)dt bs(t)dw(t) where a ad b are costats, ad w(t) is a Browia motio. Let the stock price take o a geometric Browia motio, where the chage i stock price is proportioal to the curret stock price, thas ds = as(t)dt bs(t)dw(t). Note that by Ito s lemma, dd = V t asv S b S ] V SS dt bsv S dw(t) = V t b S ] V SS dt V S ds Cosider a ivestor who holds a portfolio of the stock ad its optio, P (t) = N 1 (t)s(t) N (t)d(t). The differetial is dp = N 1 ds N dd = N 1 ds N V t b S ] V SS dt N V S ds Malliaris5] the makes the clever argumet of holdig a ratio of stock to derivative of N1 N = V S (this is kow as a delta hedge), so that N 1 ds N V S ds =. We are left with dp = N V t b S ] V SS dt which is completely idepedet of Browia motio (there is o dw(t) term, explicit or implicit). As a result ca be cosidered riskless. By the efficiet market hypothesis, the retur o this riskless asset must be equal to that o ay other riskless asset, more specifically a govermet bod. Let the retur o the govermet bod be r(t). The we have dp P = N V t b S V SS N 1 S N ] dt = rdt If we rearrage ad ormalize so that N = 1, thus makig N 1 = V S, we get or (V t b S V SS)dt = ( V S S V )rdt V t (t, S) b S V SS(t, S) SV S (t, S) rv (t, S) = Which is the Black-Scholes differetial equatio for optio pricig. 5 Coclusio Almgre ad Malliaris both serve to elucidate the coectio betwee stochastic processes ad fiacial asset valuatio ad deepe the isight provided origially by Black ad Scholes. The crux of the argumet lies with Ito s lemma, which allows oe to value a asset whose value is a radom Browia fuctio of aother asset. While Ito s origial formula was developed for more scietific fields, it has foud a iche i fiacial aalysis. I their origial thesis, Black ad Scholes further solve their differtial equatio with coditio that V (t, S = ) =, ad F (T, S) = max, S E], where T is the exercise date for the optio, ad E is the excercise date idicated i the cotract. As a result of the Black Scholes equatio, the applicatio of stochastics to fiace has bee reivigorated ad today it has bee applied to a plethora of fiacial assets. 6
7 A Appedix Lemma A.1 As max(t k1 t k ), 1. (t k1 t k ). (t k1 t k )(w(t k1 ) w(t k )) Proof 1. Without loss of geerality, let (t k1 t k ) be the largest partitio of the space, t f ]. Sice we are partitioig ito segmets, the average partitio will have size t f. Let our largest partitio have size c t f. The ( (t k1 t k ) c t ) ( f c t ) f as max(t k1 t k ), or equivaletly as.. Similar to the previous example, ote that (t k1 t k )(w(t k1 ) w(t k )) c(t f )(w(t k1 ) w(t k )) As a Browia process is almost surely cotiuous, c(t f )(w(t k1 ) w(t k )) with probability 1. Lemma A. If f(t) is i M, T ], the there exists a sequece of step fuctios f (t) i M, T ] such that with probability 1, lim f(t) f (t) dt = Proof Let us first cosider a bouded fuctio g(t). As is bouded, at each poit t, there is a sequece g (t) to the value of g(t) with probability 1. A arbitrary fuctio i M,T] ca be approximated by a bouded fuctio to a arbitrary degree of accuracy. Thus, the sequece of step fuctios ca also be approximated ad are dese i the set of all fuctios. Refereces 1] R. Almgre. Fiacial derivatives ad partial differetial equatios. The America Mathematical Mothly, 19(1):1 1,. ] F. Black ad M. Scholes. The pricig of optios ad corporate liabilities. The Joural of Political Ecoomy, 81(3): , ] I. Gikhma ad A. Skorokhod. Itroductio to the Theory of Radom Processes. W. B. Sauders Compay, ] K. Ito. O stochastic differetial equatios. Memoirs, America Mathematical Society, (4):1 51, ] A. Malliaris. Ito s calculus i fiacial decisio makig. SIAM Review, 5(4): ,
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