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CH12WIENER PROCESSES AND ITÔ'S LEMMA OUTLINE • The Markov Property .*. • Continuous-Time Stochastic Processes • The Process For a Stock Price • The Parameters • Itô's Lemma • The Lognormal Property 1 Continuous Variable the value of the variable changes only at certain fixed time point .*. Stochastic Process Discrete Variable only limited values are possible for the variable 2 12.1 THE MARKOV PROPERTY A Markov process is a particular type of stochastic process . where only the present value of a variable is relevant for predicting the future. The past history of the variable and the way that the present has emerged from the past are irrelevant. 4 CONTINUOUS-TIME STOCHASTIC PROCESSES Suppose $10(now), change in its value •In Markov processes changes in during 1 yearN~(μ=0, is f(0,1). σ=1) successive periods of time are independent •This means that variances are Whatadditive. is the probability distribution of theσ=2) N~(μ=0, stock price atdeviations the endare of not 2 years? •Standard additive. f(0,2) 6 months? f(0,0.5) 3 months? f(0,0.25) Dt years? f(0, Dt) 5 A WIENER PROCESS (1/3) It is a particular type of Markov stochastic process with a mean change of zero and a variance rate of 1.0 per year. .*. Wiener Process A variable z follows a Wiener Process if it has the following two properties: (Property 1.) The change Δz during a small period of time Δt is Dz Dt where is f (0,1) Δz~normal distribution 7 A WIENER PROCESS (2/3) (Property 2.) The values of Δz for any two different short intervals of time, Δt, are independent. Mean of Dz is 0 Dt Variance of Dz is Dt Standard deviation of Dz is 7 A WIENER PROCESS (3/3) Mean of [z (T ) – z (0)] is 0 Consider the change in the value of z during a relatively time, Variancelong of [zperiod (T ) – of z (0)] isT. T This can be denoted by z(T)–z(0). deviation of [zsum (T ) –ofzthe (0)]changes is ItStandard can be regarded as the Tin z in N small time intervals of length Dt, where T N Dt n z (T ) z (0) i Dt i 1 9 EXAMPLE12.1(WIENER PROCESS) Ex:Initially $25 and time is measured in years. Mean:25, Standard deviation :1. At the end of 5 years, what is mean and Standard deviation? Our uncertainty about the value of the variable at a certain time in the future, as measured by its standard deviation, increases as the square root of how far we are looking ahead. 10 GENERALIZED WIENER PROCESSES(1/3) A Wiener process, dz, that has been developed so far has a drift rate (i.e. average change per unit time) of 0 and Drift rate →DR , variance rate →VR a variance rate of 1 DR=0 means that the expected value of z at any future time is equal to its current value. DR=0 , VR=1 VR=1 means that the variance of the 11 GENERALIZED WIENER PROCESSES (2/3) A generalized Wiener process for a variable x can be defined in terms of dz as dx = a dt + b dz DR VR 12 GENERALIZED WIENER PROCESSES(3/3) In a small time interval Δt, the change Δx in the value of x is given by equations Dx aDt b Dt Mean of Δx is Variance of Δx is aDt b 2 Dt Standard deviation of Δx is b Dt 13 EXAMPLE 12.2 Follow a generalized Wiener process 1. DR=20 (year) VR=900(year) 2. Initially , the cash position is 50. 