Chapter 13 Wiener Processes and Itô’s Lemma Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 1 Stochastic Processes Describes the way in which a variable such as a stock price, exchange rate or interest rate changes through time Incorporates uncertainties Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 2 Example 1 Each day a stock price increases by $1 with probability 30% stays the same with probability 50% reduces by $1 with probability 20% Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 3 Example 2 Each day a stock price change is drawn from a normal distribution with mean $0.2 and standard deviation $1 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 4 Markov Processes (See pages 280-81) In a Markov process future movements in a variable depend only on where we are, not the history of how we got to where we are Is the process followed by the temperature at a certain place Markov? We assume that stock prices follow Markov processes Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 5 Weak-Form Market Efficiency This asserts that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices. In other words technical analysis does not work. A Markov process for stock prices is consistent with weak-form market efficiency Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 6 Example A variable is currently 40 It follows a Markov process Process is stationary (i.e. the parameters of the process do not change as we move through time) At the end of 1 year the variable will have a normal probability distribution with mean 40 and standard deviation 10 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 7 Questions What is the probability distribution of the stock price at the end of 2 years? ½ years? ¼ years? Dt years? Taking limits we have defined a continuous stochastic process Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 8 Variances & Standard Deviations In Markov processes changes in successive periods of time are independent This means that variances are additive Standard deviations are not additive Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 9 Variances & Standard Deviations (continued) In our example it is correct to say that the variance is 100 per year. It is strictly speaking not correct to say that the standard deviation is 10 per year. Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 10 A Wiener Process (See pages 282-84) Define f(m,v) as a normal distribution with mean m and variance v A variable z follows a Wiener process if The change in z in a small interval of time Dt is Dz Dz Dt where is f(0,1) The values of Dz for any 2 different (nonoverlapping) periods of time are independent Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 11 Properties of a Wiener Process Mean of [z (T ) – z (0)] is 0 Variance of [z (T ) – z (0)] is T Standard deviation of [z (T ) – z (0)] is Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 T 12 Generalized Wiener Processes (See page 284-86) A Wiener process has a drift rate (i.e. average change per unit time) of 0 and a variance rate of 1 In a generalized Wiener process the drift rate and the variance rate can be set equal to any chosen constants Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 13 Generalized Wiener Processes (continued) Dx a Dt b Dt Mean change in x per unit time is a Variance of change in x per unit time is b2 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 14 Taking Limits . . . What does an expression involving dz and dt mean? It should be interpreted as meaning that the corresponding expression involving Dz and Dt is true in the limit as Dt tends to zero In this respect, stochastic calculus is analogous to ordinary calculus Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 15 The Example Revisited A stock price starts at 40 and has a probability distribution of f(40,100) at the end of the year If we assume the stochastic process is Markov with no drift then the process is dS = 10dz If the stock price were expected to grow by $8 on average during the year, so that the year-end distribution is f(48,100), the process would be dS = 8dt + 10dz Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 16 Itô Process (See pages 286) In an Itô process the drift rate and the variance rate are functions of time dx=a(x,t) dt+b(x,t) dz The discrete time equivalent Dx a ( x, t )Dt b( x, t ) Dt is true in the limit as Dt tends to zero Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 17 Why a Generalized Wiener Process Is Not Appropriate for Stocks For a stock price we can conjecture that its expected percentage change in a short period of time remains constant (not its expected actual change) We can also conjecture that our uncertainty as to the size of future stock price movements is proportional to the level of the stock price Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 18 An Ito Process for Stock Prices (See pages 286-89) dS mS dt sS dz where m is the expected return s is the volatility. The discrete time equivalent is DS mSDt sS Dt The process is known as geometric Brownian motion Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 19 Interest Rates What would be a reasonable stochastic process to assume for the short-term interest rate? Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 20 Monte Carlo Simulation We can sample random paths for the stock price by sampling values for Suppose m= 0.15, s= 0.30, and Dt = 1 week (=1/52 or 0.192 years), then ΔS 0.15 0.0192 S 0.30 0.0192 ε or DS 0.00288 S 0.0416 S Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 21 Monte Carlo Simulation – Sampling one Path (See Table 13.1, page 289) Week Stock Price at Random Start of Period Sample for Change in Stock Price, DS 0 100.00 0.52 2.45 1 102.45 1.44 6.43 2 108.88 −0.86 −3.58 3 105.30 1.46 6.70 4 112.00 −0.69 −2.89 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 22 Correlated Processes Suppose dz1 and dz2 are Wiener processes with correlation r Then Dz1 1 Dt Dz 2 2 Dt where 1 and 2 are random samples from a bivariate standard normal distributi on where correlation is r Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 23 Itô’s Lemma (See pages 291) If we know the stochastic process followed by x, Itô’s lemma tells us the stochastic process followed by some function G (x, t ) Since a derivative is a function of the price of the underlying asset and time, Itô’s lemma plays an important part in the analysis of derivatives Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 24 Taylor Series Expansion A Taylor’s series expansion of G(x, t) gives G G 2G DG Dx Dt ½ 2 Dx 2 x t x 2G 2G 2 Dx Dt ½ 2 Dt xt t Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 25 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 2G 2 DG Dx Dt ½ Dx 2 x t x because Dx has a component which is of order Dt Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 26 Substituting for Dx Suppose dx a( x, t )dt b( x, t )dz so that Dx = a Dt + b Dt Then ignoring terms of higher order than Dt G G 2G 2 2 DG Dx Dt ½ b Dt 2 x t x Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 27 The 2Dt Term Since f(0,1), E () 0 E ( 2 ) [ E ()]2 1 E ( 2 ) 1 It follows that E ( 2 Dt ) Dt 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 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 28 Taking Limits Taking limits : Substituti ng : We obtain : G G 2G 2 dG dx dt ½ 2 b dt x t x dx a dt b dz G G 2G 2 G dG a ½ 2 b dt b dz t x x x This is Ito' s Lemma Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 29 Application of Ito’s Lemma to a Stock Price Process The stock price process is d S mS dt sS d z For a function G of S and t G G 2G 2 2 G dG mS ½ 2 s S dt sS dz t S S S Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 30 Examples 1. The forward price of a stock for a contract maturing at time T G S e r (T t ) dG (m r )G dt sG dz 2. The log of a stock price G ln S s2 dt s dz dG m 2 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 31
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