3. At the end of 1 year the cash position will have a normal distribution with a mean of ★★ and standard deviation of ●● ANS:★★=70, ●●=30 15 ITÔ PROCESS Itô Process is a generalized Wiener process in which the parameters a and b are functions of the value of the underlying variable x and time t. dx=a(x,t) dt+b(x,t) dz The discrete time equivalent is only true in the limit as Dt tends Dtox zero a( x, t )Dt b( x, t ) Dt 16 12.3 THE PROCESS FOR STOCKS The assumption of constant expected drift rate is inappropriate and needs to be replaced by assumption that the expected reture is constant. This means that in a short interval of time,Δt, the expected increase in S is μSΔt. A stock price does exhibit volatility. 15 AN ITO PROCESS FOR STOCK PRICES where m is the expected return and s is the volatility. dS mS dt sS dz dS mdt sdz S The discrete time equivalent is DS mSDt sS Dt 16 EXAMPLE dS 12.3 mS dt sS dz Suppose m= 0.15, s= 0.30, then dS dS mdt sdz 0.15dt 0.3dz S S DS 0.15Dt 0.3 Dt S Consider a time interval of 1 week(0.0192)year, so that Dt =0.0192 ΔS=0.00288 S + 0.0416 S 17 MONTE CARLO SIMULATION MCSDofS astochastic mSDt sS process Dt is a procedure for sampling random outcome for the process. DS m= 0.0.14, 0014s= S 0.2, 0.02 S Dt = 0.01 then Suppose and The first time period(S=20 =0.52 ): DS=0.0014*20 +0.02*20*0.52=0.236 The second time period: DS'=0.0014*20.236 18 MONTE CARLO SIMULATION – ONE PATH Week Stock Price at Random Start of Period Sample for Change in Stock Price, DS 0 20.00 0.52 0.236 1 20.236 1.44 0.611 2 20.847 -0.86 -0.329 3 20.518 1.46 0.628 4 21.146 -0.69 -0.262 DS mSDt sS Dt 19 12.4 THE PARAMETERS We do not have to concern ourselves with the determinants of μin any detail because the value of a derivative dependent on a stock is, in general, independent of μ. .*. μ、σ We will discuss procedures for estimating σ in Chaper 13 22 12.5 ITÔ'S LEMMA If we know the stochastic process followed by x, Itô's lemma tells us the stochastic process followed by some function G (x, t ) dx=a(x,t)dt+b(x,t)dz Itô's lemma a functions G of x G Gshows 1 2G that G 2 dG ( a b )dt bdz and t follows the process 2 X t 2 X X 21 DERIVATION OF ITÔ'S LEMMA(1/2) IfDx is a small change in x and D G is the resulting small change in G dG DG DX dX dG 1 d 2G 1 d 3G 3 2 DG Dx Dx Dx ... 2 3 dX 2 dx 6 dx G G DG Dx Dy x y Taylor series G G 1 2G 2G 1 2G 2 2 DG Dx Dy D x D x D y D y ... 2 2 x y 2 x xy 2 y G G dG dx dy x y 22 DERIVATION OF ITÔ'S LEMMA(2/2) A Taylor's series expansion of G (x, t) gives dx a ( x, t ) dt b( x, t ) dz G G 2G 2 DG Dx Dt ½ D x x t x 2 2G 2G 2 Dx Dt ½ D t 2 xt t 23 IGNORING TERMS OF HIGHER ORDER THAN DT In ordinary calculus we have G G DG Dx Dt x t In stochastic calculus this becomes G G G 2 DG Dx Dt ½ Dx 2 x t x because Dx has a component which is 2 of order Dt 24 SUBSTITUTING FOR ΔX Suppose dx a ( x, t )dt b( x, t )dz so that Dx = a Dt + b Dt Then ignoring terms of higher order tha n Dt G G 2G 2 2 DG Dx Dt ½ b Dt 2 x t x 25 2 E ΔT THE TERM Since f (0,1), E ( ) 0 E ( 2 ) [ E ( )]2 1 E ( 2 ) 1 It follows that E ( Dt ) Dt 2 The variance of Dt is proportion al to Dt 2 and can be ignored. Hence G G 1 2G 2 DG Dx Dt b Dt 2 x t 2 x 26 LEMMA TO A STOCK PRICE PROCESS The stock price process is d S m S dt s S d z For a function G of S and t G G 2G 2 2 G dG mS ½ s S dt s S dz 2 t S S S 27 APPLICATION TO FORWARD CONTRACTS F0 S 0 e rT F Se r (T t ) F e r (T t ) G G 2G 2 2 G S dG mS ½ s S dt s S dz 2 S t S S 2F 0 2 S F rSe r (T t ) t dF e r (T t ) mS rSe r (T t ) dt e r (T t )sSdz dF ( m r ) Fdt sFdz 28 THE LOGNORMAL PROPERTY We define: G ln S G 1 2G 1 G , 2 2, 0 S S S S t s2 dt s dz dG m 2 G G 2G 2 2 G dG mS ½ s S dt s S dz 2 t S S S 29 THE LOGNORMAL PROPERTY s2 dt s dz dG m 2 s2 2 ln ST ln S 0 ~ ( m )T , s T 2 s2 2 ln ST ~ ln S 0 ( m )T , s T 2 s T The standard deviation of the logarithm of the stock price is 30